How to Square Numbers Near 100 in Your Head

How do you square a number near 100?

To square a number just below 100, take its distance from 100 off the number, then attach the square of that distance as two digits. For 96², the distance is 4, so 96 − 4 = 92 and 4² = 16, giving 9216.

It works because (100 − d)² is 100 × (100 − 2d) + d², and 100 − 2d is the number with its distance taken off a second time.

A small distance still takes two digits: 97² gives 94 and 09, so 9409. From a distance of 10 onward, the square of the distance has three digits, so add it on: 88 is 12 below 100, so 88 − 12 = 76 and 12² = 144, giving 88² = 7600 + 144 = 7744. The same trick works for a number just above 100, with the distance added: 103² gives 106 and 09, so 10609. Check the last digit: 96² must end in 6, because 6 × 6 = 36.

What about numbers just above 100?

The distance is added instead of subtracted. 104 is 4 above 100, so 104 + 4 = 108 starts the answer and $4^2=16$4² = 16 finishes it: $104^2=10816$104² = 10816. 112 is 12 above 100, so 112 + 12 = 124, and $12^2=144$12² = 144 has three digits, so add it on: 12400 + 144 = 12544.

Why does it work?

Write the number as 100 − d. Squaring gives $10000-200d+d^2$10000 − 200d + d², and 10000 − 200d is 100 × (100 − 2d): a hundred times the number with its distance taken off once more. So the answer is that number followed by two zeros, plus $d^2$d². For a number 100 + d the sign flips, and the answer is $100\times \left(100+2d\right)+d^2$100 × (100 + 2d) + d².