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 × 6 = 36   _ _ 6
   
2. Multiply the second digit by 2 and
   
   add the carry: 2 × 6 + 3 = 15   _ 5 _
   
3. The first digit is one plus the carry:
   
   1 + 1 = 2   2 _ _
   
4. So 16 × 16 = 256.

   See the pattern?

1. For 19 × 19, square the second digit:
   
   9 × 9 = 81   _ _ 1
   
2. Multiply the second digit by 2 and
   
   add the carry: 2 × 9 + 8 = 26   _ 6 _
   
3. The first digit is one plus the carry:
   
   1 + 2 = 3   3 _ _
   
4. So 19 × 19 = 361.

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 × 4 = 24 (six times the first digit): _ _ 4 9.
3. 4 × 4 = 16 (square the first digit), 16 + 2 = 18 
   
   (add carry): 1 8 4 9.
   
4. So 43 × 43 = 1849.

   See the pattern?

1. For 83 × 83, the last digit is _ _ _ 9.
2. 6 × 8 = 48 (six times the first digit): _ _ 8 9.
3. 8 × 8 = 64 (square the first digit), 64 + 4 = 68 
   
   (add carry): 6 8 8 9.
   
4. So 83 × 83 = 6889.

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 9

one 8                                8

the same number of zeros as 9's        0 0 0

a final 1                                    1

3. So 9999 × 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 9

one 8                                   8

the same number of zeros as 9's           0 0 0 0 0

a final 1                                           1

3. So 999999 × 999999 = 999998000001.

This is not a very demanding mental math exercise

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