Squares to 99

This is a list of all squares from 13 to 99. The point is to practice the various mental methods for calculation of these squares. Squares 13 to 22 really should be learned by heart.


Method 1) when a number ends in 5
For all 2-digit numbers ending on 5, e.g. 15, 45, 95 etc, there is a really simple trick for this:
- all the answers end with -25
- to get the leftmost digits, take the first digit, add 1, and multiply it by the first.

Example: 35 x 35
- add 1 to 3, this makes 4.
- do 3 x 4, this makes 12.
- bung a "25" on the end to give: 1225


Method 1) when a number is close to 10 or 100
- move up or down to find the nearest "easy" number (like a multiple of 10)
- add the distance to this to the number to square
- multiply 10 and the result above
- finally add on the distance squared

Example: 13 x 13

- Find the distance d to the nearest 10. This is 3.
- Go up by that distance: 13 + 3 = 16
    ( basically move in the opposite direction so the new numbers 'surround' the square )
- Multiply those results: 16 x 10 = 160
- Add the square of the distance d² = 9 to give : 160 + 9 = 169


Example: 96 x 96

- nearest number is 100, distance is 4, so factors are 100 and 92
- 100 x 92 = 9200
- distance squared is 16. Add this on: 9216


There do exist other ways to do these multiplications but they involve extra steps. You can use:

- geometric partitioning ( based on the expansion of (a-b)²
- the binomial expansion (a + b)² = a² + 2ab + b²
- or you can use Trachtenberg cross-multiplication.


Example: 96 x 96 via cross-multiplication:

- multiply 6 x 6 = 36. Remember it with a memory image, say 'mash' ( see the memory section of this website )
- cross-multiply and add: 9 x 6 = 54 and 6 x 9 = 54, add them : 108
- the 8 needs to be added to the 3 of the 36. This makes 16 with a carry. Remember 16 with a trick: F-16
- add the carry to the '10' of the cross result: 11
- 9 x 9 - 81. 81 plus 11 makes 92. The answer is 9216


Although cross-multiplication is the standard method for multiplying two 2-digit numbers, it is slower than the
specific 'tricks' when squaring numbers ( i.e. the two numbers are the same.)

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This is a set list, so as with all lists in the system, it is level-independent.