Interesting Motivating funda

Motivations

Funda#1
The single most important thing that effects our lives is who we spend time with. Who we spend time with is who we become. Basically we in our lives become one of two types of people. A person who creates or a person who destroys. It's easier to destroy than it is to create. It's easier to tear down than it is to build up. What takes years to build can be burned down in minutes. But the joy one can have in destroying something is temporary, because you know you are not a creator, you're not adding value, and your life reflects it. Seek those people who are living life at the highest level. And your life will change. If you want to play tennis, you don't play with someone worse than you are. You find someone better and you have to rise just to be in their presence. One last thing. Remember that proximity is power. If you want to be a dancer and you are living in Iowa, it's going to be hard to be the best. Take action and surround yourself with people who are playing at the level you want to play. Often talented people never achieve their goals, because they put themselves in an environment where their skills can't be maximized. Even a great sprinter can't succeed if he races in mud. Go where the action is for what it is you are focusing on. If it's farming, go to a farming town, if it's media, go to a media location. Go where it's happening and the resources you need will appear. ------- Anthony Robins


Funda#2 :-Completion
There are certain people who avoid completion. Completion offers you the chance to fail. The two primary fears all human beings have is the fear that we are not enough, and therefore number two fear pops up, that we will not be loved.Every type of avoidance behavior is merely a way, even if it feels negative, to prevent ourselves from the pain of feeling insignificant and unloved.The secret is to turn this pain on itself, turn fear on itself. We all do whatever we can do avoid pain so what you've got to do is not only make a list, but get emotionally associated with what it's going to cost you if you don't change this pattern now.In addition to getting that leverage to make change a must, and by the way that is the secret, changing has to become a must not just a should If it's a must you'll find a way, if it's a should, you'll keep talking about it. But secondly, you need to set yourself up for some small wins. Complete a chapter of the book, or a paragraph, or a page. Lose five pounds, instead of the fifty, and see that as the completion, and you'll build the muscle and the belief in yourself that will create the behavior to change yourself long term.------- Anthony Robins


Funda#3:-Decisions
All change happens with a real decision, it happens in a moment. Some people make small changes some people make big decisions.It's totally determined by can you make the tough decisions or not. In my own life most of the success I've had has come from making small decisions regularly that were successful. But the greatest changes honestly was when I was faced with the most difficult decisions, decisions that could have cost me everything financially or emotionally. When you are willing to make those tough decisions, you have a life you are in control of, and the changes you make have no limit. Remember it's in your moments of decision that your destiny is shaped.
------- Anthony Robins


Funda#4:-Some things to Learn
I've learned that no matter what happens, or how bad it seems today, life does go on, and it will be better tomorrow. I've learned that you can tell a lot about a person by the way he/she handles these three things: a rainy day, lost luggage, and tangled Christmas tree lights. I've learned that regardless of your relationship with your parents, you'll miss them when they're gone from your life. I've learned that making a "living" is not the same thing as making a "life." I've learned that life sometimes gives you a second chance. I've learned that you shouldn't go through life with a catcher's mitt on both hands. You need to be able to throw something back. I've learned that if you pursue happiness, it will elude you. But if you focus on your family, your friends, the needs of others, your work and doing the very best you can, happiness will find you. I've learned that whenever I decide something with an open heart, I usually make the right decision. I've learned that even when I have pains, I don't have to be one. I've learned that every day you should reach out and touch someone. People love that human touch -- holding hands, a warm hug, or just a friendly pat on the back. I've learned that I still have a lot to learn. People will forget what you said; people will forget what you did, but people will never forget how you made them feel.


Funda#5:-The 90 - 10 Principle
"Distance between success and failure can only be measured by one'sdesire.."Discover the 90/10 Secret: It will change your life The 90/10 secret is incredible! Very few know and apply this secret. The result? Millions of people are suffering undeserved stress, trials,problems, and heartache. They never seem to be a success in life.Bad days follow bad days. Terrible things seem to be constantlyhappening.Theirs is constant stress, lack of joy, and broken relationships.Worry consumes time, anger breaks friendships, and life seems dreary and is not enjoyed to the fullest.. Friends are lost. Life is a bore andoften seems cruel.Does this describe you? If so, do not be discouraged. You can be different! Understand and apply the 90/10 secret. It will change your life!What is this secret?10% of life is made up of what happens to you.90% of life is decided by how you react.What does this mean? We really have no control over 10% of what happens to us. We cannot stop the car from breaking down. The plane may be late arriving, which throws our whole schedule off. A drivermay cut us off in traffic. We have no control over this 10%. The other 90% is different. You determine the other 90%! How? By your reaction. You cannot control a red light, but you can control your reaction. Don't let people fool you; YOU can control how you react!--------- Anthony Robins

Funda#6:-Do it yourself
" LIFE IS WHAT YOU MAKE OF IT " You have the power to succeed. The day you take complete responsibility for yourself, the day you stop making any excuses, that's the day you start moving to your goal. Nobody is going to do it for you. Only you can make it happen. You're the only one that has to live your life. Success on any major scale requires you to accept responsibility. Begin to choose the thoughts and actions that will lead you to success. Nothing will happen by itself. It will all come your way once you understand that you have to make it come your way. Next year can be so much better if you begin to act upon your dreams. The power to fulfill them all is within you.

Funda#7:-Just Do it
Enjoy having it done One of the biggest burdens you can create for yourself is to put off something that needs to be done. When there's a task that you've put off until later, it hangs over everything you do. It drags you down and brings unnecessary stress.Is there something you've been putting off, something you could do right now, something that needs to be done? Why wait any longer? Go ahead and free yourself from its burden. Go ahead and get it done. It will free your energy. It will free your time. It will free your mind and your spirit. You're going to have to do it eventually, so get it done as soon as possible and avoid letting it drag you down any longer. Don't let procrastination drain the life out of you. Get it done now and start enjoying the fruits of your labor. You'll feel great to have it done.
-- Ralph Marston

Funda#8:-Buried or Lifted
If you find yourself at the bottom of a deep hole in the ground, and somebody starts shoveling dirt into the hole, what are you going to do?Are you going to give up and resign yourself to being buried, or are you going to step up on top of each fresh mound of dirt until you're finally able to step out of the hole? It all depends on whether you look at the dirt as something that will bury you or as something that can lift you higher. In short, it's a matter of attitude and perspective. Every day, problems, frustrations, challenges and uncertainty are heaped upon you. They can bury you if you let them. Or,they can lift you if you'll step up to them .Which will it be? Don't stand still and let yourself get buried when, with a bit of effort,you can use the very same situation to lift yourself up.
-- Ralph Marston

Funda#9:- Keep Going
If you've already achieved some success today, keep going. Build on that momentum and you'll achieve even more. If you've known nothing but frustration and disappointment today, keep going. Change your approach based on what you've learned and you'll get yourself where you want to go. Through it all, keep going. The only way to lose is to give up. The only way to win is to persist. It's a clear choice. You can give up now, you can settle for less, or you can continue for as long as it takes to achieve what you've set out to achieve. Time will pass no matter what you do. Time will keep on going. Your best strategy is to keep on going, too. Make use of every moment. When you're exhausted, get some rest. Then get back up and keep on going. You are destined to make things happen. You are here to make a difference. Keep on going, and in the persistence of your efforts, you'll find a level of joy and fulfillment that you otherwise could not know.
-- Ralph Marston

Funda#10:-Why not today?
There is something you've always wanted to do. There is some desire that you've always told yourself you would someday pursue. Well, why not today? Today is the perfect day for you to begin. Of all the days you have available to choose from, today will give you the earliest start and the best advantage. There is an ideal version of you that you've always dreamed of becoming. You can start right now to work toward becoming that person, toward living that life you've always wanted. Taking the first step is not difficult. It is well within your reach. It's your life we're talking about here. Don't you want to do everything possible to make it as great Time is constantly passing by. Starting right now to pursue the life you've always wanted will immediately put time on your side. As soon as you make the commitment to move forward, every moment will bring you closer to your goal.Get up and go for it. Start right now to create for yourself the life you deserve, the very best life you can imagine.
-- Ralph Marston

Magic Squaring

Squaring a 2-digit number beginning with 1
1. Take a 2-digit number beginning with 1.
2. Square the second digit (keep the carry) _ _ X
3. Multiply the second digit by 2 and add the carry (keep the carry) _ X _
4. The first digit is one (plus the carry) X _ _
Example:
1. If the number is 16, square the second digit:6 x 6 = 36 _ _ 6
2. Multiply the second digit by 2 andadd the carry: 2 x 6 + 3 = 15 _ 5 _
3. The first digit is one plus the carry:1 + 1 = 2 2 _ _
4. So 16 x 16 = 256.
See the pattern?
1. For 19 x 19, square the second digit:9 x 9 = 81 _ _ 1
2. Multiply the second digit by 2 andadd the carry: 2 x 9 + 8 = 26 _ 6 _
3. The first digit is one plus the carry:1 + 2 = 3 3 _ _
4. So 19 x 19 = 361.
Squaring a 2-digit number beginning with 5
1. Take a 2-digit number beginning with 5.
2. Square the first digit.
3. Add this number to the second number to find the first part of the answer.
4. Square the second digit: this is the last part of the answer.
Example:
1. If the number is 58, multiply 5 x 5 = 25 (square the first digit).
2. 25 + 8 = 33 (25 plus second digit).
3. The first part of the answer is 33 3 3 _ _
4. 8 x 8 = 64 (square second digit).
5. The last part of the answer is 64 _ _ 6 4
6. So 58 x 58 = 3364.
See the pattern?
1. For 53 x 53, multiply 5 x 5 = 25 (square the first digit).
2. 25 + 3 = 28 (25 plus second digit).
3. The first part of the answer is 28 2 8 _ _
4. 3 x 3 = 9 (square second digit).
5. The last part of the answer is 09 _ _ 0 9
6. So 53 x 53 = 2809
Squaring a 2-digit number beginning with 9
1. Take a 2-digit number beginning with 9.
2. Subtract it from 100.
3. Subtract the difference from the original number: this is the first part of the answer.
4. Square the difference: this is the last part of the answer.
Example:
1. If the number is 96, subtract: 100 - 96 = 4, 96 - 4 = 92.
2. The first part of the answer is 92 _ _ .
3. Take the first difference (4) and square it: 4 x 4 = 16.
4. The last part of the answer is _ _ 16.
5. So 96 x 96 = 9216.
See the pattern?
1. For 98 x 98, subtract: 100 - 98 = 2, 98 - 2 = 96.
2. The first part of the answer is 96 _ _.
3. Take the first difference (2) and square it: 2 x 2 = 4.
4. The last part of the answer is _ _ 04.
5. So 98 x 98 = 9604
Squaring a 2-digit number ending in 1
1. Take a 2-digit number ending in 1.
2. Subtract 1 from the number.
3. Square the difference.
4. Add the difference twice to its square.
5. Add 1.
Example:
1. If the number is 41, subtract 1: 41 - 1 = 40.
2. 40 x 40 = 1600 (square the difference).
3. 1600 + 40 + 40 = 1680 (add the difference twice to its square).
4. 1680 + 1 = 1681 (add 1).
5. So 41 x 41 = 1681.
See the pattern?
1. For 71 x 71, subtract 1: 71 - 1 = 70.
2. 70 x 70 = 4900 (square the difference).
3. 4900 + 70 + 70 = 5040 (add the difference twice to its square).
4. 5040 + 1 = 5041 (add 1).
5. So 71 x 71 = 5041.
Squaring a 2-digit number ending in 2
1. Take a 2-digit number ending in 2.
2. The last digit will be _ _ _ 4.
3. Multiply the first digit by 4: the 2nd number will be the next to the last digit: _ _ X 4.
4. Square the first digit and add the number carried from the previous step: X X _ _.
Example:
1. If the number is 52, the last digit is _ _ _ 4.
2. 4 x 5 = 20 (four times the first digit): _ _ 0 4.
3. 5 x 5 = 25 (square the first digit), 25 + 2 = 27 (add carry): 2 7 0 4.
4. So 52 x 52 = 2704.
See the pattern?
1. For 82 x 82, the last digit is _ _ _ 4.
2. 4 x 8 = 32 (four times the first digit): _ _ 2 4.
3. 8 x 8 = 64 (square the first digit), 64 + 3 = 67 (add carry): 6 7 2 4.
4. So 82 x 82 = 6724
Squaring a 2-digit number ending in 3
1. Take a 2-digit number ending in 3.
2. The last digit will be _ _ _ 9.
3. Multiply the first digit by 6: the 2nd number will be the next to the last digit: _ _ X 9.
4. Square the first digit and add the number carried from the previous step: X X _ _.
Example:
1. If the number is 43, the last digit is _ _ _ 9.
2. 6 x 4 = 24 (six times the first digit): _ _ 4 9.
3. 4 x 4 = 16 (square the first digit), 16 + 2 = 18 (add carry): 1 8 4 9.
4. So 43 x 43 = 1849.
See the pattern?
1. For 83 x 83, the last digit is _ _ _ 9.
2. 6 x 8 = 48 (six times the first digit): _ _ 8 9.
3. 8 x 8 = 64 (square the first digit), 64 + 4 = 68 (add carry): 6 8 8 9.
4. So 83 x 83 = 6889.
Squaring a 2-digit number ending in 4
1. Take a 2-digit number ending in 4.
2. Square the 4; the last digit is 6: _ _ _ 6 (keep carry, 1.)
3. Multiply the first digit by 8 and add the carry (1); the 2nd number will be the next to the last digit: _ _ X 6 (keep carry).
4. Square the first digit and add the carry: X X _ _.
Example:
1. If the number is 34, 4 x 4 = 16 (keep carry, 1); the last digit is _ _ _ 6.
2. 8 x 3 = 24 (multiply the first digit by 8), 24 + 1 = 25(add the carry): the next digit is 5: _ _ 5 6. (Keep carry, 2.)
3. Square the first digit and add the carry, 2: 1 1 5 6.
4. So 34 x 34 = 1156.
See the pattern?
1. For 84 x 84, 4 x 4 = 16 (keep carry, 1); the last digit is _ _ _ 6.
2. 8 x 8 = 64 (multiply the first digit by 8),64 + 1 = 65 (add the carry): the next digit is 5: _ _ 5 6. (Keep carry, 6.)
3. Square the first digit and add the carry, 6: 7 0 5 6.
4. So 84 x 84 = 7056.
Squaring a 2-digit number ending in 5
1. Choose a 2-digit number ending in 5.
2. Multiply the first digit by the next consecutive number.
3. The product is the first two digits: XX _ _.
4. The last part of the answer is always 25: _ _ 2 5.
Example:
1. If the number is 35, 3 x 4 = 12 (first digit times next number). 1 2 _ _
2. The last part of the answer is always 25: _ _ 2 5.
3. So 35 x 35 = 1225.
See the pattern?
1. For 65 x 65, 6 x 7 = 42 (first digit times next number): 4 2 _ _.
2. The last part of the answer is always 25: _ _ 2 5.
3. So 65 x 65 = 4225.
Squaring a 2-digit number ending in 6
1. Choose a 2-digit number ending in 6.
2. Square the second digit (keep the carry): the last digit of the answer is always 6: _ _ _ 6
3. Multiply the first digit by 2 and add the carry (keep the carry): _ _ X _
4. Multiply the first digit by the next consecutive number and add the carry: the product is the first two digits: XX _ _.
Example:
1. If the number is 46, square the second digit : 6 x 6 = 36; the last digit of the answer is 6 (keep carry 3): _ _ _ 6
2. Multiply the first digit (4) by 2 and add the carry (keep the carry): 2 x 4 = 8, 8 + 3 = 11; the next digit of the answer is 1: _ _ 1 6
3. Multiply the first digit (4) by the next number (5) and add the carry: 4 x 5 = 20, 20 + 1 = 21 (the first two digits): 2 1 _ _
4. So 46 x 46 = 2116.
See the pattern?
1. For 76 x 76, square 6 and keep the carry (3):6 x 6 = 36; the last digit of the answer is 6: _ _ _ 6
2. Multiply the first digit (7) by 2 and add the carry:2 x 7 = 14, 14 + 3 = 17; the next digit of the answer is 7 (keep carry 1): _ _ 7 6
3. Multiply the first digit (7) by the next number (8)and add the carry: 7 x 8 = 56, 56 + 1 = 57 (the first two digits: 5 7 _ _
4. So 76 x 76 = 5776.
Squaring a 2-digit number ending in 7
1. Choose a 2-digit number ending in 7.
2. The last digit of the answer is always 9: _ _ _ 9
3. Multiply the first digit by 4 and add 4 (keep the carry): _ _ X _
4. Multiply the first digit by the next consecutive number and add the carry: the product is the first two digits: XX _ _.
Example:
1. If the number is 47:
2. The last digit of the answer is 9: _ _ _ 9
3. Multiply the first digit (4) by 4 and add 4 (keep the carry): 4 x 4 = 16, 16 + 4 = 20; the next digit of the answer is 0 (keep carry 2): _ _ 0 9
4. Multiply the first digit (4) by the next number (5) and add the carry (2):4 x 5 = 20, 20 + 2 = 22 (the first two digits): 2 2 _ _
5. So 47 x 47 = 2209.
See the pattern?
1. For 67 x 67
2. The last digit of the answer is 9: _ _ _ 9
3. Multiply the first digit (6) by 4 and add 4 (keep the carry): 4 x 6 = 24, 24 + 4 = 28; the next digit of the answer is 0 (keep carry 2): _ _ 8 9
4. Multiply the first digit (6) by the next number (7) and add the carry (2):6 x 7 = 42, 42 + 2 = 44 (the first two digits): 4 4 _ _
5. So 67 x 67 = 4489.
Squaring a 2-digit number ending in 8
1. Choose a 2-digit number ending in 8.
2. The last digit of the answer is always 4: _ _ _ 4
3. Multiply the first digit by 6 and add 6 (keep the carry): _ _ X _
4. Multiply the first digit by the next consecutive number and add the carry: the product is the first two digits: XX _ _.
Example:
1. If the number is 78:
2. The last digit of the answer is 4: _ _ _ 4
3. Multiply the first digit (7) by 6 and add 6 (keep the carry): 7 x 6 = 42, 42 + 6 = 48; the next digit of the answer is 8 (keep carry 4): _ _ 8 4
4. Multiply the first digit (7) by the next number (8) and add the carry (4):7 x 8 = 56, 56 + 4 = 60 (the first two digits): 6 0 _ _
5. So 78 x 78 = 6084.
See the pattern?
1. For 38 x 38
2. The last digit of the answer is 4: _ _ _ 4
3. Multiply the first digit (3) by 6 and add 6 (keep the carry): 3 x 6 = 18, 18 + 6 = 24; the next digit of the answer is 4 (keep carry 2): _ _ 4 4
4. Multiply the first digit (3) by the next number (4) and add the carry (2):3 x 4 = 12, 12 + 2 = 14 (the first two digits): 1 4 _ _
5. So 38 x 38 = 1444
Learn the pattern, practice other examples, and you will be a whiz at giving these squares.
Squaring a 2-digit number ending in 9
1. Choose a 2-digit number ending in 9.
2. The last digit of the answer is always 1: _ _ _ 1
3. Multiply the first digit by 8 and add 8 (keep the carry): _ _ X _
4. Multiply the first digit by the next consecutive number and add the carry: the product is the first two digits: XX _ _.
Example:
1. If the number is 39:
2. The last digit of the answer is 1: _ _ _ 1
3. Multiply the first digit (3) by 8 and add 8 (keep the carry): 8 x 3 = 24, 24 + 8 = 32; the next digit of the answer is 2 (keep carry 3): _ _ 2 1
4. Multiply the first digit (3) by the next number (4) and add the carry (3): 3 x 4 = 12, 12 + 3 = 15 (the first two digits): 1 5 _ _
5. So 39 x 39 = 1521.
See the pattern?
1. For 79 x 79
2. The last digit of the answer is 1: _ _ _ 1
3. Multiply the first digit (7) by 8 and add 8 (keep the carry): 8 x 7 = 56, 56 + 8 = 64; the next digit of the answer is 4 (keep carry 6): _ _ 4 1
4. Multiply the first digit (7) by the next number (8) and add the carry (6): 7 x 8 = 56, 56 + 6 = 62 (the first two digits): 6 2 _ _
5. So 79 x 79 = 6241.
Squaring numbers made up of ones
1. Choose a a number made up of ones (up to nine digits).
2. The answer will be a series of consecutive digits beginning with 1, up to the number of ones in the given number, and back to 1.
Example:
1. If the number is 11111, (5 digits) -
2. The square of the number is 123454321.(Begin with 1, up to 5, then back to 1.)
See the pattern?
1. If the number is 1111111, (7 digits) -
2. The square of the number is 1234567654321.(Begin with 1, up to 7, then back to 1.).
Squaring numbers made up of threes
1. Choose a a number made up of threes.
2. The square is made up of:
a. one fewer 1 than there are repeating 3's
b. zero
c. one fewer 8 than there are repeating 3's (same as the 1's in the square)
d. nine.
Example:
1. If the number to be squared is 3333:
2. The square of the number has:
three 1's (one fewer than digits in number) 1 1 1 _ _ _ _ _next digit is 0 _ _ _ 0 _ _ _ _three 8's (same number as 1's) _ _ _ _ 8 8 8 _a final 9 _ _ _ _ _ _ _ 9
3. So 3333 x 3333 = 11108889.
See the pattern?
1. If the number to be squared is 333:
2. The square of the number has:
two 1's 1 1 _ _ _ _ _next digit is 0 _ _ _ 0 _ _ _ two 8's _ _ _ _ 8 8 _a final 9 _ _ _ _ _ _ 9
3. So 333 x 333 = 110889.
Squaring numbers made up of sixes
1. Choose a a number made up of sixes.
2. The square is made up of:
a. one fewer 4 than there are repeating 6's
b. 3
c. same number of 5's as 4's
d. 6
Example:
1. If the number to be squared is 666
2. The square of the number has:
4's (one less than digits in number) 4 43 35's (same number as 4's) 5 56 6
3. So 666 x 3666333 = 443556.
See the pattern?
1. If the number to be squared is 66666
2. The square of the number has:
4's (one less than digits in number) 4 4 4 43 35's (same number as 4's) 5 5 5 56 6
3. So 66666 x 66666 = 4444355556.
Squaring numbers made up of nines
1. Choose a a number made up of nines (up to nine digits).
2. The answer will have one less 9 than the number, one 8, the same number of zeros as 9's, and a final 1
Example:
1. If the number to be squared is 9999
2. The square of the number has:
one less nine than the number 9 9 9one 8 8the same number of zeros as 9's 0 0 0a final 1 1
3. So 9999 x 9999 = 99980001.
See the pattern?
1. If the number to be squared is 999999
2. The square of the number has:
one less nine than the number 9 9 9 9 9one 8 8the same number of zeros as 9's 0 0 0 0 0a final 1 1
3. So 999999 x 999999 = 999998000001.
Squaring numbers in the 20s
1. Square the last digit (keep the carry) _ _ X
2. Multiply the last digit by 4, add the carry _ X _
3. The first digit will be 4 plus the carry: X _ _
Example:
If the number to be squared is 24:
1. Square the last digit (keep the carry): 4 x 4 = 16 (keep 1) _ _ 6
2. Multiply the last digit by 4, add the carry:4 x 4 = 16, 16 + 1 = 17 _ 7 _
3. The first digit will be 4 plus the carry: 4 (+ carry): 4 + 1 = 5 5 _ _
4. So 24 x 24 = 576.
See the pattern?
If the number to be squared is 26:
1. Square the last digit (keep the carry): 6 x 6 = 36 (keep 3) _ _ 6
2. Multiply the last digit by 4, add the carry:4 x 6 = 24, 24 + 3 = 27 (keep 2) _ 7 _
3. The first digit will be 4 plus the carry: 4 (+ carry): 4 + 2 = 6 6 _ _.
4. So 26 x 26 = 676.
Squaring numbers in the 30s
1. Square the last digit (keep the carry) _ _ _ X
2. Multiply the last digit by 6, add the carry _ _ X _
3. The first digits will be 9 plus the carry: X X _ _
Example:
If the number to be squared is 34:
1. Square the last digit (keep the carry): 4 x 4 = 16 (keep 1) _ _ _ 6
2. Multiply the last digit by 6, add the carry:6 x 4 = 24, 24 + 1 = 25 _ _ 5 _
3. The first digits will be 4 plus the carry: 9 (+ carry): 9 + 2 = 11 1 1 _ _
4. So 34 x 34 = 1156.
See the pattern?
If the number to be squared is 36:
1. Square the last digit (keep the carry): 6 x 6 = 36 (keep 3) _ _ _ 6
2. Multiply the last digit by 6, add the carry:6 x 6 = 36, 36 + 3 = 39 (keep 3) _ _ 9 _
3. The first digits will be 9 plus the carry: 9 (+ carry): 9 + 3 = 12 1 2 _ _.
4. So 36 x 36 = 1296.
Squaring numbers in the 40s
1. Square the last digit (keep the carry) _ _ X
2. Multiply the last digit by 8, add the carry _ X _
3. The first digits will be 16 plus the carry: X X _ _
Example:
If the number to be squared is 42:
1. Square the last digit: 2 x 2 = 4 _ _ _ 4
2. Multiply the last digit by 8:8 x 2 = 16 _ _ 6 _
3. The first digits will be 16 plus the carry: 16 (+ carry): 16 + 1 = 17 1 7 _ _
4. So 42 x 42 = 1764.
See the pattern?
If the number to be squared is 48:
1. Square the last digit (keep the carry): 8 x 8 = 64 (keep 6) _ _ _ 4
2. Multiply the last digit by 8, add the carry:8 x 8 = 64, 64 + 6 = 70 (keep 7) _ _ 0 _
3. The first digits will be 16 plus the carry: 16 (+ carry): 16 + 7 = 23 2 3 _ _
4. So 48 x 48 = 2304.
Squaring numbers in the 50s
1. Square the last digit (keep the carry) _ _ _ X
2. Multiply the last digit by 10, add the carry _ _ X _
3. The first digits will be 25 plus the carry: X X _ _
Example:
If the number to be squared is 53:
1. Square the last digit (keep the carry): 3 x 3 = 9 (keep 3) _ _ _ 9
2. Multiply the last digit by 10, add the carry:10 x 3 = 30 (keep 3) _ _ 0 _
3. The first digits will be 25 plus the carry: 25 (+ carry): 25 + 3 = 28 2 8 _ _
4. So 53 x 53 = 2809.
See the pattern?
If the number to be squared is 56:
1. Square the last digit (keep the carry): 6 x 6 = 36 (keep 3) _ _ _ 6
2. Multiply the last digit by 10, add the carry:10 x 6 = 60, 60 + 3 = 63 _ _ 3 _
3. The first digits will be 25 plus the carry: 25 (+ carry): 25 + 6 = 31 3 1 _ _
4. So 53 x 53 = 3136.
Practice and you will soon be producing these products quickly and accurately.

Squaring numbers in the 60s
1. Square the last digit (keep the carry) _ _ _ X
2. Multiply the last digit by 12, add the carry _ _ X _
3. The first digits will be 36 plus the carry: X X _ _
Example:
If the number to be squared is 63:
1. Square the last digit (keep the carry): 3 x 3 = 9 (keep 3) _ _ _ 9
2. Multiply the last digit by 12, add the carry:12 x 3 = 36 (keep 3) _ _ 6 _
3. The first digits will be 36 plus the carry: 36 (+ carry): 36 + 3 = 39 3 9 _ _
4. So 63 x 63 = 3969.
See the pattern?
If the number to be squared is 67:
1. Square the last digit (keep the carry): 7 x 7 = 49 (keep 4) _ _ _ 9
2. Multiply the last digit by 12, add the carry:12 x 7 = 84, 84 + 4 = 88 _ _ 8 _
3. The first digits will be 36 plus the carry: 36 (+ carry): 36 + 8 = 44 4 4 _ _
4. So 67 x 67 = 4489.
Use this pattern and you will be squaring these numbers with ease.
Squaring numbers in the 70s
1. Square the last digit (keep the carry) _ _ _ X
2. Multiply the last digit by 14, add the carry _ _ X _
3. The first digits will be 49 plus the carry: X X _ _
Example:
If the number to be squared is 72:
1. Square the last digit: 2 x 2 = 4 _ _ _ 4
2. Multiply the last digit by 14:14 x 2 = 28 (keep the carry) _ _ 8 _
3. The first digits will be 49 plus the carry: 49 (+ carry): 49 + 2 = 51 5 1 _ _
4. So 72 x 72 = 5184.
See the pattern?
If the number to be squared is 78:
1. Square the last digit (keep the carry): 8 x 8 = 64 (keep 6) _ _ _ 4
2. Multiply the last digit by 14, add the carry:14 x 8 = 80 + 32 = 112112 + 6 = 118 (keep 11) _ _ 8 _
3. The first digits will be 49 plus the carry (11): 49 (+ carry): 49 + 11 = 60 6 0 _ _.
4. So 78 x 78 = 6084
Squaring numbers in the 80s
1. Square the last digit (keep the carry) _ _ X
2. Multiply the last digit by 16, add the carry _ X _
3. The first digits will be 64 plus the carry: X X _ _
Example:
If the number to be squared is 83:
1. Square the last digit: 3 x 3 = 9 _ _ _ 9
2. Multiply the last digit by 16:16 x 3 = 30 + 18 = 48 _ _ 8 _
3. The first digits will be 64 plus the carry: 64 (+ carry): 64 + 4 = 68 6 8 _ _
4. So 83 x 83 = 6889.
See the pattern?
If the number to be squared is 86:
1. Square the last digit (keep the carry): 6 x 6 = 36 (keep 3) _ _ _ 6
2. Multiply the last digit by 16, add the carry:16 x 6 = 60 + 36 = 96 96 + 3 = 99 (keep 9) _ _ 9 _
3. The first digits will be 64 plus the carry: 64 (+ carry): 64 + 9 = 73 7 3 _ _
4. So 86 x 86 = 7396.
Squaring numbers in the hundreds
1. Choose a number over 100 (keep it low for practice,then go higher when expert).
2. The last two places will be the square of the last two digits (keep any carry) _ _ _ X X.
3. The first three places will be the number plus the last two digits plus any carry: X X X _ _.
Example:
1. If the number to be squared is 106:
2. Square the last two digits (no carry): 6 x 6 = 36: _ _ _ 3 6
3. Add the last two digits (06) to the number: 106 + 6 = 112: 1 1 2 _ _
4. So 106 x 106 = 11236.
See the pattern?
1. If the number to be squared is 112:
2. Square the last two digits (keep carry 1): 12 x 12 = 144: _ _ _ 4 4
3. Add the last two digits (12) plus the carry (1) to the number: 112 + 12 + 1 = 125: 1 2 5 _ _
4. So 112 x 112 = 12544.
With a little practice your only limit will be your ability to square the last two digits!
Squaring numbers in the 200s
1. Choose a number in the 200s (practice with numbers under 210, then progress to larger ones).
2. The first digit of the square is 4: 4 _ _ _ _
3. The next two digits will be 4 times the last 2 digits: _ X X _ _
4. The last two places will be the square of the last digit: _ _ _ X X
Example:
1. If the number to be squared is 206:
2. The first digit is 4: 4 _ _ _ _
3. The next two digits are 4 times the last digit: 4 x 6 = 24: _ 2 4 _ _
4. Square the last digit: 6 x 6 = 36: _ _ _ 3 6
5. So 206 x 206 = 42436.
For larger numbers work right to left:
1. Square the last two digits (keep the carry): _ _ _ X X
2. 4 times the last two digits + carry: _ X X _ _
3. Square the first digit + carry: X _ _ _ _
See the pattern?
1. If the number to be squared is 225:
2. Square last two digits (keep carry): 25x25 = 625 (keep 6): _ _ _ 2 5
3. 4 times the last two digits + carry: 4x25 = 100; 100+6 = 106 (keep 1): _ 0 6 _ _
4. Square the first digit + carry: 2x2 = 4; 4+1 = 5: 5 _ _ _ _
5. So 225 x 225 = 50625.
Squaring numbers in the 300s
1. Choose a number in the 300s (practice with numbers under 310, then progress to larger ones).
2. The first digit of the square is 9: 9 _ _ _ _
3. The next two digits will be 6 times the last 2 digits: _ X X _ _
4. The last two places will be the square of the last digit: _ _ _ X X
Example:
1. If the number to be squared is 309:
2. The first digit is 9: 9 _ _ _ _
3. The next two digits are 6 times the last digit: 6 x 9 = 54: _ 5 4 _ _
4. Square the last digit: 9 x 9 = 81: _ _ _ 8 1
5. So 309 x 309 = 95481.
For larger numbers reverse the steps:
1. Square the last two digits (keep the carry): _ _ _ X X
2. 6 times the last two digits + carry: _ X X _ _
3. Square the first digit + carry: X _ _ _ _
See the pattern?
1. If the number to be squared is 325:
2. Square last two digits (keep carry): 25x25 = 625 (keep 6): _ _ _ 2 5
3. 6 times the last two digits + carry: 6x25 = 150; 150+6 = 156 (keep 1): _ 5 6 _ _
4. Square the first digit + carry: 3x3 = 9; 9+1 = 10: 1 0 _ _ _ _
5. So 325 x 325 = 105625.
Squaring numbers in the 400s
1. Choose a number in the 400s (keep the numbers low at first; then progress to larger ones).
2. The first two digits of the square are 16: 1 6 _ _ _ _
3. The next two digits will be 8 times the last 2 digits: _ _ X X _ _
4. The last two places will be the square of the last two digits: _ _ _ _ X X
Example:
1. If the number to be squared is 407:
2. The first two digits are 16: 1 6 _ _ _ _
3. The next two digits are 8 times the last 2 digits: 8 x 7 = 56: _ _ 5 6 _ _
4. Square the last digit: 7 x 7 = 49: _ _ _ 4 9
5. So 407 x 407 = 165,649.
For larger numbers reverse the steps:
1. Square the last two digits (keep the carry): _ _ _ _ X X
2. 8 times the last two digits + carry: _ _ X X _ _
3. 16 + carry: X X _ _ _ _
See the pattern?
1. If the number to be squared is 425:
2. Square the last two digits (keep the carry): 25 x 25 = 625 (keep 6): _ _ _ _ 2 5
3. 8 times the last two digits + carry: 8 x 25 = 200; 200 + 6 = 206 (keep 2): _ _ 0 6 _ _
4. 16 + carry: 16 + 2 = 18: 1 8 _ _ _ _
5. So 425 x 425 = 180,625.
Squaring numbers in the 500s
1. Choose a number in the 500s (start with low numbers at first; then graduate to larger ones).
2. The first two digits of the square are 25: 2 5 _ _ _ _
3. The next two digits will be 10 times the last 2 digits: _ _ X X _ _
4. The last two places will be the square of the last two digits: _ _ _ _ X X
Example:
1. If the number to be squared is 508:
2. The first two digits are 25: 2 5 _ _ _ _
3. The next two digits are 10 times the last 2 digits: 10 x 8 = 80: _ _ 8 0 _ _
4. Square the last digit: 8 x 8 = 64: _ _ _ 6 4
5. So 508 x 508 = 258,064.
For larger numbers reverse the steps:
1. Square the last two digits (keep the carry): _ _ _ _ X X
2. 10 times the last two digits + carry: _ _ X X _ _
3. 25 + carry: X X _ _ _ _
See the pattern?
1. If the number to be squared is 525:
2. Square the last two digits (keep the carry): 25 x 25 = 625 (keep 6): _ _ _ _ 2 5
3. 10 times the last two digits + carry: 10 x 25 = 250; 250 + 6 = 256 (keep 2): _ _ 5 6 _ _
4. 25 + carry: 25 + 2 - 27: 2 7 _ _ _ _
5. So 425 x 425 = 275,625.
Squaring numbers in the 600s
1. Choose a number in the 600s (practice with smaller numbers, then progress to larger ones).
2. The first two digits of the square are 36: 3 6 _ _ _ _
3. The next two digits will be 12 times the last 2 digits: _ _ X X _ _
4. The last two places will be the square of the last two digits: _ _ _ _ X X
Example:
1. If the number to be squared is 607:
2. The first two digits are 36: 3 6 _ _ _ _
3. The next two digits are 12 times the last 2 digits: 12 x 07 = 84: _ _ 8 4 _ _
4. Square the last 2 digits: 7 x 7 = 49: _ _ _ _ 4 9
5. So 607 x 607 = 368,449.
For larger numbers reverse the steps:
1. If the number to be squared is 625:
2. Square the last two digits (keep carry): 25x25 = 625 (keep 6): _ _ _ _ 2 5
3. 12 times the last 2 digits + carry: 12x25 = 250 + 50 = 300 + 6 = 306: _ _ 0 6 _ _
4. 36 + carry: 36 + 3 = 39: 3 9 _ _ _ _
5. So 625 x 625 = 390,625.
Squaring numbers in the 700s
1. Choose a number in the 700s (practice with smaller numbers, then progress to larger ones).
2. Square the last two digits (keep the carry): _ _ _ _ X X
3. Multiply the last two digits by 14 andadd the carry: _ _ X X _ _
4. The first two digits will be 49 plus the carry: X X _ _ _ _
Example:
1. If the number to be squared is 704:
2. Square the last two digits (keep the carry): 4 x 4 = 16: _ _ _ _ 1 6
3. Multiply the last two digits by 14 andadd the carry: 14 x 4 = 56: _ _ 5 6 _ _
4. The first two digits will be 49 plus the carry: 4 9 _ _ _ _
5. So 704 x 704 = 495,616.
See the pattern?
1. If the number to be squared is 725:
2. Square the last two digits (keep the carry): 25 x 25 = 625: _ _ _ _ 2 5
3. Multiply the last two digits by 14 andadd the carry: 14 x 25 = 10 x 25 + 4 x 25= 250 + 100 = 350. 350 + 6 = 356: 56: _ _ 5 6 _ _
4. The first two digits will be 49 plus the carry: 49 + 3 = 52: 5 2 _ _ _ _
5. So 725 x 725 = 525,625.
Squaring numbers between 800 and 810
1. Choose a number between 800 and 810.
2. Square the last two digits:_ _ _ _ X X
3. Multiply the last two digits by 16(keep the carry): _ _ X X _ _
4. Square 8, add the carry: X X _ _ _ _
Example:
1. If the number to be squared is 802:
2. Square the last two digits:2 x 2 = 4: _ _ _ _ 0 4
3. Multiply the last two digits by 16:16 x 2 = 32: _ _ 3 2 _ _
4. Square 8: 6 4 _ _ _ _
5. So 802 x 802 = 643,204.
See the pattern?
1. If the number to be squared is 807:
2. Square the last two digits:7 x 7 = 49: _ _ _ _ 4 9
3. Multiply the last two digits by 16(keep the carry): 16 x 7 = 112: _ _ 1 2 _ _
4. Square 8, add the carry (1): 6 5 _ _ _ _
5. So 807 x 807 = 651, 249.
Squaring numbers in the 900s
1. Choose a number in the 900s - start out easy with numbers near 1000; then go lower when expert.
2. Subtract the number from 1000 to get the difference.
3. The first three places will be the number minus the difference: X X X _ _ _.
4. The last three places will be the square of the difference: _ _ _ X X X(if 4 digits, add the first digit as carry).
Example:
1. If the number to be squared is 985:
2. Subtract 1000 - 985 = 15 (difference)
3. Number - difference: 985 - 15 = 970: 9 7 0 _ _ _
4. Square the difference: 15 x 15 = 225: _ _ _ 2 2 5
5. So 985 x 985 = 970225.
See the pattern?
1. If the number to be squared is 920:
2. Subtract 1000 - 920 = 80 (difference)
3. Number - difference: 920 - 80 = 840: 8 4 0 _ _ _
4. Square the difference: 80 x 80 = 6400: _ _ _ 4 0 0
5. Carry first digit when four digits: 8 4 6 _ _ _
6. So 920 x 920 = 846400

Interesting IQ test

You have to work out what the letters mean. See No 0 as an example.
According to MENSA, if you get 23 of these, you are a "genius".
Only 2 MENSA members achieved full marks. See how well you do.

Type in your answer

No.
Cryptic
Answer
0. 24 H in a D : 24 hours in a day

1. 26 L of the A

2. 7 D of the W

3 7 W of the W

4 12 S of the Z

5 66 B of the B

6 52 C in a P (WJs)

7 13 S in the USF

8 18 H on a G C

9 39 B of the O T

10 5 T on a F

11 90 D in a R A

12 3 B M (S H T R)

13 32 is the T in D F at which W F

14 15 P in a R T

15 3 W on a T

16 100 C in a R

17 11 P in a F (S) T

18 12 M in a Y

19 13=UFS

20 8 T on a O

21 29 D in F in a L Y

22 27 B in the N T

23 365 D in a Y

24 13 L in a B D

25 52 W in a Y

26 9 L of a C

27 60 M in a H

28 23 P of C in the H B

29 64 S on a C B

30 9 P in S A

31 6 B to an O in C

32 1000 Y in a M

33 15 M on a D M C

Interesting Brain Teasers

I am taken from a mine, and shut up in a wooden case, from which I am never released, and yet I am used by almost everybody. Pencil lead
What goes round the house and in the house but never touches the house? The sun
What is it that you can keep after giving it to someone else? Your word
What walks all day on its head? A nail in a horseshoe
What gets wet when drying? A towel
What comes once in a minute, twice in a moment, but never in a thousand years? The letter M
What is round as a dishpan, deep as a tub, and still the oceans couldn't fill it up? A sieve
There were five men going to church and it started to rain. The four that ran got wet and the one that stood still stayed dry. Body in coffin, and bearers
The more you take, the more you leave behind. What are they? Footsteps
He who has it doesn't tell it. He who takes it doesn't know it. He who knows it doesn't want it. What is it? Counterfeit money
Brothers and sisters have I none but that man's father is my father's son. My son
Who spends the day at the window, goes to the table for meals and hides at night? A fly
I bind it and it walks. I loose it and it stops. A sandal
What goes round and round the wood but never goes into the wood? The bark of a tree
I went to the city, I stopped there, I never went there, and I came back again. A watch
I have a little house in which I live all alone. It has no doors or windows, and if I want to go out I must break through the wall. A chicken in an egg
Scarcely was the father in this world when the son could be found sitting on the roof. Fire, smoke
There are four brothers in this world that were all born together. The first runs and never wearies. The second eats and is never full. The third drinks and is always thirsty. The fourth sings a song that is never good. Water, fire, earth, wind
A cloud was my mother, the wind is my father, my son is the cool stream, and my daughter is the fruit of the land. A rainbow is my bed, the earth my final resting place, and I'm the torment of man. Rain
Poke your fingers in my eyes and I will open wide my jaws. Linen cloth, quills, or paper, my greedy lust devours them all. Shears (or scissors)
What is that which goes with a carriage, comes with a carriage, is of no use to a carriage, and yet the carriage cannot go without it? Noise
It stands on one leg with its heart in its head. A cabbage
It's been around for millions of years, but it's no more than a month old. What is it? The moon
A white dove flew down by the castle. Along came a king and picked it up handless, ate it up toothless, and carried it away wingless. Snow melted by the sun
As I went across the bridge, I met a man with a load of wood which was neither straight nor crooked. What kind of wood was it? Sawdust
What belongs to you but others use it more than you do? Your name
What goes up the chimney down, but can't go down the chimney up? An umbrella
What is is that you will break even when you name it? Silence
What fastens two people yet touches only one? A wedding ring
What is it the more you take away the larger it becomes? A hole
I am the beginning of sorrow, and the end of sickness. You cannot express happiness without me, yet I am in the midst of crosses. I am always in risk, yet never in danger. You may find me in the sun, but I am never out of darkness. The letter S
What is put on a table, cut, but never eaten? A pack of cards
Who are the two brothers who live on opposite sides of the road yet never see each other? Eyes
What holds water yet is full of holes? A sponge
Though it is not an ox, it has horns; though it is not an ass, it has a pack-saddle; and wherever it goes it leaves silver behind. What is it? A snail
Lives without a body, hears without ears, speaks without a mouth, to which the air alone gives birth. An echo
A hundred-year-old man and his head one night old. Snow on a tree stump
What goes into the water red and comes out black? A red-hot poker
What goes into the water black and comes out red? A lobster
When one does not know what it is, then it is something; but when one knows what it is, then it is nothing. A riddle
A
· Able was I ere I saw Elba.
· A dog! A panic in a pagoda!
· Age, irony, Noriega.
· Ah, Aristides opposed it, sir, aha!
· Ah, Satan sees Natasha.
· All erotic, I lose lame female solicitor Ella.
· Al lets Della call Ed, Stella.
· A man, a plan, a canal - Panama!
· "Amen!" I call if I fill a cinema.
· Amore? Roma!
· A new order began, a more Roman age bred Rowena.
· Anne, I stay a day at Sienna.
· Anne, I vote more cars race Rome-to-Vienna.
· A nut for a jar of tuna.
· A pain, a blast; ah, that's Albania, Pa!
· A pre-war dresser drawer, Pa.
· Are we not drawn onwards, we Jews, drawn onward to new era?
· Are we not, Rae, near to new era?
· A rod, not a bar, a baton, Dora.
· A Toyota: race fast, safe car: a Toyota.
· A Toyota's a Toyota.
B (Top)
· Ban campus motto: "Bottoms up, MacNab".
· Bob: "Did Anna peep?" Anna: "Did Bob?"
· Bog dirt up a sidetrack carted is a putrid gob.
· Borrow or rob.
· But Anita sat in a tub.
C (Top)
· Cain: a maniac.
· Cigar? Toss it in a can, it is so tragic!
· Civic sin, Alan, is civic.
· Cora sees a roc.
D (Top)
· Damn! I, Agassi, miss again! Mad!
· Damosel, a poem? A carol? Or a cameo pale? So mad!
· Deer breed.
· Deer flee freedom in Oregon? No, Geronimo - deer feel freed.
· Deer frisk, sir, freed.
· Degenerate Moslem, a cad! Eva saved a camel so Meta reneged.
· Deirdre wets altar of St. Simon's - no mists, for at last ewer dried.
· Delia and Edna ailed.
· Delia failed.
· Delia, here we nine were hailed.
· Delia sailed as sad Elias ailed.
· Delia sailed, Eva waved, Elias ailed.
· Delia's debonair dahlias, poor, drop or droop. Sail, Hadrian; Obed sailed.
· Deliver, Eva, him I have reviled.
· "Deliver desserts," demanded Nemesis, "emended, named, stressed, reviled."
· Dennis and Edna sinned.
· Dennis, Nell, Edna, Leon, Nedra, Anita, Rolf, Nora, Alice, Carol, Leo, Jane, Reed, Dena, Dale, Basil, Rae, Penny, Lana, Dave, Denny, Lena, Ida, Bernadette, Ben, Ray, Lila, Nina, Jo, Ira, Mara, Sara, Mario, Jan, Ina, Lily, Arne, Bette, Dan, Reba, Diane, Lynn, Ed, Eva, Dana, Lynne, Pearl, Isabel, Ada, Ned, Dee, Rena, Joel, Lora, Cecil, Aaron, Flora, Tina, Arden, Noel and Ellen sinned.
· Dennis, no misfit, can act if Simon sinned.
· Deny me not; atone, my Ned.
· Depardieu, go rap a rogue I draped.
· Desserts I desire not, so long no lost one rise distressed.
· Diana saw Dr. Awkward was an aid.
· Did Dean aid Diana? Ed did.
· Did Hannah say as Hannah did?
· Did Hannah see bees? Hannah did.
· Di, did I as I said I did?
· Did Ione take Kate? No, I did.
· Did I do, O God, did I as I said I'd do? Good, I did!
· Did I draw Della too tall, Edward? I did?
· Doc, note, I dissent, A fast never prevents a fatness. I diet on cod.
· Dog, as a devil deified, lived as a god.
· Do geese see God?
· Dogma: I am God.
· Do Good's deeds live on? No, Evil's deeds do, O God.
· "Do nine men interpret?" "Nine men," I nod.
· Do not start at rats to nod.
· Don't nod.
· Doom an evil deed, liven a mood.
· Doom, royal panic, I mimic in a play or mood.
· Dora tendered net, a rod.
· Drab as a fool, as aloof as a bard.
· Drab Reg, no longer bard.
· Drat Saddam! A mad dastard!
· Draw - aye, no melody - dole-money award.
· Draw no dray a yard onward.
· Draw, O Caesar, erase a coward.
· Draw, O coward!
· Draw pupil's pup's lip upward.
· Drowsy baby's word.
E (Top)
· E. Borgnine drags Dad's gardening robe.
· Ed: a general, a renegade.
· Ed, I saw Harpo Marx ram Oprah W. aside.
· Ed is on no side.
· Edith, cold-eyed, eyed loch tide.
· Egad, a base life defiles a bad age.
· Egad, a base tone denotes a bad age.
· Egad, an adage!
· Egad! Loretta has Adams as mad as a hatter. Old age!
· Egad! No bondage!
· Emil asleep, Allen yodelled "Oy!" Nella peels a lime.
· Emil asleep, Hannah peels a lime.
· Enid and Edna dine.
· Ere hypocrisies or poses are in, my hymn I erase. So prose I, sir, copy here.
· Euston saw I was not Sue.
· Euston sees not Sue.
· Eva, can I pose as Aesop in a cave?
· Eva, can I stab bats in a cave?
· Evade me, Dave.
· Eve damned Eden, mad Eve.
· Eve saw diamond, erred. No maid was Eve.
· Evil is a name of a foeman, as I live.
· Evil - a diamond, a cad - no maid alive!
· Evil odes or prose do live.
G (Top)
· Flee to me, remote elf.
· Flesh? Saw I Mimi wash self?
G (Top)
· Gate-man sees name, garage-man sees name-tag.
· Gert, I saw Ron avoid a radio-van - or was it Reg?
· Gnu dung.
· God, a red nugget! A fat egg under a dog!
· God, a slap! Paris, sir, appals a dog!
· Goddesses so pay a possessed dog.
· "Go, droop aloof," sides reversed, is "fool a poor dog".
· Go hang a salami! I'm a lasagna hog!
· Golf? No, sir, prefer prison-flog.
H (Top)
· Ha! I rush to my lion oily moths, Uriah!
· Ha! Jar level Rajah!
· Harass selfless Sarah.
· Harass sensuousness, Sarah.
· Ha! Robed Selim smiles, Deborah!
· Ha! Robed rats deliver reviled star, Deborah!
· He lived as a devil, eh?
· Hell! A spacecraft farce caps all, eh?
· Help Max, Enid, in example "H".
· Here so long? No loser, eh?
· He won a Toyota now, eh?
I (Top)
· If I had a hi-fi...
· I, madam, I made radio. So I dared! Am I mad? Am I?
· I made border bard's drowsy swords; drab, red-robed am I.
· I maim nine men in Saginaw; wan, I gas nine men in Miami.
· I maim nine more hero-men in Miami.
· I, man, am regal; a German am I.
· I, Marian, I too fall; a foot-in-air am I.
· I moan, "Live on, O evil Naomi!"
· In airy Sahara's level, Sarah, a Syrian, I.
· In a regal age ran I.
· I prefer pi.
· I, Rasputin, knit up sari.
· I roamed under it as a tired, nude Maori.
· I saw desserts. I'd no lemons, alas no melon. Distressed was I.
· I saw I was I.
· I saw thee, madame, eh? 'Twas I.
· "Is Don Adams mad?" A nod. "Si!"
· I tip away a wapiti.
· I told Edna how to get a mate: "Go two-handed." Loti.
J (Top)
· Jar a tonga, nag not a Raj.
K (Top)
· Kayak salad: Alaska yak.
· Kay, a red nude, peeped under a yak.
· Kay dated a cadet, a Dyak.
· "Knight, I ask nary rank," saith gink.
L (Top)
· La, not atonal!
· Lager, sir, is regal.
· Lapp, Mac? No, sir, prison-camp pal.
· Last fig - as a gift, Sal.
· Lay a wallaby baby ball away, Al.
· Lee had a heel.
· Lepers repel.
· Let O'Hara gain an inn in a Niagara hotel.
· Lew, Otto has a hot towel.
· Lid off a daffodil.
· Live dirt up a sidetrack carted is a putrid evil.
· Live not on evil.
· Live not on evil deed, live not on evil.
· Live not on evil, madam, live not on evil.
· Live, O Devil, revel ever, live, do evil.
· Live on, Time; emit no evil.
· Live was I ere I saw Evil.
· Loot: slate, metal plate, metal stool.
M (Top)
· Madame, not one man is selfless; I name not one, Madam.
· Madam, I'm Adam.
· Madam, in Eden I'm Adam.
· Ma handed Edna ham.
· Ma is a nun, as I am.
· Ma is as selfless as I am.
· "Ma," Jerome raps pot top, "spare more jam!"
· Man, Eve let an irate tar in at eleven a.m.
· Man, Oprah's sharp on A.M.
· Marge let a moody baby doom a telegram.
· Marge lets Norah see Sharon's telegram.
· Marge, let's "went". I await news telegram.
· Max, I stay away at six a.m.
· May a moody baby doom a yam?
· Milestones? Oh, 'twas I saw those, not Selim.
· Mirth, sir, a gay asset? No, don't essay a garish trim.
· Moorgate got nine men in to get a groom.
· Moors dine, nip - in Enid's room.
· Mother at song no star, eh Tom?
· Mother Eve's noose we soon sever, eh Tom?
· Mr. Owl ate my metal worm.
· Murder for a jar of red rum.
· Must sell at tallest sum.
N (Top)
· Name no-one man.
· Name now one man.
· Name tarts? No, medieval slave, I demonstrate Man!
· Naomi, did I moan?
· Ned, go gag Ogden.
· Ned, I am a maiden.
· Nella, demand a lad named Allen.
· Nella risks all: "I will ask Sir Allen."
· Nella's simple hymn: "I attain my help, Miss Allen."
· Nella won't set a test now, Allen.
· Nemo, we revere women.
· Never a foot too far, even.
· Never odd or even.
· Niagara, O roar again!
· No benison, no sin, Ebon.
· "No cab, eh, Ted?" I sat up. I put aside the bacon.
· No Dot nor Ottawa "legal age" law at Toronto, Don.
· Noel, did I not rub Burton? I did, Leon.
· Noel, let's egg Estelle on.
· Noel sees Leon.
· No evils Shahs live on.
· No garden, one dragon.
· No, Hal, I led Delilah on.
· No ham came, sir, now siege is won. Rise, MacMahon.
· No, I save on final perusal, a sure plan if no evasion.
· No, is Ivy's order a red rosy vision?
· No, it can assess an action.
· No, it is open on one position.
· No, it is opposed; Art sees Trade's opposition.
· No, it is opposition.
· No, it never propagates if I set a gap or prevention.
· No, it's a bar of gold, a bad log for a bastion.
· No lemons, no melon.
· No, Mel Gibson is a casino's big lemon.
· No miss, it is Simon.
· No Misses ordered roses, Simon.
· No mists or frost, Simon.
· Nomists reign at Tangier, St. Simon.
· Nora, alert, saws goldenrod-adorned logs. Wastrel Aaron!
· "Nora, a raft!" "Is it far, Aaron?"
· Norah's foes order red rose of Sharon.
· "Norah's moods," Naomi moans, "doom Sharon."
· Noriega can idle, held in a cage - iron!
· Nor I, fool, ah no! We won halo - of iron.
· Nor I nor Emma had level'd a hammer on iron.
· Norma is as selfless as I am, Ron.
· No, set a maple here; help a mate, son.
· "Not for Cecil?" asks Alice Crofton.
· Not I, no hotel; cycle to Honiton.
· "Not New York," Roy went on.
· Not nil, Clinton.
· Not seven on a mere man - one vest on.
· Not so, Boston.
· No waste, grab a bar, get saw on.
· "Now dine," said I as Enid won.
· Now do I repay a period won.
· Now ere we nine were held idle here, we nine were won.
· Now, Eve, we're here, we've won.
· Nowise I bury rubies I won.
· Now, Ned, I am a maiden nun; Ned, I am a maiden won.
· No word, no bond; row on.
· Now saw ye no mosses or foam, or aroma of roses. So money was won.
· Now, sir, a war is won.
· No "x" in "Mr. R. M. Nixon"?
· Nurse, save rare vases, run!
· Nurse, I spy gypsies, run!
· Nurse's onset abates, noses run.
O (Top)
· O desirable Melba, rise, do!
· O Geronimo, no minor ego!
· O gnats, tango!
· O had I nine more hero-men in Idaho!
· Oh, cameras are macho.
· O.J. nabs Bob's banjo.
· Oh who was it I saw, oh who?
· On tub, Edward imitated a cadet; a timid raw debut, no?
· O render gnostic illicit song, red Nero.
· O, stone, be not so.
· Otto made Ned a motto.
P (Top)
· Paget saw an Irish tooth, sir, in a waste gap.
· Paget saw a wasp in a waste gap.
· Part of U.S. is UFO trap.
· Party boobytrap.
· Pa's a sap.
· Pat and Edna tap.
· Peel's lager on red rum did murder no regal sleep.
· Plan no damn Madonna LP.
· Pooh, roll a ball or hoop.
· "Pooh," smiles Eva, "have Selim's hoop."
· Poor Dan is in a droop.
· Pull a bat! I held a ladle, hit a ball up.
· Pull up, Eva, we're here; wave, pull up.
· Pull up if I pull up.
· Pupils roll a ball or slip up.
· Pupils slip up.
· Pusillanimity obsesses Boy Tim in "All Is Up".
· Puss, a legacy! Rat in a snug, unsanitary cage, lass, up!
R (Top)
· "Rats gnash teeth," sang star.
· Rats live on no evil star.
· Raw was I ere I saw war.
· Red lost case, Ma. Jesse James acts older.
· Red Nevada vendor.
· Red now on level - no wonder.
· Redraw a warder.
· Red robber gazes not on S.E. Zagreb border.
· Red root put up to order.
· Red roses run no risk, sir, on nurse's order.
· Red rum, eh? 'Twas I saw the murder.
· Red rum, sir, is murder.
· Refasten gipsy's pig-net safer.
· Regard a mere mad rager.
· Reg, no lone car won, now race no longer.
· Re hypocrisy: as I say, sir, copy her.
· Remit Rome cargo to go to Grace Mortimer.
· Repel evil as a live leper.
· Resume so pacific a pose, muser.
· Retracting, I sign it, Carter.
· Revenge, Bill? I won! Will I beg? Never!
· Revenge my baby, Meg? Never!
· Revered now I live on. O did I do no evil, I wonder ever?
· "Reviled did I live," said I, "as evil I did deliver."
· "Revolt, love," raved Eva. "Revolt, lover!"
· Revolt on Yale, Democrats edit "Noon-Tide Star". Come, delay not, lover.
· Rise, morning is red, no wonder-sign in Rome, sir.
· Rise, take lame female Kate, sir.
· Rise to vote, sir.
· Ron, Eton mistress asserts I'm no tenor.
· Rot can rob a born actor.
· Roy Ames, I was a wise mayor.
· Roy, am I mayor?
S (Top)
· Sad? I'm Midas!
· Sail on, game vassal! Lacy callas save magnolias.
· Saladin enrobes a baroness, Señora, base-born Enid, alas!
· Salisbury Moor, sir, is roomy. Rub Silas.
· "Sal is not in?" Ruth asks. "Ah, turn it on, Silas."
· Satan, oscillate my metallic sonatas.
· Sat in a taxi, left Felix at Anita's.
· Satire: Veritas.
· Saw tide rose? So red it was!
· See few owe fees.
· See, slave, I demonstrate yet arts no medieval sees.
· Selim's tired; no wonder, it's miles.
· Semite, be sure! Damn a man-made ruse betimes!
· Senile felines.
· Set a broom on no moor, Bates.
· Sh! Tom sees moths.
· Sir, I demand, I am a maid named Iris.
· Sir, I'm Iris.
· Sir, I soon saw Bob was no Osiris.
· "Sirrah! Deliver deified desserts detartrated!" stressed deified, reviled Harris.
· Sis, ask Costner to not rent socks "As Is"!
· Sis, Sargasso moss a grass is.
· Sit on a potato pan, Otis.
· Si, we'll let Dad tell Lewis.
· Six at party; no pony-trap, taxis.
· "slang is not suet, is it?" Euston signals.
· Slap-dab set-up, Mistress Ann asserts, imputes bad pals.
· Snug satraps eye Sparta's guns.
· Snug was I ere I saw guns.
· So, Ida, adios!
· "So I darn on," a canon radios.
· Solo gigolos.
· So many dynamos.
· So may Apollo pay Amos.
· So may get Arts award. Draw a strategy, Amos.
· So may Obadiah aid a boy, Amos.
· So may Obadiah, even in Nineveh, aid a boy, Amos.
· Some men interpret nine memos.
· So remain a mere man. I am Eros.
· Sore was I ere I saw Eros.
· Spots tie - it's tops!
· Star? Come, Donna Melba, I'm an amiable man - no Democrats!
· Star comedy by Democrats.
· Stella won no wallets.
· Stephen, my hat! Ah, what a hymn, eh, pets?
· Step on hosepipes? Oh no, pets.
· Step on no pets!
· Stop! Murder us not, tonsured rumpots!
· "Stop!" nine myriad murmur. "Put up rum, rum, dairymen, in pots!"
· Stop, Syrian, I start at rats in airy spots.
· St. Simon sees no mists.
· Strategy: get arts.
· Straw? No, too stupid a fad. I put soot on warts.
· Sue, dice, do, to decide us.
· "Sue," Tom smiles, "Selim smote us!"
· "Suit no regrets." A motto, Master Gerontius.
· Sums are not set as a test on Erasmus.
· Swept pews? Yes, Sam Massey swept pews.
T (Top)
· Tarzan raised Desi Arnaz' rat.
· Telegram, Margelet!
· Ten animals I slam in a net.
· Ten dip a rapid net.
· Tenet C is a basis, a basic tenet.
· Tennis set won now Tess in net.
· Tennis tips: saliva. Vilas spits in net.
· Ten? No bass orchestra tarts, eh? Cross a bonnet.
· Tense, I snap Sharon roses or Norah's pansies net.
· Tessa's in Italy, Latin is asset.
· Tide-net safe soon, all in - a manilla noose fastened it.
· 'Tis Ivan on a visit.
· To last, Carter retracts a lot.
· To nets, ah no, son, haste not.
· Too bad, I hid a boot.
· Too far away, no mere clay or royal ceremony, a war afoot.
· Too far, Edna, we wander afoot.
· Too hot to hoot.
· Top step - Sara's pet spot.
· Top step's pup's pet spot.
· Tracy, no panic in a pony-cart.
· Trade ye no mere moneyed art.
· Trap all afoot; I too fall apart.
· Trap a rat! Stare, piper, at star apart!
· Trash? Even interpret Nineveh's art.
· Tulsa nightlife: filth, gin, a slut.
W (Top)
· War-distended nets I draw.
· Ward nurses run draw.
· Warsaw was raw.
· Was it a rat I saw?
· Was it Eliot's toilet I saw?
· Was it felt? I had a hit left, I saw.
· Was raw tap ale not a reviver at one lap at Warsaw?
· We freer few.
· We'll let Mom tell Lew.
· We name opera, rare poem, anew.
· We panic in a pew.
· We seven, Eve, sew.
· We, so to get a mate, go to sew.
· Wonders in Italy: Latin is "red" now.
· Won race, so loth to lose car now.
· Won't I repaper? Repaper it now.
· Won't lovers revolt now?
· Wonton on salad? Alas, no, not now!
X (Top)
· Xerxes was stunned! Eden nuts saw sex, Rex!
Y (Top)
· Yacht notes radar set on th' cay.
· Yawn a more Roman way.
· Yawn! Madonna Fan? No damn way!
· Ye slew Wesley.
· Yes, Mark, cable to hotel: "Back, Ramsey!"
· Yes, Syd, Owen saved Eva's new Odyssey.
· Yo! Bad anaconda had no Canada boy.
· Yo! Basil is a boy!
· Yo! Bottoms up! U.S. motto, boy!
· Yo! Breed deer, boy!
Z (Top)
· Zeus was deified, saw Suez.
The Damaged Engine
A Classic Puzzle by Henry Ernest Dudeney

We were going by train from Anglechester to Clinkerton, and an hour after starting an accident happened to the engine.We had to continue the journey at three-fifths of the former speed. It made us two hours late at Clinkerton, and the driver said that if only the accident had happened fifty miles farther on the train would have arrived forty minutes sooner. Can you tell from that statement just how far it is from Anglechester to Clinkerton?

The Damaged Engine
Solution

The distance from Anglechester to Clinkerton must be 200 miles.
The train went 50 miles at 50 m.p.h. and 150 miles at 30 m.p.h.
If the accident had occurred 50 miles farther on, it would have gone 100 miles at 50 m.p.h. and 100 miles at 30 m.p.h.
The Man and the Dog
A Classic Puzzle by Henry Ernest Dudeney

"Yes, when I take my dog for a walk," said a mathematical friend, "he frequently supplies me with some interesting puzzle to solve. One day, for example, he waited, as I left the door, to see which way I should go, and when I started he raced along to the end of the road, immediately returning to me; again racing to the end of the road and again returning. He did this four times in all, at a uniform speed, and then ran at my side the remaining distance, which according to my paces measured 27 yards. I afterwards measured the distance from my door to the end of the road and found it to be 625 feet. Now, if I walk 4 miles per hour, what is the speed of my dog when racing to and fro?"
The Man and the Dog
Solution

The dog's speed was 16 miles per hour.
The following facts will give the reader clues to the general solution. The distance remaining to be walked side by side with the dog was 81 feet, the fourth power of 3 (for the dog returned four times), and the distance to the end of the road was 625 feet, the fourth power of 5. Then the difference between the speeds (in miles per hour) of man and dog (that is, 12) and the sum of the speeds (20) must be in the same ratio, 3 to 5, as is the case.

Crossing the River
A Classic Puzzle by Henry Ernest Dudeney

During the Turkish stampede in Thrace, a small detachment found itself confronted by a wide and deep river. However, they discovered a boat in which two children were rowing about. It was so small that it would only carry the two children, or one grown person.How did the officer get himself and his 357 soldiers across the river and leave the two children finally in joint possession of their boat? And how many times need the boat pass from shore to shore?

Crossing the River
Solution

The two children row to the opposite shore. One gets out and the other brings the boat back. One soldier rows across; soldier gets out, and child returns with boat. Thus it takes four crossings to get one man across and the boat brought back. Hence it takes four times 358, or 1432 journeys, to get the officer and his 357 men across the river and the children left in joint possession of their boat.

Interesting Maths

CHAPTER - 1
Diamond Of Vedic Mathematics- Number “9”
Friends, I am sure that the ‘Magical Methods’
explained in this book are very easy to work with and you
will be thrilled after learning & understanding these
methods. Try to teach these methods to as many people as
you can.
To develop the interest in ‘Vedic Mathematics’, I
take an example of ‘Table - 99’.
Have a view at following example :-
99 - Multiplicand
x 20 - Multiplier
19/80 - Product
The clue to the answer is found by the multiplier 20 only.
The product is divided into two parts by a slash ( / ) sign.
L.H.S =19 & R.H.S. = 80
By more observation you can see that L.H.S = Multiplier-1
(i.e.20-1) & R.H.S. = 100-20 (Multiplicand 99 is close to
100, so we take 100 as base).
If you have understood the above principle hats off, you
can now Mentally Calculate 99 into any 2 digit number.
Let us see some more example.
99 99 99
X 30 x 85 x 63
29/70 84/15 62/37
9 (30-1) (100-30) 9 (85-1) (100-85) 9 (63-1)
(100-63)
With two example you will master the table 99.
99 99
X 08 similarly x 95
07/92 94/05
L.h.s =8-1=07 where o is called zero deficiency r.h.s. =
95-1=94
R.h.s. =100-8=92
r.h.s. =100-5=05
Thus to conclude i may, vedic mathematics is magic until.
You understood & it is mathematics there after. The sutra
on which the above problem works is explained in the
chapter multiplication. Now we go to individual
applications of each sutras.
99999 999
9999
X12345 x678
x 2566
12344/87655 677/322
2566/7435
Nikhilam sutra :- all from 9 & last from10
Chapter
Cube roots of exact cubes
The technique used in this chapter for extraction of cube
roots is very simple & interesting. You just need to
memories the following cubes & rest all is easy for you.
cube last digit
13 = 1 1
23 = 8 8
33 = 27 7
43 = 64 4
53 = 125 5
63 = 216 6
73 = 343 3
83 = 512 2
93 = 729 9
03 = 0 0
Silent features for finiding cube root:-
1. The cube of numbers ending in 0,1,4,5,6 &9
have their cube roots also ending in the same digits
respectively.(b)
2. The cubes ending in2,3,7& 8 have their cube
roots also ending in 8,7,3,&2 respectively.(b)
3. The numbers og digits in a cube root of a
numbers in the same as like number of 3 digit groups in the
given numbers including a single or a two digit group if
there in any.
Thus we start from the right hand side of the cube &put a
comma when the 3 digits are over.
Eg:- thus 117649 will be written as 117,649 & thus will
have 2 digits in its cube-root
12167 will be written as12,167 & thus will have 2 digits in
its cube-root
12977875 will be written as12,977,875 & thus will have 3
digits in its cube-root
125 has 3 digits only, 20 there will be 1 digit in its cube
root i.e.5 as you see above.
Thus to summaries we say that the grouping of digits is
done from right to left for non decimal & from left of the
decimal towards right in the case of decimals.
Let us take a example to find cube root.
59319
Starting from the right hand side of the numbers we put
comma after every three digits are over. The numbers of
groups formed specify the number digits in the cube – root.
Thus we writ 59,319 & so there are 2 digits in the cube –
root.
Now first see the last digits of the numbers i.e 9 going
though the first two points of salient features & the table
provided above, you can now easily guess. The last digit of
the cube root. Thus we get 9 as last digit of cube root
Now take the first group i.e. 59
From the above table just find out cube of number which
is less than 59. Since 27 (cube of 3) in ten than 59 we get 3
as the first digit of cube root.
Thus the equation looks as follows
59 319
3 9
Cube root is 39.
Case ii:-
474, 552
We write the numbers as 474, 552.
Since there are 2 groups so there are two digits in cube
root.
The last digit of the numbers is 2, reading the first two
points of salient features & the table provided. We can now
easily judge that 8 is the last digit of cube root.
The first groups is 474. From the table provided, we see
that 373 (cube of 7) lets below 474. Thus 7 is the first digit
of cube root.
474 , 552
7 8
Cube root is 78.
Now practice the following examples on your own.
Case iii:-
614 , 125
8 5
Cube root is 85.
Case 4:-
2 , 197
1 3
Cube root of 2197 is 13.
Case v:-
19 , 683
2 7
Cube root of 19683 is 27
Case vi:-
42, 875
3 5
Cube root of42875 is 35.
Case vii:-
884, 736
9 6
Cube root of 884736 is 96.
Friends , i believe that you are now in a condition to easily
calculate mentally the cube root of a 6 digit number in just
6 records!!! To make a note above technique is valid only
for exact cubes. But do not worry, the technique for
calculating cube roots of any numbers is get to some & will
be expounded at a later stage.
When you might be wondering , whether there is a one line
method to find cube –root of any general numbers (may or
may not be a exact cube) bt the vedic sutras, for your
information the answer ia yes! And will be dealt with, at an
appropriate place, in a later stage.
Simultaneous equations
Chapter-
Students & professionals, both come across simultaneous
equations frequently. The current method of simultaneous
equations method by which we frame new equations
involving only x or y coefficients is tiresome & can load
to manual errors. The vedic system use the cross
multiplication method, which gives one line mental answer
for the coefficient thus saving time & errors .
Let us start with an example.
3x – 4y = 4
2x – 3y = 6
Here we apply the ‘paravartya’ rule which means transpose
& divide. It enable us to calculate value of ‘x’ by mere
mental dritrmetic. X= numerator for numerator.
Denominator
We adopt the following procedures .
3x – 4y = 4
2x – 3y = 6
Numerator=(coefficients of y in 1st row x constant in 2nd
row)-
(coefficients of y in 2nd row x constant in 1st row)
Note: the coefficients are taken along with this signs (+or-)
intact
\numerator =(-4x6) – (-3x4) = -24 +12 =-12
For denominator we adopt like following procedure.
3x – 4y = 4
2x – 3y = 6
Denominator = (coefficients of y in 1st row x coefficient
of x 2nd row) –( coefficients of y in 2nd row x coefficients
of x 1st row)
Note: the coefficients are taken along with this signs (+ or -
) intact.
Denominator = (-4x2) – (-3x3) =
= -8 + 9
=1
\x=numerator =-12 = -12
denominator 1
Ituting the value of x in 1st eqn we can now find the value
of y.
3(-12) -4y=4
-36-4y=4
-4=4+36=40
Y=-10
\x=-12 & y =-10
Ii) 6x-3y=3
4x+2y=14
X = (-3x14) – (2x3) = -42-6 = -48 =2
(-3x4) – (2x6) = -12-12 = -24
x in 1st eqn we get value for y.
6 x 2-3 x y =3
\y= -9 =3
-3
\x=2 & y=3
Iii) 2x-8y=20
3x+3y=-3
X = (-8x-3) – (-3x-20) = +24-60 = -36 = 2
(-8x-3 ) – (-3 x 2) = -24+6 = -18
. In 1st eqn
2 x 2 -8y = -20
-8y - -24
Y = 3
\x=2 & y=3
Sunyam anyat: if one is in ratio, the other one is zero. Lets
us clear the above sutra by an example.
3x + 4y = 6
6x + 3y = 12
In the above two equators, we see that the x-coefficients
are the same ratio to each other as the independent terms
are to each other. Thus by the above sutra, if one
coefficient is in ratio, the coefficient in ‘ 0.’
Thus y=0
value of y in 1st eqn we getx
3x + 4 x 0 =6
\x =2
Ii) 43x + 86y =43
86x + 72y = 86
Since x is in same ratio to each other as the independent
terms are to each other, y=0.
\ 43xx + 86 x 0 =43
\ x=1
\ x=1 & y=0
Iii) 142 x + 72y = 216
799x + 216 = 648
Since y is in same ratio to each other as the independent
terms are to each other we conclude that x=0
valve of x in 1st eqn
142 x 0 + 72y = 216
\y =3
\ x=0 & y=3
Sankalana – vyavakalanalhyam :- by addition &
subtraction
The above up sutra is helpful wherever the x& y
coefficients are found interchanged by simply adding or
subtracting the two equations give the values of (x+y) &
(x-y)
Repeating the above process one more time gives us values
of x & y.
23x – 33y = -53 (1)
33x – 23y = -3 (2)
By adding the above two equation 1 &2 we get
56x – 56y = -56
\56 (x-y) = -56
X- = -1 (3)
& by subtracting the above two equation 1&2 we get
-10x – 10y = -50
-10 (x+y) = -50
\x+y = 5 (4)
Adding equation 3 & 4 we get
2x = 4
\ x = 2 & y = 3
Introduction about vedic mathematics:-
The main research on vedic mathematics is done by his
holies jagadguru sankaracarya , sri bharti krishna , tirthaji
maharaja of govardhana matha puri (1884-1960). He had
used 16 sutras and sub sutras (corollaries) which are listed
in this chapter.
Sutra literally means ‘ thread’ but tirthaji maharaja
employs the word ‘ aphorism’ because it does not show
how a calculation is to be made but only throws up a
pointer or direction in which the calculation can proceed.
Thus the same sutra can be employed for a variety of
applications. The sutras are original in sanskrit and so the
english version of one which are used in this book are
given below
Sutra (word-formula) used in
Sub-sutra (corollary)
1. Ekadhikena pu rvena multiplication,
divisibility, recurring
“by one more than the decimals.
privious one”
anurupyrna multiplication,
division, cubing
(corollary)
“proportionately”
2.nikiilam multiplication,
division
navataxaranam
dasatah
“all from nine & last ten”
Sir multiplication, division
(corollary) multiplication
‘remainder remains
Constant
3. Urdhava-tiryagbhyam multiplication
‘vertically & cross- division
wise’
4. Paravartya yojayet division
‘transpose & divide
5.yavadunam multiplication
tavadunam(corollary)
“whatever the extent
of its deficiency lessen
It stil further to that
Very extent.
6.antyayordasake api multiplication
(corollary)
A.squaring of a number
whose last digits add
to 10 and whose
previous part is
exactly the same
7. Yavadunam squaring
‘deficiency’ cubing
8. Desanyankena recurring decimals
caramena
“the remainder
by the last digit”
vilokanam (corollary)
‘by inspection or division
observation’
9.ekany-------- multiplication
‘multiplication
whenever the
multiplied- digit
entirely of 9’5”
10.dhvajanka division
‘on top of the flag’
Ekadhikena purvena – “by one more than the previous
one”
Valgar fractions.
By more observation of the sutra we see that it has used the
preparation “by” at the start at indicate that the arithmetical
operation prescribed is either multiplication or division.
For in the case of addition & multiplication, ‘to’ and aaa
respectively would have been the appropriate preposition
to use thus as the matter of selection we use ‘division’
technique for
take example of 1/19
=1 dividend & 19 divisor.
The last digit of the denominator in this case being 9 & the
previous one being 1 “one more than the previous one”
evidently means 2.
By the vedic one line method we write the answer as
follows-
1 = 0.0526315789473 68421 in vedic form we
write as follows
19
1 = 0.1 = 10 05 12 06 03 11 15 17 18 quotient
reminder
quotient
09 14 7 13 16 8 4 2 1 reminder
Can you believe it !!! Even the calculation through which
you are checking has ten digit answer & we re upto 18
digit answer.
Explanation:-
I. Since the denominator has a single 9, we shift the
decimal in the numerator by one place to the left making
number = 0.1
II. Drop 9 from the denominator and increase the
penultimate digit (i,e.1) of denominator by one so that the
vulgar fraction now reads 0.1.
2
III. We now divided 0.1 by 2 which is a very simple and
working divisor.
IV. On dividing 0.1 by 2 we get quotient q=0 &
remainder r=1. We therefore, set 0 down as the first digit of
the quotient and prefix the remainder 1 to that very digit of
the quotient and thus obtain 10 as our next dividend. (10)
V. Dividing this 10 by 2, we get 5 as the record digit of
the quotient and there is no remainder to the prefixed there
to, we take up that digit 5 itself as our next dividend (1.05)
VI. Do the next quotient digit is 2 and the remainder is
1. We therefore put 2 down as the third digit of quotient
and prefix the remainder 1 to that quotient digit 2 and thus
have 12 as our next dividend (1 .05 1 2)
VII. This gives us 6 as quotient digit and remainder is 0.
So we set 6 down as the fourth digit of the quotient and as
there is no remainder to the prefixed thereto, we take 6
itself as our next digit for division which gives the next
quotient digit as 3.(105 12 6 311)
VIII. Dividing by 2 & prefixed to quotient we get new
dividing by 2 we get 5 as quotient and 1 as remainder. So
the new dividend is 15. 1 0 5 1 2 6 3 1 1 1 5
IX. Carrying this process of straight, continuous
division by 2, we get 2 as the 17th quotient-digit and 0 as
remainder.
X. Dividing 2 by 2, we get 1 as 18th quotient digit and 0
as remainder but it is repletion of what we started with.
Thus the decimal begins to repeat itself from here. So we
stop the mental division process and put down the usual
recurring symbols (dots) on the 1st & 18th digit to show that
the whole of it is a circulating decimal.
A further short-cut :- let us put down the first 9 digit of
the answer in one horizontal row above and the other 9
digits exactly below the first 9 digits.
0.052631578
947368421
999999999
By more observation we see that each set of digits in the
upper row and lower row total 9. Thus it means that when
just half the work has been completed by the great vedic
one line method the other half is mechanically available to
us by subtracting from 9 each digits already obtained and
this means a reduction of work still further by 50%
But now you should know when the t is exactly half
finished. Do here it is as soon as we reach the difference
between the numerator & denominator (i,e. 19-1=18), we
shall have completed exactly half the work!!!. So in above
example when we reached 18 as dividend we stop the
work, thus if you see now, the vulgar fractions such as 1 ,
1
19 29
1 are solved in one simple line & oven young boys man
do it.
19
Case 2:- 1 = 0.1 = 0.10131424 8 22 17 25 18 6 22 20 628
9 6 5 5 1 7 2 4 1 3 8 9 3 1
1. Drop 9 from the denominator and increase the
penultimate digit (i.e. 2) of denominator by 1 so that the
vulgar fraction now reads 0.1/3.
2. Dividing 0.1 by 3 we get q=0 & r=1. Thus the
new dividing is 10 which is again divided by 3.
3. Continue the process up to we get the dividend
as 28 because numerator – denominator = 29-1=28
and now we know that after we get 28, the further digit are
obtained by merely subtracting the 14 digits each by 9.
Case 3:-
1 = 0.1 = 0.1011213253854945055055167187696747
89 9
25720880887 9 8 8 7 6 4 0 4 4 9 4 3 8 2 0 2 2
47191
Note:- if you notice that in the above examples the last
answer is found to be 1. Product of the last digit of the
denominator and the last digit of the decimal equivalent of
the fraction in question must invariably and in 9.
Therefore, as the last digit of the denominator in this case
is 9, it automatically follows that the last digit of decimal
equivalent is found to be 1 ( so that the product of the
multiplicand and the multiplies concerned may end in 9)
Let us consider some more auxiliary fractions
63 = 6.3 = 6.3 = 0.74 45 33 52 103
139 13.9 14
Friends, if you have carefully studied the earlier case
studies, then the above example is self explanatory. Do
practice the following examples.
65 = 6.5 = 6.5 0.94 106 87 36 82
139 13.9 14
83 = 8.3 = 8.3 0.85 105 07 70 104 146
149 14.9 15
All the above cases have denominator ending in 9 this does
not mean that the above rule is applicable only for above
cases. For your information & pleasure the name rule
applies for digits in denominator ending with 8,7,6 etc, but
with a plight change.
Denominator ending with 8:-
Please observe carefully the following example & you will
understand the rule on your own.
+4 +2 +5 +6
63 = 6.3 = 6.3 = 0.34 82 95 106 quotient
148 1 4.8 15
remainders
In case of denominator digits ending with 8 (1less than9),
the steps are as follows:
1. Placing of the remainder in front of the quotient
remains the same as explained in the earlier cases.
2. In the quotient digit, 1 time (9-8=1) of the
quotient digit is added to every step and divided by the
divisor.as in the above case. We found our first quotient
q1=4 and remainder r1=3. Our gross dividend comes out to
be 34 in which we add the quotient digit as per rule 2, to
make it 38 and then divide it by 15.
In the next step q2=2 & r2= 8, thus our 2nd gross dividend
will be 82 + q2 = 84, divide this by 15. Continue the above
steps to find the solution to the required number of decimal
places you desire.
Some more examples will make you more comfortable
with rule.
+5 +1 + 6 + 9
61 = 6.1 = 6.1 = 0. 15 81 106 49
118 11.8 12
89 = 8.9 = 8.9 = +4 +4 +9 +4
198 19.8 20 0.94 184 89 184
Denominator ending with 7 :-
Let me clear the rule by a simple example
+8 +10 +2 +9
53 = 5.3 = 5.3 = 0. 54 25 112 69 quotient
117 11.7 12 remainder
By more observation you can guess that for the above case
the quotient digit is multiplied by 2 (9-7=2) and added to
the quotient
Q1 = 4
R1 = 5
New divided = 54 + 201 = 54 + 8 = 62
And now divided 62 by 12 & proceed as earlier method.
Denominator ending with 6 :-
+15 +27 +15 + 6
75 = 7.5 = 7.5 = 0. 105 39 15 42
126 12.6 13
For the above case, the quotient digit is multiplied by 3 (:9-
6=3)
Q1 = 5, r1 = 10
:new dividend = 105 + 3q = 105 + 15 = 120
Rest all the steps are same as before.
Denominators ending in 1:-
Let us take an example
3 4 0 4 2
93 = 93-1 = 92 = 9.2 0. 86 135 89 105 67
141 141-1 140 14
We find reduce 1 from both numerator & denominator. The
gross dividend will be prefixing the remainder to the
compliment from 9 of each quotient digit.
Thus in the above example we get 8 as remainder & 6 as f
quotient. Compliment of 6 from 9 is 3 and then the gross
dividend is 83 and then to be divided by 14.
Now we get r2 = 13 & q2 = 5 . Compliment of q2 from 9 is
4 thus giving second gross dividend as 134
2 4 3 2 4 3
84 = 84-1 = 83 = 8.3 = 0. 67 75 86 67 75 86
111 111-1 110 11
Friends, we have discussed in detail the auxiliary & vulgar
fraction & i belive that you might be thrilled by doing the
above examples, to add to your pleasure & information, the
next chapter de with division technique of large numbers
and it is my word that if you do there both chapter
thoroughly then you people will never use calculators for
division in yours day to day practice.
Division “gowning gem of vedic mathematics”
The magical method explained in this chapter in one of the
crowning beauties of vedic mathematics. It is a ap
beautiful & very easy method to understood and thus will
make you put your fingers in mouth seeing the “vedic one
line mental answer”
The conventional format for division you people are using
is as mentioned below.
division ) dividend ( quotient
remainder
The vedic format for division is as follows.
Flag
Divisor
Dividend
Remainder
Quotient
Let me take a example to explain the above vedic format
which will reduce considerably the lab of operation as
well as time of the conventional format.
Eg. Divide 2,35,796 by 46
As per vedic division format we get
6
4
23
5
5
3
11
7
1
27
9
3
:
:
0
6
3-remainder
5 1 2 6 . 0- quotient
Please study the following steps carefully.
1. The divisor 46 is split into two parts 4 & 6.
We put down only the first digit i.e.4 in the divisor column
and put the other digit i.e. 6 “on top of the flag” by the
“dhvajanka sutra” as shown above.
2. Thus doing the above step, we are left by a
single digit divisor i.e. 4 and the entire division is to be
done by 4!
3. Number of digit on the remainders side is
always equal to the number of digits in the flag. This
decides the location of decimal point! As one digit has
been put on the top, we allot one place at the right end of
the dividend to the remainder portion of the answer and
mark it off from the digits by a vertical dot.
4. Since 2 cannot be divided by 4, we take 23
of the dividend to be divided by 4. Thus we get the
quotient q1=5 & remainder r1=3. We put 5 down as first
quotient digit and just prefix the remainder 3 up before the
5 of dividend.
5. Thus we get the new dividend as 35 from
this, we however , deduct the product of the indexed 6 (
flag digit) and the first quotient digit (q1) 5 i.e. 35- (6x5) =
5. The remainder 5 as circled in our actual net dividend.
6. Now 5 is divided by 4 which gives us q2 = 1
& r2=1, to be placed in their respective places as done in
above steps. Now we get 17 as gross dividend through
which we subtract product of flag digit & record quotient
digit i.e.1. I.e. 17 – (6 x 1 ) = 11 thus like 2nd net dividend
in 11 to be divided by 4
7. On dividing 11 by 4 we get
q3 = 2 & r3 = 3 dividend = 39
: 3rd net dividend = 39 – ( 6x 2) =27
8. Divide 27 by 4, we get
q4=6, r4= 3 & dividend = 36
: 4th net dividend =36 – (6 x 6) =0
9. Divide 0 by 4, we get
q5=0, r5 = 0.
as remainder is 0 we say that 2,25,796 is exactly divisible
by 46 & the answer is 5126.
Divide
25,116 by 42
41 33 0
2 25 1 1 : 6
4 5 5 : 1
5 9 8 . 0
1) Divisor 42 split into two parts. First digit
4 is taken as divisor & second digit 2 in kept as flag.
2) Since 2 cannot be divided by 4 so 25
taken as first dividend.
3) Divide 25 by 4. We get 01=6 & r1 = 1.
Now dividend is 11.
\ net 1st dividend = 11-(6x2) = -1.
Rule:- if the net divided obtained is a negative numbers
then reduce 1 from the previous quotient digit & work
again.
4) Thus the first quotient 6 is reduced by 1 for the
above example to give
Q1 = 5 & r1 = 5 \ new dividend = 51
Thus net 1st dividend = 51 – (5x2) = 41.
5) divide 41 by 4
Rule:- the quotient should always be a single digit even it it
comes a two digit number. Thus the largest quotient
number will be 9.
Now 4 goes 10 times for 41, but as per the above said rule
take the quotient as 9.
\q2= 9, r2 = 5 – dividend = 51
Thus net 2nd dividend = 51- (9x2) = 33.
6) divide 33 by 4. Q3 = 8 & r3 = 1 dividend = 16
\3rd net dividend = 16 –(8x2) = 0.
\25,116 = 598.0
Note: the best part of vedic divisor, is that it gives it gives
one line mental answer in which actual division is done by
only 1 digit divisor.
Divide:- 7,92,456 by 93
48 19 9 3 30 21 24 54
3 79 2 4 5 : 6 0 0 0 0
9 7 3 1 0 3 3 3 6
8 5 2 1 . 0 3 2 2
1) 93 split into two parts. The first digit 9 is divisor &
second digit 2 in flag digit.
2)since 7 cannot be divided by 9 so take first dividend as
79. On dividing 79 by 9 we get first quotient * remainder.
Q1 = 8 & r1 = 7 giving new dividend as 72
\net 2nd dividend = 72-8 x 3 = 48
3) on dividing 48 by 9 we get
Q2 = 5 & r2 = 3 \dividend = 35
\net 3rd dividend = 34-5x3 = 19
4) on dividing 19 by 9, we get
Q3 = 2 & r3 = 1 giving dividend = 15
\net 4th dividend = 15-2x3=9
5) on dividing 9 by 9 we get
Q4 = 1 & r4 = 0 - dividend = 6
\net 5th dividend = 6-1x3 = 3
6) on dividing 3 by 9 we get
Q5 = 0 & r5 = 3 - dividend = 30
\net 6th dividend = 30-3x0 = 30
7) on dividing 30 by 9 ,we get
Q6 = 3 & r6 = 3 \dividend = 30
\net 7th dividend = 30-3x3 =21
\ 7,92,456 = 8521.032258
93
Divide :- 1,50,381by 651
20 6 0 0
51 15 0 3 : 8 1
6 3 2 : 0 0
2 3 1 . 0 0
1) the divisor 651 is divided into two parts. The first digit 6
is taken as divisor & other two digit i.e.51 taken as flag
digits.
2) as two parts digit are taken as flag digits, we allot two
places at the right end of the dividend and mark it off by a
semi colon. The decimal is immediately marked in quotient
row below the semi-colon.
3) since 1 cannot be divided by 6, so we take 15 as our first
dividend on dividing 15 by 6 we get
Q1=2 & r1=3 giving next dividend 30.
Rule:- from 30, we subtract 10 the product of the first flag
digit (5) & first quotient digit (2) and get the net 2nd
dividend as 20.
4) on dividing 20 by 6, we get
Q2=3 & r2=2 giving next dividend as 23,
Rule:- from 23 we deduct 17 ( the cross – products of two
flag digits 51 and two quotient digits 2 & 3 i.e.)
5 1 = 5 x 3 + 1 x 2 = 17
2 3
Thus 3rd net dividend = 23-17 =6
5) on dividing the 6 by divisor 6, we get
Q3=1 & r3 = 0 giving dividend as 08
Rule:- same above step, from 08 we deduct 8 (crossproduct
of two flag digits 3 & 1 )
5 1 = 5 + 3 = 8
3 1
Thus 4th net dividend = 0
6) on dividing 0 by 6, we get
R4=0 & q4=0 giving next dividend as 01
Rule:- same as 4 th step, from 01 we deduct 01 ( the cross
– products of flag digits& quotient digits 1&0)
5 1 = 0x5 + 1 = 1
1 0
5th net dividend = 0
Thus we conclude 1,50,381 = 231.00
651
Now i take a very hard problem which you people will face
rarely & so i am sure that if you understand it properly
then you will become master in vedic division technique.
Divide:- 202272 by 258
27 16 3 0
58 2 0 2 2 : 7 2
2 6 11 : 8 3
7 8 4 . 0 0
(1) (9) (7) (1)
1) The divisor 258 is divided into two parts. The first
digit (2) is taken as divisor & other two digit i.e. 58 kept as
flag digits.
2) As two digits are taken as flag digits, we allot two
places at the right end of dividend and mark it off by a
semi – colon. The decimal is immediately marked below
the semi- colon in the quotient row.
3) Since 2 can be divided by 2, so we take our first
dividend as 2. On dividing 2 by, we get
Q1 = 1 & r = 0 giving next dividend as 0.
Now we deduct 5 x 1 = 5 from 0, & we get 2nd net
dividend as -5.
sanket
Digitally signed
by sanket
DN: cn=sanket,
o=Reliance,
c=IN
Date:
2005.05.21
12:17:15 +
05'30'
the end