Some square patterns are arranged. Study these. Perhaps
you will get another easiest way to find the square of a number.
Squaring a 2-digit number beginning with 1
Take a 2-digit number beginning with 1.
Square the second digit
(keep the carry) _ _ X
Multiply the second digit by 2 and
add the carry (keep the carry) _ X _
The first digit is one
(plus the carry) X _ _
Example:
If the number is 16, square the second digit:
6 × 6 = 36 _ _ 6
Multiply the second digit by 2 and
add the carry: 2 × 6 + 3 = 15 _ 5 _
The first digit is one plus the carry:
1 + 1 = 2 2 _ _
So 16 × 16 = 256.
See the pattern?
For 19 × 19, square the second digit:
9 × 9 = 81 _ _ 1
Multiply the second digit by 2 and
add the carry: 2 × 9 + 8 = 26 _ 6 _
The first digit is one plus the carry:
1 + 2 = 3 3 _ _
So 19 × 19 = 361.
Squaring a 2-digit number beginning with 5
Take a 2-digit number beginning with 5.
Square the first digit.
Add this number to the second number to find the first part of the answer.
Square the second digit: this is the last part of the answer.
Example:
If the number is 58, multiply 5 × 5 = 25 (square the first digit).
25 + 8 = 33 (25 plus second digit).
The first part of the answer is 33 3 3 _ _
8 × 8 = 64 (square second digit).
The last part of the answer is 64 _ _ 6 4
So 58 × 58 = 3364.
See the pattern?
For 53 × 53, multiply 5 × 5 = 25 (square the first digit).
25 + 3 = 28 (25 plus second digit).
The first part of the answer is 28 2 8 _ _
3 × 3 = 9 (square second digit).
The last part of the answer is 09 _ _ 0 9
So 53 × 53 = 2809.
Squaring a 2-digit number beginning with 9
Take a 2-digit number beginning with 9.
Subtract it from 100.
Subtract the difference from the original number:
this is the first part of the answer.
Square the difference: this is the last part of the answer.
Example:
If the number is 96, subtract: 100 - 96 = 4, 96 - 4 = 92.
The first part of the answer is 92 _ _ .
Take the first difference (4) and square it: 4 × 4 = 16.
The last part of the answer is _ _ 16.
So 96 × 96 = 9216.
See the pattern?
For 98 × 98, subtract: 100 - 98 = 2, 98 - 2 = 96.
The first part of the answer is 96 _ _.
Take the first difference (2) and square it: 2 × 2 = 4.
The last part of the answer is _ _ 04.
So 98 × 98 = 9604.
Squaring a 2-digit number ending in 1
Take a 2-digit number ending in 1.
Subtract 1 from the number.
Square the difference.
Add the difference twice to its square.
Add 1.
Example:
If the number is 41, subtract 1: 41 - 1 = 40.
40 × 40 = 1600 (square the difference).
1600 + 40 + 40 = 1680 (add the difference twice
to its square).
1680 + 1 = 1681 (add 1).
So 41 × 41 = 1681.
See the pattern?
For 71 × 71, subtract 1: 71 - 1 = 70.
70 × 70 = 4900 (square the difference).
4900 + 70 + 70 = 5040 (add the difference twice
to its square).
5040 + 1 = 5041 (add 1).
So 71 × 71 = 5041.
Squaring a 2-digit number ending in 2
Take a 2-digit number ending in 2.
The last digit will be _ _ _ 4.
Multiply the first digit by 4: the 2nd number will be
the next to the last digit: _ _ X 4.
Square the first digit and add the number carried from
the previous step: X X _ _.
Example:
If the number is 52, the last digit is _ _ _ 4.
4 × 5 = 20 (four times the first digit): _ _ 0 4.
5 × 5 = 25 (square the first digit), 25 + 2 = 27 (add carry): 2 7 0 4.
So 52 × 52 = 2704.
See the pattern?
For 82 × 82, the last digit is _ _ _ 4.
4 × 8 = 32 (four times the first digit): _ _ 2 4.
8 × 8 = 64 (square the first digit), 64 + 3 = 67 (add carry): 6 7 2 4.
So 82 × 82 = 6724.
Squaring a 2-digit number ending in 3
Take a 2-digit number ending in 3.
The last digit will be _ _ _ 9.
Multiply the first digit by 6: the 2nd number will be
the next to the last digit: _ _ X 9.
Square the first digit and add the number carried from
the previous step: X X _ _.
Example:
If the number is 43, the last digit is _ _ _ 9.
6 × 4 = 24 (six times the first digit): _ _ 4 9.
4 × 4 = 16 (square the first digit), 16 + 2 = 18
(add carry): 1 8 4 9.
So 43 × 43 = 1849.
See the pattern?
For 83 × 83, the last digit is _ _ _ 9.
6 × 8 = 48 (six times the first digit): _ _ 8 9.
8 × 8 = 64 (square the first digit), 64 + 4 = 68
(add carry): 6 8 8 9.
So 83 × 83 = 6889
Squaring a 2-digit number ending in 4
Take a 2-digit number ending in 4.
Square the 4; the last digit is 6: _ _ _ 6
(keep carry, 1.)
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).
Square the first digit and add the carry: X X _ _.
Example:
If the number is 34, 4 × 4 = 16 (keep carry, 1);
the last digit is _ _ _ 6.
8 × 3 = 24 (multiply the first digit by 
, 24 + 1 = 25
(add the carry):
the next digit is 5: _ _ 5 6. (Keep carry, 2.)
Square the first digit and add the carry, 2: 1 1 5 6.
So 34 × 34 = 1156.
See the pattern?
For 84 × 84, 4 × 4 = 16 (keep carry, 1);
the last digit is _ _ _ 6.
8 × 8 = 64 (multiply the first digit by ,
64 + 1 = 65 (add the carry):
the next digit is 5: _ _ 5 6. (Keep carry, 6.)
Square the first digit and add the carry, 6: 7 0 5 6.
So 84 × 84 = 7056.
Squaring a 2-digit number ending in 5
Choose a 2-digit number ending in 5.
Multiply the first digit by the next consecutive number.
The product is the first two digits: XX _ _.
The last part of the answer is always 25: _ _ 2 5.
Example:
If the number is 35, 3 × 4 = 12 (first digit
times next number). 1 2 _ _
The last part of the answer is always 25: _ _ 2 5.
So 35 × 35 = 1225.
See the pattern?
For 65 × 65, 6 × 7 = 42 (first digit
times next number): 4 2 _ _.
The last part of the answer is always 25: _ _ 2 5.
So 65 × 65 = 4225.
Squaring a 2-digit number ending in 6
Choose a 2-digit number ending in 6.
Square the second digit (keep the carry): the last digit
of the answer is always 6: _ _ _ 6
Multiply the first digit by 2 and add the carry
(keep the carry): _ _ X _
Multiply the first digit by the next consecutive number and
add the carry: the product is the first two digits:
XX _ _.
Example:
If the number is 46, square the second digit :
6 × 6 = 36; the last digit of the answer is 6
(keep carry 3): _ _ _ 6
Multiply the first digit (4) by 2 and add the carry
(keep the carry): 2 × 4 = 8, 8 + 3 = 11; the next digit
of the answer is 1: _ _ 1 6
Multiply the first digit (4) by the next number (5)
and add the carry: 4 × 5 = 20, 20 + 1 = 21
(the first two digits): 2 1 _ _
So 46 × 46 = 2116.
See the pattern?
For 76 × 76, square 6 and keep the carry (3):
6 × 6 = 36; the last digit of the answer is 6: _ _ _ 6
Multiply the first digit (7) by 2 and add the carry:
2 × 7 = 14, 14 + 3 = 17; the next digit of the answer
is 7 (keep carry 1): _ _ 7 6
Multiply the first digit (7) by the next number (8)
and add the carry: 7 × 8 = 56, 56 + 1 = 57
(the first two digits: 5 7 _ _
So 76 × 76 = 5776.
Squaring a 2-digit number ending in 7
Choose a 2-digit number ending in 7.
The last digit of the answer is always 9: _ _ _ 9
Multiply the first digit by 4 and add 4
(keep the carry): _ _ X _
Multiply the first digit by the next consecutive number and
add the carry: the product is the first two digits:
XX _ _.
Example:
If the number is 47:
The last digit of the answer is 9: _ _ _ 9
Multiply the first digit (4) by 4 and add 4
(keep the carry): 4 × 4 = 16, 16 + 4 = 20; the next
digit of the answer is 0 (keep carry 2): _ _ 0 9
Multiply the first digit (4) by the next number (5)
and add the carry (2):
4 × 5 = 20, 20 + 2 = 22 (the first two digits): 2 2 _ _
So 47 × 47 = 2209.
See the pattern?
For 67 × 67
The last digit of the answer is 9: _ _ _ 9
Multiply the first digit (6) by 4 and add 4 (keep the
carry): 4 × 6 = 24, 24 + 4 = 28; the next digit of the
answer is 0 (keep carry 2): _ _ 8 9
Multiply the first digit (6) by the next number (7)
and add the carry (2):
6 × 7 = 42, 42 + 2 = 44 (the first two digits): 4 4 _ _
So 67 × 67 = 4489.
Squaring a 2-digit number ending in 8
Choose a 2-digit number ending in 8.
The last digit of the answer is always 4: _ _ _ 4
Multiply the first digit by 6 and add 6 (keep the
carry): _ _ X _
Multiply the first digit by the next consecutive
number and add the carry: the product is the first
two digits: XX _ _.
Example:
If the number is 78:
The last digit of the answer is 4: _ _ _ 4
Multiply the first digit (7) by 6 and add 6 (keep the
carry): 7 × 6 = 42, 42 + 6 = 48; the next digit of the
answer is 8 (keep carry 4): _ _ 8 4
Multiply the first digit (7) by the next number (8)
and add the carry (4):
7 × 8 = 56, 56 + 4 = 60 (the first two digits): 6 0 _ _
So 78 × 78 = 6084.
See the pattern?
For 38 × 38
The last digit of the answer is 4: _ _ _ 4
Multiply the first digit (3) by 6 and add 6 (keep the
carry): 3 × 6 = 18, 18 + 6 = 24; the next digit of the
answer is 4 (keep carry 2): _ _ 4 4
Multiply the first digit (3) by the next number (4)
and add the carry (2):
3 × 4 = 12, 12 + 2 = 14 (the first two digits): 1 4 _ _
So 38 × 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
Choose a 2-digit number ending in 9.
The last digit of the answer is always 1: _ _ _ 1
Multiply the first digit by 8 and add 8 (keep the carry): _ _ X _
Multiply the first digit by the next consecutive number and
add the carry: the product is the first two
digits: XX _ _.
Example:
If the number is 39:
The last digit of the answer is 1: _ _ _ 1
Multiply the first digit (3) by 8 and add 8 (keep the
carry): 8 × 3 = 24, 24 + 8 = 32; the next digit of
the answer is 2 (keep carry 3): _ _ 2 1
Multiply the first digit (3) by the next number (4)
and add the carry (3): 3 × 4 = 12, 12 + 3 = 15
(the first two digits): 1 5 _ _
So 39 × 39 = 1521.
See the pattern?
For 79 × 79
The last digit of the answer is 1: _ _ _ 1
Multiply the first digit (7) by 8 and add 8 (keep the
carry): 8 × 7 = 56, 56 + 8 = 64; the next digit of
the answer is 4 (keep carry 6): _ _ 4 1
Multiply the first digit (7) by the next number (8)
and add the carry (6): 7 × 8 = 56, 56 + 6 = 62
(the first two digits): 6 2 _ _
So 79 × 79 = 6241.
Practice other examples using this pattern, and in no time you'll be able to give these squares faster than someone using a calculator.
Squaring numbers made up of ones
Choose a a number made up of ones (up to nine digits).
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:
If the number is 11111, (5 digits) -
The square of the number is 123454321.
(Begin with 1, up to 5, then back to 1.)
See the pattern?
If the number is 1111111, (7 digits) -
The square of the number is 1234567654321.
(Begin with 1, up to 7, then back to 1.).
This is an easy one, but it should be good for a quick example of your mental math abilities. Challenge a friend and BEAT THE CALCULATOR!
Squaring numbers made up of threes
Choose a a number made up of threes.
The square is made up of:
one fewer 1 than there are repeating 3's
zero
one fewer 8 than there are repeating 3's
(same as the 1's in the square)
nine.
Example:
If the number to be squared is 3333:
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
So 3333 × 3333 = 11108889.
See the pattern?
If the number to be squared is 333:
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
So 333 × 333 = 110889.
Squaring numbers made up of sixes
Choose a a number made up of sixes.
The square is made up of:
one fewer 4 than there are repeating 6's
3
same number of 5's as 4's
6
Example:
If the number to be squared is 666
The square of the number has:
4's (one less than digits
in number) 4 4
3 3
5's (same number as 4's) 5 5
6 6
So 666 × 3666333 = 443556.
See the pattern?
If the number to be squared is 66666
The square of the number has:
4's (one less than digits
in number) 4 4 4 4
3 3
5's (same number as 4's) 5 5 5 5
6 6
So 66666 × 66666 = 4444355556
Squaring numbers made up of nines
Choose a a number made up of nines (up to nine digits).
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:
If the number to be squared is 9999
The square of the number has:
one less nine than the number 9 9 9
one 8 8
the same number of zeros as 9's 0 0 0
a final 1 1
So 9999 × 9999 = 99980001.
See the pattern?
If the number to be squared is 999999
The square of the number has:
one less nine than the number 9 9 9 9 9
one 8 8
the same number of zeros as 9's 0 0 0 0 0
a final 1 1
So 999999 × 999999 = 999998000001.
This is not a very demanding mental math exercise, but it is an interesting pattern.
Squaring numbers in the 20s
Square the last digit (keep the carry) _ _ X
Multiply the last digit by 4, add the carry _ X _
The first digit will be 4 plus the carry: X _ _
Example:
If the number to be squared is 24:
Square the last digit (keep the carry):
4 × 4 = 16 (keep 1) _ _ 6
Multiply the last digit by 4, add the carry:
4 × 4 = 16, 16 + 1 = 17 _ 7 _
The first digit will be 4 plus the carry:
4 (+ carry): 4 + 1 = 5 5 _ _
So 24 × 24 = 576.
See the pattern?
If the number to be squared is 26:
Square the last digit (keep the carry):
6 × 6 = 36 (keep 3) _ _ 6
Multiply the last digit by 4, add the carry:
4 × 6 = 24, 24 + 3 = 27 (keep 2) _ 7 _
The first digit will be 4 plus the carry:
4 (+ carry): 4 + 2 = 6 6 _ _.
So 26 × 26 = 676.
Squaring numbers in the 30s
Square the last digit (keep the carry) _ _ _ X
Multiply the last digit by 6, add the carry _ _ X _
The first digits will be 9 plus the carry: X X _ _
Example:
If the number to be squared is 34:
Square the last digit (keep the carry):
4 × 4 = 16 (keep 1) _ _ _ 6
Multiply the last digit by 6, add the carry:
6 × 4 = 24, 24 + 1 = 25 _ _ 5 _
The first digits will be 4 plus the carry:
9 (+ carry): 9 + 2 = 11 1 1 _ _
So 34 × 34 = 1156.
See the pattern?
If the number to be squared is 36:
Square the last digit (keep the carry):
6 × 6 = 36 (keep 3) _ _ _ 6
Multiply the last digit by 6, add the carry:
6 × 6 = 36, 36 + 3 = 39 (keep 3) _ _ 9 _
The first digits will be 9 plus the carry:
9 (+ carry): 9 + 3 = 12 1 2 _ _.
So 36 × 36 = 1296.
With some practice you will be giving these squares quickly.
Squaring numbers in the 40s
Square the last digit (keep the carry) _ _ X
Multiply the last digit by 8, add the carry _ X _
The first digits will be 16 plus the carry: X X _ _
Example:
If the number to be squared is 42:
Square the last digit:
2 × 2 = 4 _ _ _ 4
Multiply the last digit by 8:
8 × 2 = 16 _ _ 6 _
The first digits will be 16 plus the carry:
16 (+ carry): 16 + 1 = 17 1 7 _ _
So 42 × 42 = 1764.
See the pattern?
If the number to be squared is 48:
Square the last digit (keep the carry):
8 × 8 = 64 (keep 6) _ _ _ 4
Multiply the last digit by 8, add the carry:
8 × 8 = 64, 64 + 6 = 70 (keep 7) _ _ 0 _
The first digits will be 16 plus the carry:
16 (+ carry): 16 + 7 = 23 2 3 _ _
So 48 × 48 = 2304.
With some practice you will be giving these squares quickly.
Squaring numbers in the 50s
Square the last digit (keep the carry) _ _ _ X
Multiply the last digit by 10, add the carry _ _ X _
The first digits will be 25 plus the carry: X X _ _
Example:
If the number to be squared is 53:
Square the last digit (keep the carry):
3 × 3 = 9 (keep 3) _ _ _ 9
Multiply the last digit by 10, add the carry:
10 × 3 = 30 (keep 3) _ _ 0 _
The first digits will be 25 plus the carry:
25 (+ carry): 25 + 3 = 28 2 8 _ _
So 53 × 53 = 2809.
See the pattern?
If the number to be squared is 56:
Square the last digit (keep the carry):
6 × 6 = 36 (keep 3) _ _ _ 6
Multiply the last digit by 10, add the carry:
10 × 6 = 60, 60 + 3 = 63 _ _ 3 _
The first digits will be 25 plus the carry:
25 (+ carry): 25 + 6 = 31 3 1 _ _
So 53 × 53 = 3136.
Practice and you will soon be producing these products quickly and accurately.
Squaring numbers in the 60s
Square the last digit (keep the carry) _ _ _ X
Multiply the last digit by 12, add the carry _ _ X _
The first digits will be 36 plus the carry: X X _ _
Example:
If the number to be squared is 63:
Square the last digit (keep the carry):
3 × 3 = 9 (keep 3) _ _ _ 9
Multiply the last digit by 12, add the carry:
12 × 3 = 36 (keep 3) _ _ 6 _
The first digits will be 36 plus the carry:
36 (+ carry): 36 + 3 = 39 3 9 _ _
So 63 × 63 = 3969.
See the pattern?
If the number to be squared is 67:
Square the last digit (keep the carry):
7 × 7 = 49 (keep 4) _ _ _ 9
Multiply the last digit by 12, add the carry:
12 × 7 = 84, 84 + 4 = 88 _ _ 8 _
The first digits will be 36 plus the carry:
36 (+ carry): 36 + 8 = 44 4 4 _ _
So 67 × 67 = 4489.
Use this pattern and you will be squaring these numbers with ease.
Squaring numbers in the 70s
Square the last digit (keep the carry) _ _ _ X
Multiply the last digit by 14, add the carry _ _ X _
The first digits will be 49 plus the carry: X X _ _
Example:
If the number to be squared is 72:
Square the last digit:
2 × 2 = 4 _ _ _ 4
Multiply the last digit by 14:
14 × 2 = 28 (keep the carry) _ _ 8 _
The first digits will be 49 plus the carry:
49 (+ carry): 49 + 2 = 51 5 1 _ _
So 72 × 72 = 5184.
See the pattern?
If the number to be squared is 78:
Square the last digit (keep the carry):
8 × 8 = 64 (keep 6) _ _ _ 4
Multiply the last digit by 14, add the carry:
14 × 8 = 80 + 32 = 112
112 + 6 = 118 (keep 11) _ _ 8 _
The first digits will be 49 plus the carry (11):
49 (+ carry): 49 + 11 = 60 6 0 _ _.
So 78 × 78 = 6084.
Squaring numbers in the 80s
Square the last digit (keep the carry) _ _ X
Multiply the last digit by 16, add the carry _ X _
The first digits will be 64 plus the carry: X X _ _
Example:
If the number to be squared is 83:
Square the last digit:
3 × 3 = 9 _ _ _ 9
Multiply the last digit by 16:
16 × 3 = 30 + 18 = 48 _ _ 8 _
The first digits will be 64 plus the carry:
64 (+ carry): 64 + 4 = 68 6 8 _ _
So 83 × 83 = 6889.
See the pattern?
If the number to be squared is 86:
Square the last digit (keep the carry):
6 × 6 = 36 (keep 3) _ _ _ 6
Multiply the last digit by 16, add the carry:
16 × 6 = 60 + 36 = 96 96 + 3 = 99 (keep 9) _ _ 9 _
The first digits will be 64 plus the carry:
64 (+ carry): 64 + 9 = 73 7 3 _ _
So 86 × 86 = 7396.
Squaring numbers in the hundreds
Choose a number over 100 (keep it low for practice,
then go higher when expert).
The last two places will be the square of
the last two digits (keep any carry) _ _ _ X X.
The first three places will be the number plus
the last two digits plus any carry: X X X _ _.
Example:
If the number to be squared is 106:
Square the last two digits (no carry): 6 × 6 = 36: _ _ _ 3 6
Add the last two digits (06) to the number: 106 + 6 = 112: 1 1 2 _ _
So 106 × 106 = 11236.
See the pattern?
If the number to be squared is 112:
Square the last two digits (keep carry 1): 12 × 12 = 144: _ _ _ 4 4
Add the last two digits (12) plus the carry (1) to the number:
112 + 12 + 1 = 125: 1 2 5 _ _
So 112 × 112 = 12544.
With a little practice your only limit will be your ability to square the last two digits!
Squaring numbers in the 200s
Choose a number in the 200s (practice with numbers under 210, then progress to larger ones).
The first digit of the square is 4: 4 _ _ _ _
The next two digits will be 4 times the last 2 digits: _ X X _ _
The last two places will be the square of the last digit: _ _ _ X X
Example:
If the number to be squared is 206:
The first digit is 4: 4 _ _ _ _
The next two digits are 4 times the last digit:
4 × 6 = 24: _ 2 4 _ _
Square the last digit: 6 × 6 = 36: _ _ _ 3 6
So 206 × 206 = 42436.
For larger numbers work right to left:
Square the last two digits (keep the carry): _ _ _ X X
4 times the last two digits + carry: _ X X _ _
Square the first digit + carry: X _ _ _ _
See the pattern?
If the number to be squared is 225:
Square last two digits (keep carry):
25x25 = 625 (keep 6): _ _ _ 2 5
4 times the last two digits + carry:
4x25 = 100; 100+6 = 106 (keep 1): _ 0 6 _ _
Square the first digit + carry:
2x2 = 4; 4+1 = 5: 5 _ _ _ _
So 225 × 225 = 50625.
Squaring numbers in the 300s
Choose a number in the 300s (practice with numbers under 310, then progress to larger ones).
The first digit of the square is 9: 9 _ _ _ _
The next two digits will be 6 times the last 2 digits: _ X X _ _
The last two places will be the square of the last digit: _ _ _ X X
Example:
If the number to be squared is 309:
The first digit is 9: 9 _ _ _ _
The next two digits are 6 times the last digit:
6 × 9 = 54: _ 5 4 _ _
Square the last digit: 9 × 9 = 81: _ _ _ 8 1
So 309 × 309 = 95481.
For larger numbers reverse the steps:
Square the last two digits (keep the carry): _ _ _ X X
6 times the last two digits + carry: _ X X _ _
Square the first digit + carry: X _ _ _ _
See the pattern?
If the number to be squared is 325:
Square last two digits (keep carry):
25x25 = 625 (keep 6): _ _ _ 2 5
6 times the last two digits + carry:
6x25 = 150; 150+6 = 156 (keep 1): _ 5 6 _ _
Square the first digit + carry:
3x3 = 9; 9+1 = 10: 1 0 _ _ _ _
So 325 × 325 = 105625.
Squaring numbers in the 400s
Choose a number in the 400s (keep the numbers low at first; then progress to larger ones).
The first two digits of the square are 16: 1 6 _ _ _ _
The next two digits will be 8 times the last 2 digits: _ _ X X _ _
The last two places will be the square of the last two digits: _ _ _ _ X X
Example:
If the number to be squared is 407:
The first two digits are 16: 1 6 _ _ _ _
The next two digits are 8 times the last 2 digits:
8 × 7 = 56: _ _ 5 6 _ _
Square the last digit: 7 × 7 = 49: _ _ _ 4 9
So 407 × 407 = 165,649.
For larger numbers reverse the steps:
Square the last two digits (keep the carry): _ _ _ _ X X
8 times the last two digits + carry: _ _ X X _ _
16 + carry: X X _ _ _ _
See the pattern?
If the number to be squared is 425:
Square the last two digits (keep the carry):
25 × 25 = 625 (keep 6): _ _ _ _ 2 5
8 times the last two digits + carry:
8 × 25 = 200; 200 + 6 = 206 (keep 2): _ _ 0 6 _ _
16 + carry: 16 + 2 = 18: 1 8 _ _ _ _
So 425 × 425 = 180,625.
Squaring numbers in the 500s
Choose a number in the 500s (start with low numbers at first; then graduate to larger ones).
The first two digits of the square are 25: 2 5 _ _ _ _
The next two digits will be 10 times the last 2 digits: _ _ X X _ _
The last two places will be the square of the last two digits: _ _ _ _ X X
Example:
If the number to be squared is 508:
The first two digits are 25: 2 5 _ _ _ _
The next two digits are 10 times the last 2 digits:
10 × 8 = 80: _ _ 8 0 _ _
Square the last digit: 8 × 8 = 64: _ _ _ 6 4
So 508 × 508 = 258,064.
For larger numbers reverse the steps:
Square the last two digits (keep the carry): _ _ _ _ X X
10 times the last two digits + carry: _ _ X X _ _
25 + carry: X X _ _ _ _
See the pattern?
If the number to be squared is 525:
Square the last two digits (keep the carry):
25 × 25 = 625 (keep 6): _ _ _ _ 2 5
10 times the last two digits + carry:
10 × 25 = 250; 250 + 6 = 256 (keep 2): _ _ 5 6 _ _
25 + carry: 25 + 2 - 27: 2 7 _ _ _ _
So 425 × 425 = 275,625.
Squaring numbers in the 600s
Choose a number in the 600s (practice with smaller numbers, then progress to larger ones).
The first two digits of the square are 36: 3 6 _ _ _ _
The next two digits will be 12 times the last 2 digits: _ _ X X _ _
The last two places will be the square of the last two digits: _ _ _ _ X X
Example:
If the number to be squared is 607:
The first two digits are 36: 3 6 _ _ _ _
The next two digits are 12 times the last 2 digits:
12 × 07 = 84: _ _ 8 4 _ _
Square the last 2 digits: 7 × 7 = 49: _ _ _ _ 4 9
So 607 × 607 = 368,449.
For larger numbers reverse the steps:
If the number to be squared is 625:
Square the last two digits (keep carry):
25x25 = 625 (keep 6): _ _ _ _ 2 5
12 times the last 2 digits + carry:
12x25 = 250 + 50 = 300 + 6 = 306: _ _ 0 6 _ _
36 + carry: 36 + 3 = 39: 3 9 _ _ _ _
So 625 × 625 = 390,625.
Squaring numbers in the 700s
Choose a number in the 700s (practice with smaller numbers, then progress to larger ones).
Square the last two digits (keep the carry): _ _ _ _ X X
Multiply the last two digits by 14 and
add the carry: _ _ X X _ _
The first two digits will be 49 plus the carry: X X _ _ _ _
Example:
If the number to be squared is 704:
Square the last two digits (keep the carry):
4 × 4 = 16: _ _ _ _ 1 6
Multiply the last two digits by 14 and
add the carry: 14 × 4 = 56: _ _ 5 6 _ _
The first two digits will be 49 plus the carry: 4 9 _ _ _ _
So 704 × 704 = 495,616.
See the pattern?
If the number to be squared is 725:
Square the last two digits (keep the carry):
25 × 25 = 625: _ _ _ _ 2 5
Multiply the last two digits by 14 and
add the carry: 14 × 25 = 10 × 25 + 4 × 25
= 250 + 100 = 350. 350 + 6 = 356: 56: _ _ 5 6 _ _
The first two digits will be 49 plus the carry: 49 + 3 = 52: 5 2 _ _ _ _
So 725 × 725 = 525,625.
Squaring numbers between 800 and 810
Choose a number between 800 and 810.
Square the last two digits:
_ _ _ _ X X
Multiply the last two digits by 16
(keep the carry): _ _ X X _ _
Square 8, add the carry: X X _ _ _ _
Example:
If the number to be squared is 802:
Square the last two digits:
2 × 2 = 4: _ _ _ _ 0 4
Multiply the last two digits by 16:
16 × 2 = 32: _ _ 3 2 _ _
Square 8: 6 4 _ _ _ _
So 802 × 802 = 643,204.
See the pattern?
If the number to be squared is 807:
Square the last two digits:
7 × 7 = 49: _ _ _ _ 4 9
Multiply the last two digits by 16
(keep the carry): 16 × 7 = 112: _ _ 1 2 _ _
Square 8, add the carry (1): 6 5 _ _ _ _
So 807 × 807 = 651, 249.
Squaring numbers in the 900s
Choose a number in the 900s - start out easy with numbers near 1000; then go lower when expert.
Subtract the number from 1000 to get the difference.
The first three places will be the number minus the difference: X X X _ _ _.
The last three places will be the square of the difference: _ _ _ X X X
(if 4 digits, add the first digit as carry).
Example:
If the number to be squared is 985:
Subtract 1000 - 985 = 15 (difference)
Number - difference: 985 - 15 = 970: 9 7 0 _ _ _
Square the difference: 15 × 15 = 225: _ _ _ 2 2 5
So 985 × 985 = 970225.
See the pattern?
If the number to be squared is 920:
Subtract 1000 - 920 = 80 (difference)
Number - difference: 920 - 80 = 840: 8 4 0 _ _ _
Square the difference: 80 × 80 = 6400: _ _ _ 4 0 0
Carry first digit when four digits: 8 4 6 _ _ _
So 920 × 920 = 846400.
Squaring numbers between 1000 and 1100
Choose a number between 1000 and 1100.
The first two digits are: 1,0 _ _, _ _ _
Find the difference between your number and 1000.
Multiply the difference by 2: 1,0 X X, _ _ _
Square the difference: 1,0 _ _, X X X
Example:
If the number to be squared is 1007:
The first two digits are: 1,0 _ _, _ _ _
Find the difference: 1007 - 1000 = 7
Two times the difference: 2 × 7 = 14:
1,0 1 4, _ _ _
Square the difference: 7 × 7 = 49:
1,0 1 4, 0 4 9
So 1007 × 1007 = 1,014,049.
See the pattern?
If the number to be squared is 1012:
The first two digits are: 1,0 _ _, _ _ _
Find the difference: 1012 - 1000 = 12
Two times the difference: 2 × 12 = 24:
1,0 2 4, _ _ _
Square the difference: 12 × 12 = 144:
1,0 2 4, 1 4 4
So 1012 × 1012 = 1,024,144.
Start with lower numbers and then extend your expertise to all the numbers between 1000 and 1100. Remember to add the first digit as carry when the square of the difference is four digits.
Squaring numbers between 2000 and 2099
Choose a number between 2000 and 2099. (Start with numbers below 2025 to begin with, then graduate to larger numbers.)
The first two digits are: 4 0 _ _ _ _ _
The next two digits are 4 times the last two digits:
4 0 X X _ _ _
For the last three digits, square the last two digits in the number chosen (insert zeros when needed):
4 0 _ _ X X X
Example:
If the number to be squared is 2003:
The first two digits are: 4 0 _ _ _ _ _
The next two digits are 4 times the last two:
4 × 3 = 12: _ _ 1 2 _ _ _
For the last three digits, square the last two:
3 × 3 = 9: _ _ _ _ 0 0 9
So 2003 × 2003 = 4,012,009.
See the pattern?
For larger numbers, reverse the order:
If the number to be squared is 2025:
For the last three digits, square the last two:
25 × 25 = 625: _ _ _ _ 6 2 5
The middle two digits are 4 times the last two (keep the carry):
4 × 25 = 100 (keep carry of 1): _ _ 0 0 _ _ _
The first two digits are 40 + the carry:
40 + 1 = 41: 4 1 _ _ _ _ _
So 2025 × 2025 = 4,100,625.
Squaring numbers between 3000 and 3099
Choose a number between 3000 and 3099. (Start with numbers below 3025 to begin with, then graduate to larger numbers.)
The first two digits are: 9 0 _ _ _ _ _
The next two digits are 6 times the last two digits:
9 0 X X _ _ _
For the last three digits, square the last two digits in the number chosen (insert zeros when needed):
9 0 _ _ X X X
Example:
If the number to be squared is 3004:
The first two digits are: 9 0 _ _ _ _ _
The next two digits are 6 times the last two:
6 × 4 = 24: _ _ 2 4 _ _ _
For the last three digits, square the last two:
4 × 4 = 16: _ _ _ _ 0 1 6
So 3004 × 3004 = 9,024,016.
See the pattern?
For larger numbers, reverse the order:
If the number to be squared is 3025:
For the last three digits, square the last two:
25 × 25 = 625: _ _ _ _ 6 2 5
The middle two digits are 6 times the last two (keep the carry):
6 × 25 = 150 (keep carry of 1): _ _ 5 0 _ _ _
The first two digits are 90 + the carry:
90 + 1 = 91: 9 1 _ _ _ _ _
So 3025 × 3025 = 9,150,625.
Squaring numbers between 4000 and 4099
Choose a number between 4000 and 4099.
For numbers less than 4013:
The first three digits are: 1 6 0 _ _ _ _ _
The next two digits are 8 times the last two digits:
_ _ _ X X _ _ _
For the last three digits, square the last two digits in the number chosen (insert zeros when needed):
_ _ _ _ _ X X X
Example:
If the number to be squared is 4005:
The first three digits are: 1 6 0 _ _ _ _ _
The next two digits are 8 times the last two:
8 × 5 = 40: _ _ 4 0 _ _ _
For the last three digits, square the last two:
5 × 5 = 25: _ _ _ _ _ 0 2 5
So 4005 × 4005 = 16,040,025.
See the pattern?
For numbers greater than 4012, reverse the order:
If the number to be squared is 4080:
For the last three digits, square the last two:
80 × 80 = 6400, carry 6: _ _ _ _ 4 0 0
The middle two digits are 8 times the last two (keep the carry):
8 × 80 = 640 (keep carry of 6), 40 + 6:
_ _ _ 4 6 _ _ _
The first three digits are 160 + the carry:
160 + 6 = 166: 1 6 6 _ _ _ _ _
So 4080 × 4080 = 16,646,400.
Squaring numbers between 5000 and 5099
Choose a number between 5000 and 5099.
The first three digits are: 2 5 0 _ _ _ _ _
The next two digits are 10 times the last two digits:
2 5 0 X X _ _ _
For the last three digits, square the last two digits in the number chosen (insert zeros when needed):
2 5 0 _ _ X X X
Example:
If the number to be squared is 5004:
The first three digits are: 2 5 0 _ _ _ _ _
The next two digits are 10 times the last two:
10 × 4 = 40: _ _ 4 0 _ _ _
For the last three digits, square the last two:
4 × 4 = 16: _ _ _ _ _ 0 1 6
So 5004 × 5004 = 25,040,016.
See the pattern?
For numbers greater than 5011, reverse the order:
If the number to be squared is 5012:
For the last three digits, square the last two:
12 × 12 = 144: _ _ _ _ 1 4 4
The middle two digits are 10 times the last two (keep the carry):
10 × 12 = 120 (keep carry of 1):
_ _ _ 2 0 _ _ _
The first three digits are 150 + the carry:
250 + 1 = 251: 2 5 1 _ _ _ _ _
So 5012 × 5012 = 25,120,144.
Squaring numbers between 6000 and 6099
Choose a number between 6000 and 6099.
The first three digits are: 3 6 0 _ _ _ _ _
The next two digits are 12 times the last two digits:
_ _ _ X X _ _
For the last three digits, square the last two digits in the number chosen (insert zeros when needed):
_ _ _ _ _ X X X
Example:
If the number to be squared is 6004:
The first three digits are: 3 6 0 _ _ _ _ _
The next two digits are 12 times the last two:
12 × 4 = 48: _ _ _ 4 8 _ _ _
For the last three digits, square the last two:
4 × 4 = 16: _ _ _ _ 0 1 6
So 6004 × 6004 = 36,048,016.
See the pattern?
For numbers greater than 6008, reverse the order:
If the number to be squared is 6020:
For the last three digits, square the last two:
20 × 20 = 400: _ _ _ _ 4 0 0
The middle two digits are 12 times the last two:
12 × 20 = 240 (keep carry): _ _ _ 4 0 _ _ _
The first digits are 360 + the carry:
360 + 2 = 362: 3 6 2 _ _ _ _ _
So 3025 × 3025 = 36,240,400.
Squaring numbers between 7000 and 7099
Choose a number between 7000 and 7099.
The first three digits are: 4 9 0 _ _ _ _ _
The next two digits are 4 times the last two digits,
with zero added: _ _ _ X X _ _
For the last three digits, square the last two digits in the number chosen (insert zeros when needed):
_ _ _ _ _ X X X
Example:
If the number to be squared is 7004:
The first three digits are: 4 9 0 _ _ _ _ _
The next two digits are 4 times the last two, with zero added:
4 × 4 = 16; 16 + 40 = 56: _ _ _ 5 6 _ _ _
For the last three digits, square the last two:
4 × 4 = 16: _ _ _ _ 0 1 6
So 7004 × 7004 = 49,056,016.
See the pattern?
For numbers greater than 7007, reverse the order:
If the number to be squared is 7025:
For the last three digits, square the last two:
25 × 25 = 625: _ _ _ _ 6 2 5
For the middle two digits, add zero to the last two,
then add 4 times the last two:
250 + 4 × 25: 250 + 100 = 350 (keep carry):
_ _ _ 5 0 _ _ _
The first three digits are 490 + the carry:
490 + 3 = 493: 4 9 3 _ _ _ _ _
So 7025 × 7025 = 49,350,625.
Squaring special numbers (3's and final 1)
Choose a number with repeating 3's and a final 1.
The square is made up of:
· One fewer 1 than there are repeating 3's
· 09
· The same number of 5's as there are 1's in the square;
· A final 61
Example:
If the number to be squared is 3331:
The square has:
Two 1's (one fewer than
repeating 3's) 1 1
Next digits: 09 0 9
Two 5's (same as 1's in square) 5 5
A final 61 6 1
So the square of 3331 is 11,095,561.
See the pattern?
If the number to be squared is 333331:
The square has:
Four 1's (one fewer than
repeating 3's) 1 1 1 1
Next digits: 09 0 9
Four 5's (same as 1's in square) 5 5 5 5
A final 61 6 1
So the square of 333331 is 111,109,555,561.
Squaring special numbers (3's and final 2)
Choose a number with repeating 3's and a final 2.
The square is made up of:
· the same number of 1's as there are repeating 3's;
· a zero;
· the same number of 2's as there are 1's in the square;
· a final 4.
Example:
If the number to be squared is 3332:
The square has:
three 1's (number of 3's
in number) 1 1 1
a zero 0
three 2's (same as 1's in square) 2 2 2
a final 4 4
So the square of 3332 is 11,102,224.
See the pattern?
If the number to be squared is 333332:
The square has:
five 1's (number of 3's
in number) 1 1 1 1 1
a zero 0
five 2's (same as 1's in square) 2 2 2 2 2
a final 4 4
So the square of 333332 is 111,110,222,224.
These big squares should be quite impressive, and difficult for others to check unless they have a huge calculator.
Squaring special numbers (3's and final 4)
Choose a number with repeating 3's and a final 4.
The square is made up of:
· the same number of 1's as there are digits in the number;
· one fewer 5;
· a final 6
Example:
If the number to be squared is 3334:
The square has:
four 1's (number of digits
in number) 1 1 1 1
three 5's (one fewer) 5 5 5
a final 6 6
So the square of 3334 is 11,115,556.
See the pattern?
If the number to be squared is 333334:
The square has:
six 1's 1 1 1 1 1 1
five 5's 5 5 5 5 5
a final 6 6
So the square of 333334 is 111,111,555,556.
Squaring special numbers (3's and final 5)
Choose a number with repeating 3's and a final 5.
The square is made up of:
· the same number of 1's as there are repeating 3's in the number;
· one more 2 than there are repeating 3's;
· a final 5.
Example:
If the number to be squared is 3335:
The square has:
three 1's (same as repeating 3's) 1 1 1
four 2's (one more than
repeating 3's) 2 2 2 2
a final 5 5
So the square of 3335 is 11,122,225.
See the pattern?
If the number to be squared is 333335:
The square has:
five 1's (same as
repeating 3's) 1 1 1 1 1
six 2's (one more than
repeating 3's) 2 2 2 2 2 2
a final 5 5
So the square of 333335 is 111,112,222,225.
Squaring special numbers (3's and final 6)
Choose a number with repeating 3's and a final 6.
The square is made up of:
· the same number of 1's as there are repeating 3's in the number;
· one 2
· one fewer 8 than there are repeating 3's;
· a final 96.
Example:
If the number to be squared is 3336:
The square has:
three 1's (same as
repeating 3's) 1 1 1
one 2 2
two 8's (one fewer than
repeating 3's) 8 8
a final 96 9 6
So the square of 3336 is 11,128,896.
See the pattern?
If the number to be squared is 333336:
The square has:
five 1's (same as
repeating 3's) 1 1 1 1 1
one 2 2
four 8's (one fewer than
repeating 3's) 8 8 8 8
a final 96 9 6
So 333336 × 3333336 = 111,112,888,896.
Squaring special numbers (3's and final 7)
Choose a number with repeating 3's and a final 7.
The square is made up of:
· the same number of 1's as there are repeating 3's in the number;
· one 3
· one fewer 5 than there are repeating 3's;
· a final 69.
Example:
If the number to be squared is 3337:
The square has:
three 1's (same as
repeating 3's) 1 1 1
one 3 3
two 5's (one fewer than
repeating 3's) 5 5
a final 69 6 9
So the square of 3337 is 11,135,569.
See the pattern?
If the number to be squared is 333337:
The square has:
five 1's (same as
repeating 3's) 1 1 1 1 1
one 2 2
four 5's (one fewer than
repeating 3's) 5 5 5 5
a final 69 6 9
So 333337 × 3333337 = 111,113,555,569.
Squaring special numbers (3's and final
Choose a number with repeating 3's and a final 8.
The square is made up of:
· the same number of 1's as there are repeating 3's in the number;
· one 4
· one fewer 2 than there are repeating 3's;
· a final 44.
Example:
If the number to be squared is 33338:
The square has:
four 1's (same as
repeating 3's) 1 1 1 1
one 4 4
three 2's (one fewer than
repeating 3's) 2 2 2
a final 44 4 4
So the square of 33338 is 1,111,422,244.
See the pattern?
If the number to be squared is 3333338:
The square has:
six 1's (same as
repeating 3's) 1 1 1 1 1 1
one 4 4
five 2's (one fewer than
repeating 3's) 2 2 2 2 2
a final 44 4 4
So 3333338 × 3333338 = 11,111,142,222,244.
Squaring special numbers (3's and final 9)
Choose a number with repeating 3's and a final 9.
The square is made up of:
· the same number of 1's as there are repeating 3's in the number;
· one 4
· two fewer 8's than there are repeating 3's;
· a final 921.
Example:
If the number to be squared is 33339:
The square has:
four 1's (same as
repeating 3's) 1 1 1 1
one 4 4
two 8's (two fewer than
repeating 3's) 8 8
a final 921 9 2 1
So 33339 × 33339 is 1,111,488,921.
See the pattern?
If the number to be squared is 3333339:
The square has:
six 1's (same as
repeating 3's) 1 1 1 1 1 1
one 4 4
four 8's (two fewer than
repeating 3's) 8 8 8 8
a final 921 9 2 1
So the square of 3333339 = 11,111,148,888,921.
Squaring special numbers (6's and final 1)
Choose a number with repeating 6's and a final 1.
The square is made up of:
· one fewer 4 than there are repeating 6's
· 36
· two fewer 8's than there are repeating 6's
· A final 921
Example:
If the number to be squared is 6661:
The square has:
two 4's (one fewer than
repeating 6's) 4 4
Next digits: 36 3 6
one 8 (two fewer than repeating 6's) 8
A final 921 9 2 1
So the square of 6661 is 44,368,921.
See the pattern?
If the number to be squared is 666661:
The square has:
four 4's (one fewer than
repeating 6's) 4 4 4 4
Next digits: 36 3 6
three 8's 8 8 8
A final 921 9 2 1
So the square of 666661 is 44,4436,888,921.
Squaring special numbers (6's and final 2)
Choose a number with repeating 6's and a final 2.
The square is made up of:
· one fewer 4 than there are repeating 6's
· 38
· same number of 2's as 4's in the square
· a final 44
Example:
If the number to be squared is 6662:
The square has:
two 4's (one fewer than
repeating 6's) 4 4
Next digits: 38 3 8
two 2's (same number as repeating 6's) 2 2
A final 44 4 4
So the square of 6662 is 44,382,244.
See the pattern?
If the number to be squared is 666662:
The square has:
four 4's (one fewer than
repeating 6's) 4 4 4 4
Next digits: 38 3 8
four 2's (same as repeating 6's) 2 2 2 2
A final 44 4 4
So the square of 666662 is 444,438,222,244.
Squaring special numbers (6's and final 3)
Choose a number with repeating 6's and a final 3.
The square is made up of:
· one fewer 4 than there are repeating 6's
· 39
· same number of 5's as 4's in the square
· a final 69
Example:
If the number to be squared is 6663:
The square has:
two 4's (one fewer than
repeating 6's) 4 4
Next digits: 39 3 9
two 5's (same number as repeating 6's) 5 5
A final 69 6 9
So the square of 6663 is 44,395,569.
See the pattern?
If the number to be squared is 666663:
The square has:
four 4's (one fewer than
repeating 6's) 4 4 4 4
Next digits: 39 3 9
four 5's (same as repeating 6's) 5 5 5 5
A final 69 6 9
So the square of 666663 is 444,439,555,569.
Squaring special numbers (6's and final 4)
Choose a number with repeating 6's and a final 4.
The square is made up of:
· the same number of 4's as repeating 6's
· 0
· one fewer 8 than repeating 6's
· a final 96
Example:
If the number to be squared is 6664:
The square has:
three 4's (same number as
repeating 6's) 4 4 4
next digit: 38 0
two 8's (one fewer than repeating 6's) 8 8
a final 96 9 6
So the square of 6664 is 44,408,896.
See the pattern?
If the number to be squared is 666664:
The square has:
five 4's (same number as
repeating 6's) 4 4 4 4 4
next digit: 0 0
four 8's (one fewer than
repeating 6's) 8 8 8 8
a final 96 9 6
So the square of 666664 is 444,440,888,896.
Squaring special numbers (6's and final 5)
Choose a number with repeating 6's and a final 5.
The square is made up of:
· same number of 4's as repeating 6's
· same number of 2's as repeating 6's
· a final 25
Example:
If the number to be squared is 6665:
The square has:
three 4's (same number as
repeating 6's) 4 4 4
three 2's (same number as
repeating 6's) 2 2 2
A final 25 2 5
So the square of 6665 is 44,422,225.
See the pattern?
If the number to be squared is 666665:
five 4's (same number as
repeating 6's) 4 4 4 4 4
five 2's (same number as
repeating 6's) 2 2 2 2 2
A final 25 2 5
So the square of 666665 is 444,442,222,225.
Squaring special numbers (6's and final 7)
Choose a number with repeating 6's and a final 7.
The square is made up of:
· The same number of 4's as there are digits in the number;
· One fewer 8;
· A final 9.
Example:
If the number to be squared is 6667:
The square has:
four 4's (number of digits
in number) 4 4 4 4
three 8's (one fewer) 8 8 8
a final 9 9
So the square of 6667 is 44448889.
See the pattern?
If the number to be squared is 667:
The square has:
three 4's 4 4 4
two 8's 8 8
a final 9 9
So the square of 667 is 444889.
Use a big number - others will need a powerful calculator or lots of time to check your answer!
Squaring special numbers (6's and final
Choose a number with repeating 6's and a final 8.
The square is made up of:
· the same number of 4's as there are repeating 6's in the number;
· one 6
· the same number of 2's as repeating 6's;
· a final 4.
Example:
If the number to be squared is 6668:
The square has:
three 4's (same as
repeating 6's) 4 4 4
one 6 6
three 2's (same number as
repeating 3's) 2 2 2
a final 4 4
So the square of 6668 is 44,462,224.
See the pattern?
If the number to be squared is 666668:
The square has:
five 4's (same number as
repeating 6's) 4 4 4 4 4
one 6 6
five 2's (same number as
repeating 6's) 2 2 2 2 2
a final 4 4
So 666668 × 666668 = 444,446,222,224.
Squaring special numbers (6's and final 9)
Choose a number with repeating 6's and a final 9.
The square is made up of:
· the same number of 4's as there are repeating 6's;
· a 7
· one fewer 5 than there are repeating 6's;
· A final 61.
Example:
If the number to be squared is 6669:
The square has:
same number of 4's as repeating 6's: 4 4 4
a 7 7
one fewer 5 than repeating 6's 5 5
a final 61 6 1
So the square of 6669 is 44,475,561.
See the pattern?
If the number to be squared is 666669:
The square has:
same number of 4's as 6's: 4 4 4 4 4
a 7 7
one fewer 5 than repeating 6's 5 5 5 5
a final 61 6 1
So 666,669 × 666,669 = 44,447,555,561.
Use the pattern to amaze your friends with your multiplying abilities.
Squaring special numbers (9's and final 1)
Choose a number with repeating 9's and a final 1.
The square is made up of:
· one fewer 9 than there are repeating 9's
· 82
· the same number of 0's as there are 9's in the square
· A final 81
Example:
If the number to be squared is 9991:
The square has:
One fewer 9 than the
repeating 9's: 9 9
82 8 2
same number of 0's as 9's
in the square 0 0
a final 81 8 1
So 9991 × 9991 = 99820081.
See the pattern?
If the number to be squared is 999991:
The square has:
one fewer 9 than the
repeating 9's: 9 9 9 9
82 8 2
same number of 0's as 9's
in the square 0 0 0 0
a final 81 8 1
So 999991 × 999991 = 999982000081.
Those big products ought to impress your friends, and they will need a BIG calculator to keep up with you!
Squaring special numbers (9's and final 2)
Choose a number with repeating 9's and a final 2.
The square is made up of:
· one fewer 9 than there are repeating 9's
· 84
· the same number of 0's as there are 9's in the square
· A final 64
Example:
If the number to be squared is 9992:
The square has:
one fewer 9 than the
repeating 9's: 9 9
84 8 4
same number of 0's as 9's
in the square 0 0
a final 64 6 4
So 9992 × 9992 = 99840064.
See the pattern?
If the number to be squared is 999992:
The square has:
one fewer 9 than the
repeating 9's: 9 9 9 9
84 8 4
same number of 0's as 9's
in the square 0 0 0 0
a final 64 6 4
So 999992 × 999992 = 999984000064.
With a little practice you will be finding these huge products with ease.
Squaring special numbers (9's and final 3)
Choose a number with repeating 9's and a final 3.
The square is made up of:
· one fewer 9 than there are repeating 9's
· 86
· the same number of 0's as there are 9's in the square
· A final 49
Example:
If the number to be squared is 9993:
The square has:
one fewer 9 than the
repeating 9's: 9 9
86 8 6
same number of 0's as 9's
in the square 0 0
a final 49 4 9
So 9993 × 9993 = 99860049.
See the pattern?
If the number to be squared is 999993:
The square has:
one fewer 9 than the
repeating 9's: 9 9 9 9
86 8 6
same number of 0's as 9's
in the square 0 0 0 0
a final 49 4 9
So 999993 × 999993 = 999986000049.
Using this pattern you will be able to square these large numbers with ease.
Squaring special numbers (9's and final 4)
Choose a number with repeating 9's and a final 4.
The square is made up of:
· one fewer 9 than there are repeating 9's
· 88
· the same number of 0's as there are 9's in the square
· A final 36
Example:
If the number to be squared is 9994:
The square has:
one fewer 9 than the
repeating 9's: 9 9
88 8 8
same number of 0's as 9's
in the square 0 0
a final 36 3 6
So 9994 × 9994 = 99880036.
See the pattern?
If the number to be squared is 999994:
The square has:
one fewer 9 than the
repeating 9's: 9 9 9 9
88 8 8
same number of 0's as 9's
in the square 0 0 0 0
a final 36 3 6
So 999994 × 999994 = 999988000036.
Squaring special numbers (9's and final 5)
Choose a number with repeating 9's and a final 5.
The square is made up of:
· same number of 9's as there are repeating 9's
· same number of 0's
· a final 25
Example:
If the number to be squared is 9995:
The square has:
same number of 9's as
repeating 9's: 9 9 9
same number of 0's as 9's
in the square 0 0 0
a final 25 2 5
So 9995 × 9995 = 99900025.
See the pattern?
If the number to be squared is 999995:
The square has:
same number of 9's as
repeating 9's: 9 9 9 9
same number of 0's as 9's
in the square 0 0 0 0
a final 25 2 5
So 999995 × 999995 = 999990000025
Squaring special numbers (9's and final 6)
Choose a number with repeating 9's and a final 6.
The square is made up of:
· same number of 9's as there are repeating 9's
· a 2
· one fewer 0 than repeating 9's
· a final 16
Example:
If the number to be squared is 9996:
The square has:
same number of 9's
as repeating 9's: 9 9 9
a 2 2
one fewer 0 than there
are 9's in the square 0 0
a final 16 1 6
So 9996 × 9996 = 99920016.
See the pattern?
If the number to be squared is 999996:
The square has:
same number of 9's
as repeating 9's: 9 9 9 9 9
a 2 2
one fewer 0 than there
are 9's in the square 0 0 0 0
a final 16 1 6
So 999996 × 999996 = 999992000016
Squaring special numbers (9's and final 7)
Choose a number with repeating 9's and a final 7.
The square is made up of:
· the same number of 9's as there are repeating 9's
· 4
· the same number of 0's as there are 9's in the square
· A final 9
Example:
If the number to be squared is 9997:
The square has:
same number of 9's as there are
repeating 9's: 9 9 9
4 4
same number of 0's as 9's
in the square 0 0 0
a final 9 9
So 9997 × 9997 = 99940009.
See the pattern?
If the number to be squared is 999997:
The square has:
same number of 9's as there are
repeating 9's: 9 9 9 9 9
4 4
same number of 0's as 9's
in the square 0 0 0 0 0
a final 9
9
So 999997 × 999997 = 999994000009.
Learn the pattern and it's easy!
Squaring special numbers (9's and final
Choose a number with repeating 9's and a final 8.
The square is made up of:
· the same number of 9's as there are repeating 9's
· 6
· the same number of 0's as there are 9's in the square
· A final 4
Example:
If the number to be squared is 9998:
The square has:
same number of 9's as there are
repeating 9's: 9 9 9
6 6
same number of 0's as 9's
in the square 0 0 0
a final 4 4
So 9998 × 9998 = 99,960,004.
See the pattern?
If the number to be squared is 999998:
The square has:
same number of 9's as there are
repeating 9's: 9 9 9 9 9
6 6
same number of 0's as 9's
in the square 0 0 0 0 0
a final 4
4
So 999997 × 999997 = 999,996,000,004.
Squaring special numbers (1 and repeating 3's)
Choose a number with a 1 and repeating 3's.
The square is made up of:
· first digits: 1 & one fewer 7 than repeating 3's
· next digits: 6 & one fewer 8 than repeating 3's
· last digit: 9
Example:
If the number to be squared is 1333:
The square has:
first digits: 1 and one fewer
7 than 3's 1 7 7
next digits: 6 and one fewer
8 than 3's 6 8 8
last digit: 9 9
So 1333 × 1333 = 1776889.
See the pattern?
If the number to be squared is 133333:
The square has:
first digits: 1 and one fewer
7 than 3's 1 7 7 7 7
next digits: 6 and one fewer
8 than 3's 6 8 8 8 8
last digit: 9
9
So 133333 × 133333 = 17777688889.
Squaring special numbers (1 and repeating 6's)
Choose a number with a 1 and repeating 6's.
The square is made up of:
· first digits: 2 & one fewer 7 than repeating 6's
· next digits: same number of 5's as repeating 6's
· last digit: 6
Example:
If the number to be squared is 1666:
The square has:
first digits: 2 and one fewer
7 than 6's 2 7 7
next digits: same number of 5's
as 6's 5 5 5
last digit: 6 6
So 1666 × 1666 = 2775556.
See the pattern?
If the number to be squared is 166666:
The square has:
first digits: 2 and one fewer
7 than 6's 2 7 7 7 7
next digits: same number of 5's
as 6's 5 5 5 5 5
last digit: 6
6
So 166666 × 166666 = 27777555556.
Squaring special numbers (1 and repeating 9's)
Choose a number with a 1 and repeating 9's.
The square is made up of:
· first digits: 3 & one fewer 9 than repeating 9's
· next digits: 6 & one fewer 0 than repeating 9's
· last digit: 1
Example:
If the number to be squared is 1999:
The square has:
first digits: 3 and one fewer
9 than 9's 3 9 9
next digits: 6 and one fewer
0 than 9's 6 0 0
last digit: 1 1
So 1999 × 1999 = 3996001.
See the pattern?
If the number to be squared is 199999:
The square has:
first digits: 3 and one fewer
9 than 9's 3 9 9 9 9
next digits: 6 and one fewer
0 than 9's & 6 0 0 0 0
last digit: 1
1
So 199999 × 199999 = 39999600001.
Squaring special numbers (2 and repeating 3's)
Choose a number with a 2 and repeating 3's.
The square is made up of:
· first digits: 5 & one fewer 4 than repeating 3's
· next digits: 2 & one fewer 8 than repeating 3's
· last digit: 9
Example:
If the number to be squared is 2333:
The square has:
first digits: 5 and one fewer
4 than 3's 5 4 4
next digits: 2 and one fewer
8 than 3's 2 8 8
last digit: 9 9
So 2333 × 2333 = 5442889.
See the pattern?
If the number to be squared is 233333:
The square has:
first digits: 5 and one fewer
4 than 3's 5 4 4 4 4
next digits: 2 and one fewer
8 than 3's 2 8 8 8 8
last digit: 9
9
So 233333 × 233333 = 54444288889.
Squaring special numbers (2 and repeating 6's)
Choose a number with a 2 and repeating 6's.
The square is made up of:
· first digits: 7 & two fewer 1's than repeating 6's
· next digits: 07 & one fewer 5 than repeating 6's
· last digit: 6
Example:
If the number to be squared is 2666:
The square has:
first digits: 7 and two fewer
1's than 6's 7 1
next digits: 07 and one fewer
5 than 6's 0 7 5 5
last digit: 6 6
So 2666 × 2666 = 7,107,556.
See the pattern?
If the number to be squared is 266666:
The square has:
first digits: 7 and two fewer
1's than 6's 7 1 1 1
next digits: 07 and one fewer
5 than 6's 0 7 5 5 5 5
last digit: 6 6
So 266666 × 266666 = 71,110,755,556.
Squaring special numbers (2 and repeating 9's)
Choose a number with a 2 and repeating 9's.
The square is made up of:
· first digits: 8 & one fewer 9 than repeating 9's
· next digits: 4 & one fewer 0 than repeating 9's
· last digit: 1
Example:
If the number to be squared is 2999:
The square has:
first digits: 8 and one fewer
9 than 9's 8 9 9
next digits: 4 and one fewer
0 than 9's 4 0 0
last digit: 1 1
So 2999 × 2999 = 8,994,001.
See the pattern?
If the number to be squared is 299999:
The square has:
first digits: 8 and one fewer
9 than 9's 8 9 9 9 9
next digits: 4 and one fewer
0 than 9's 4 0 0 0 0
last digit: 1 1
So 299999 × 299999 = 89,999,400,001.
Squaring special numbers (3 and repeating 6's)
Choose a number with a 3 and repeating 6's.
The square is made up of:
· first digits: 13 & two fewer 4's than repeating 6's
· next digits: 39 & one fewer 5 than repeating 6's
· last digit: 6
Example:
If the number to be squared is 3666:
The square has:
first digits: 13 and two fewer
4's than 6's 1 3 4
next digits: 39 and one fewer
5 than 6's 3 9 5 5
last digit: 6 6
So 3666 × 3666 = 13,439,556.
See the pattern?
If the number to be squared is 366666:
The square has:
first digits: 13 and two fewer
4's than 6's 1 3 4 4 4
next digits: 39 and one fewer
5 than 6's 3 9 5 5 5 5
last digit: 6
6
So 366666 × 366666 = ?
Squaring special numbers (3 and repeating 9's)
Choose a number with a 3 and repeating 9's.
The square is made up of:
· first digits: 15 & one fewer 9 than repeating 9's
· next digits: 2 & one fewer 0 than repeating 9's
· last digit: 1
Example:
If the number to be squared is 3999:
The square has:
first digits: 15 and one fewer
9 than 9's 1 5 9 9
next digits: 2 and one fewer
0 than 9's 2 0 0
last digit: 1 1
So 3999 × 3999 = 15,992,001.
See the pattern?
If the number to be squared is 399999:
The square has:
first digits: 15 and one fewer
9 than 9's 1 5 9 9 9 9
next digits: 2 and one fewer
0 than 9's 2 0 0 0 0
last digit: 1
1
So 399999 × 399999 = 159,999,200,001.
34,443,955,556.
Squaring special numbers (4 and
repeating 3's)
Choose a number with a 4 and repeating 3's.
The square is made up of:
· first digits: 18 & one fewer 7 than repeating 3's
· next digits: 4 & one fewer 8 than repeating 3's
· last digit: 9
Example:
If the number to be squared is 4333:
The square has:
first digits: 18 and one fewer
7 than 3's 1 8 7 7
next digits: 4 and one fewer
8 than 3's 4 8 8
last digit: 9 9
So 4333 × 4333 = 18,774,889.
See the pattern?
If the number to be squared is 433333:
The square has:
first digits: 18 and one fewer
7 than 3's 1 8 7 7 7 7
next digits: 4 and one fewer
8 than 3's 4 8 8 8 8
last digit: 9 9
So 433333 × 433333 = 187,777,488,889.
Squaring special numbers (4 and repeating 6's)
Choose a number with a 4 and repeating 6's.
The square is made up of:
· first digits: 21 & one fewer 7 than repeating 6's
· next digits: 1 & one fewer 5 than repeating 6's
· last digit: 6
Example:
If the number to be squared is 4666:
The square has:
first digits: 21 and one fewer
7 than 6's 2 1 7 7
next digits: 1 and one fewer
5 than 6's 1 5 5
last digit: 6 6
So 4666 × 4666 = 21,771,556.
See the pattern?
If the number to be squared is 466666:
The square has:
first digits: 21 and one fewer
7 than 6's 2 1 7 7 7 7
next digits: 1 and one fewer
5 than 6's 1 5 5 5 5
last digit: 6
6
So 466666 × 466666 = 217,777,155,556.
Squaring special numbers (4 and repeating 9's)
Choose a number with a 4 and repeating 9's.
The square is made up of:
· first digits: 24 & one fewer 9 than repeating 9's in the number
· next digits: same number of 0's as repeating 9's in the number
· last digit: 1
Example:
If the number to be squared is 4999:
The square has:
first digits: 24 and one fewer
9 than 9's (in the number) 2 4 9 9
next digits: same number of 0's
as 9's (in the number) 0 0 0
last digit: 1 1
So 4999 × 4999 = 24,990,001.
See the pattern?
If the number to be squared is 499999:
The square has:
first digits: 24 and one
fewer 9 than 9's 2 4 9 9 9 9
next digits: same number
of 0's as 9's 0 0 0 0 0
last digit: 1
1
So 499999 × 499999 = 249,999,000,001.
Squaring special numbers (5 and repeating 3's)
Choose a number with a 5 and repeating 3's.
The square is made up of:
· first digits: 28 & one fewer 4 than repeating 3's
· next digits: 0 & one fewer 8 than repeating 3's
· last digit: 9
Example:
If the number to be squared is 5333:
The square has:
first digits: 28 and one fewer
4 than 3's 2 8 4 4
next digits: 0 and one fewer
8 than 3's 0 8 8
last digit: 9 9
So 5333 × 5333 = 28,440,889.
See the pattern?
If the number to be squared is 533333:
The square has:
first digits: 28 and one fewer
4 than 3's 2 8 4 4 4 4
next digits: 0 and one fewer
8 than 3's 0 8 8 8 8
last digit: 9
9
So 533333 × 533333 = 284,444,088,889.
Squaring special numbers (5 and repeating 6's)
Choose a number with a 5 and repeating 6's.
The square is made up of:
· first digits: 32 & two fewer 1's than repeating 6's
· next digits: 03 & one fewer 5 than repeating 6's
· last digit: 6
Example:
If the number to be squared is 5666:
The square has:
first digits: 32 and two fewer
1's than 6's 3 2 1
next digits: 03 and one fewer
5 than 6's 0 3 5 5
last digit: 6 6
So 5666 × 5666 = 32,103,556.
See the pattern?
If the number to be squared is 566666:
The square has:
first digits: 32 and two fewer
1's than 6's 3 2 1 1 1
next digits: 03 and one fewer
1 than 6's 0 3 5 5 5 5
last digit: 6
6
So 566666 × 566666 = 321,110,355,556.
Squaring special numbers (5 and repeating 9's)
Choose a number with a 5 and repeating 9's.
The square is made up of:
· first digits: 35 & two fewer 9's than repeating 9's
· next digits: 88 & one fewer 0 than repeating 9's
· last digit: 1
Example:
If the number to be squared is 5999:
The square has:
first digits: 35 and two fewer
9's than 9's 3 5 9
next digits: 88 and one fewer
0 than 9's 8 8 0 0
last digit: 1 1
So 5999 × 5999 = 35,988,001.
See the pattern?
If the number to be squared is 599999:
The square has:
first digits: 35 and two fewer
9's than 9's 3 5 9 9 9
next digits: 88 and one fewer
0 than 9's 8 8 0 0 0 0
last digit: 1
1
So 599999 × 599999 = 359,998,800,001.
Squaring special numbers (6 and repeating 3's)
Choose a number with a 6 and repeating 3's.
The square is made up of:
· first digits: 40 & two fewer 1's than repeating 3's
· next digits: 06 & one fewer 8 than repeating 3's
· last digit: 9
Example:
If the number to be squared is 6333:
The square has:
first digits: 40 and two fewer
1's than 3's 4 0 1
next digits: 06 and one fewer
8 than 3's 0 6 8 8
last digit: 9 9
So 6333 × 6333 = 40,106,889.
See the pattern?
If the number to be squared is 633333:
The square has:
first digits: 40 and two fewer
1's than 3's 4 0 1 1 1
next digits: 06 and one fewer
8 than 3's 0 6 8 8 8 8
last digit: 9
9
So 633333 × 633333 = 401,110,688,889.
Squaring special numbers (6 and repeating 9's)
Choose a number with a 6 and repeating 9's.
The square is made up of:
· first digits: 48 & two fewer 9's than repeating 9's in the number
· next digits: 86 & one fewer 0 than repeating 9's in the number
· last digit: 1
Example:
If the number to be squared is 6999:
The square has:
first digits: 48 and two fewer
9's than rep. 9's 4 8 9
next digits: 86 and one fewer
0 than rep. 9's 8 6 0 0
last digit: 1 1
So 6999 × 6999 = 48,986,001.
See the pattern?
If the number to be squared is 699999:
The square has:
first digits: 48 and two fewer
9's than rep. 9's 4 8 9 9 9
next digits: 86 and one fewer
0 than rep. 9's 8 6 0 0 0 0
last digit: 1
1
So 699999 × 699999 = 489,998,600,001.
Squaring special numbers (7 and repeating 3's)
Choose a number with a 7 and repeating 3's.
The square is made up of:
· first digits: 53 & one fewer 7 than repeating 3's
· next digits: 2 & one fewer 8 than repeating 3's
· last digit: 9
Example:
If the number to be squared is 7333:
The square has:
first digits: 53 and one fewer
7 than 3's 5 3 7 7
next digits: 2 and one fewer
8 than 3's 2 8 8
last digit: 9 9
So 7333 × 7333 = 53,772,889.
See the pattern?
If the number to be squared is 733333:
The square has:
first digits: 53 and one fewer
7 than 3's 5 3 7 7 7 7
next digits: 2 and one fewer
8 than 3's 2 8 8 8 8
last digit: 9 9
So 733333 × 733333 = 537,777,288,889.
Squaring special numbers (7 and repeating 6's)
Choose a number with a 7 and repeating 6's.
The square is made up of:
· first digits: 58 & two fewer 7's than repeating 6's
· next digits: 67 & one fewer 5 than repeating 6's
· last digit: 6
Example:
If the number to be squared is 7666:
The square has:
first digits: 58 and two fewer
7's than 6's 5 8 7
next digits: 67 and one fewer
5 than 6's 6 7 5 5
last digit: 6 6
So 7666 × 7666 = 58,767,556.
See the pattern?
If the number to be squared is 766666:
The square has:
first digits: 58 and two fewer
7's than 6's 5 8 7 7 7
next digits: 67 and one fewer
5 than 6's 6 7 5 5 5 5
last digit: 6 6
So 766666 × 766666 = 587,776,755,556.
Squaring special numbers (7 and repeating 9's)
Choose a number with a 7 and repeating 9's (use a mimimum of three 9's).
The square is made up of:
· first digits: 63 & two fewer 9's than repeating 9's in the number
· next digits: 84 & one fewer 0 than repeating 9's in the number
· last digit: 1
Example:
If the number to be squared is 7999:
The square has:
first digits: 63 and two fewer
9's than rep. 9's 6 3 9
next digits: 84 and one fewer
0 than rep. 9's 8 4 0 0
last digit: 1 1
So 7999 × 7999 = 63,984,001.
See the pattern?
If the number to be squared is 799999:
The square has:
first digits: 63 and two fewer
9's than rep. 9's 6 3 9 9 9
next digits: 84 and one fewer
0 than rep. 9's 8 4 0 0 0 0
last digit: 1
1
So 799999 × 799999 = 639,998,400,001.
Squaring special numbers (8 and repeating 3's)
Choose a number with an 8 and (at least 3) repeating 3's.
The square is made up of:
· first digits: 69 & two fewer 4's than repeating 3's
· next digits: 3 & same number of 8's as repeating 3's
· last digit: 9
Example:
If the number to be squared is 8333:
The square has:
first digits: 60 and two fewer
4's than 3's 6 9 4
next digits: 3 and same number of
8's as 3's 3 8 8 8
last digit: 9 9
So 8333 × 8333 = 69,438,889.
See the pattern?
If the number to be squared is 833333:
The square has:
first digits: 69 and two fewer
4's than 3's 6 9 4 4 4
next digits: 3 and same number of
8's as 3's 3 8 8 8 8 8
last digit: 9 9
So 833333 × 833333 = 694,443,888,889.
Squaring special numbers (8 and repeating 9's)
Choose a number with an 8 and repeating 9's (use a mimimum of three 9's).
The square is made up of:
· first digits: 80 & two fewer 9's than repeating 9's in the number
· next digits: 82 & one fewer 0 than repeating 9's in the number
· last digit: 1
Example:
If the number to be squared is 8999:
The square has:
first digits: 80 and two fewer
9's than rep. 9's 8 0 9
next digits: 82 and one fewer
0 than rep. 9's 8 2 0 0
last digit: 1 1
So 8999 × 8999 = 80,982,001.
See the pattern?
If the number to be squared is 899999:
The square has:
first digits: 80 and two fewer
9's than rep. 9's 8 0 9 9 9
next digits: 82 and one fewer
0 than rep. 9's 8 2 0 0 0 0
last digit: 1
1
So 899999 × 899999 = 809,998,200,001.
Squaring special numbers (9 and repeating 3's)
Choose a number with a 9 and repeating 3's (use at least three 3's).
The square is made up of:
· first digits: 87 & two fewer 1's than repeating 3's
· next digits: 04 & one fewer 8 than repeating 3's
· last digit: 9
Example:
If the number to be squared is 9333:
The square has:
first digits: 87 and two fewer
1's than 3's 8 7 1
next digits: 04 and one fewer
8 than 4's 0 4 8 8
last digit: 9 9
So 9333 × 9333 = 87,104,889.
See the pattern?
If the number to be squared is 933333:
The square has:
first digits: 87 and two fewer
1's than 3's 8 7 1 1 1
next digits: 04 and one fewer
8 than 3's 0 4 8 8 8 8
last digit: 9
9
So 933333 × 933333 = 871,110,488,889.
Squaring special numbers (9 and repeating 6's)
Choose a number with a 9 and repeating 6's (use at least three 6's).
The square is made up of:
· first digits: 93 & two fewer 4's than repeating 6's
· next digits: 31 & one fewer 5 than repeating 6's
· last digit: 6
Example:
If the number to be squared is 9666:
The square has:
first digits: 93 and two fewer
4's than 6's 9 3 4
next digits: 31 and one fewer
5 than 6's 3 1 5 5
last digit: 6 6
So 9666 × 9666 = 93,431,556.
See the pattern?
If the number to be squared is 966666:
The square has:
first digits: 93 and two fewer
4's than 6's 9 3 4 4 4
next digits: 31 and one fewer
5 than 6's 3 1 5 5 5 5
last digit: 6
6
So 966666 × 966666 = 934,443,155,556.
Squaring a repeating 6-digit number
Choose a number with repeating 6's .
The square is made up of:
· One less 4 than there are digits in the number;
· One 3;
· The same number of 5's as 4's;
· A final 6.
Example:
If the number to be squared is 666:
The square has:
one less 4 than digits in
the number 4 4
one 3 3
same number of 5's as 4's 5 5
a final 6 6
So the square of 666 is 443556.
See the pattern?
If the number to be squared is 66666:
The square has:
one less 4 than digits in
the number) 4 4 4 4
one 3 3
same number of 5's as 4's 5 5 5 5
a final 6
6
So the square of 66666 is 4444355556.
Squaring 2 2's, 3 3's, etc., then dividing by square of single digit
Choose a number with 2 repeating 2's, 3 repeating 3's, 4 repeating 4's, etc., up to 9 repeating 9's.
Square the number.
Divide that product by the square of the single digit of the selected number.
The answer is a sequence beginning with 1 and going up to the single digit of the number, and back down to 1.
Example:
· If the number to be squared is 333:
· The answer is 12321.
· If the number to be squared is 666666:
· The answer is 12345654321.
Try to vary the procedure with a last step so the answer is not so obvious. That will make this trick more interesting. You might ask that some number, perhaps 321, be added as a last step. Then the last three digits of the answer would be 642.
Now you practice. Perhaps you will be helpful.
Square Patterns
, 24 + 1 = 25
