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