I suppose many might have figured this out, but it's a neat little mind trick for calculating the product of a number between 10 - 19 with itself. I stumbled upon it as I was riding the bus back home from my math class, and thought it might come in handy for those who don't know it yet!

Multiplication:

$\displaystyle

13^2 = 169

$

Think:

$\displaystyle

1^2 = 1

$

$\displaystyle

3+3 = 6

$

$\displaystyle

3^2 = 9

$

Excellent, should take a normally gifted person about $\displaystyle \phi$ seconds!

For numbers which gives two digits for any of the operations, you do the same basically. But just remember that you need to summarize the results you get correctly.

Multiplication:

$\displaystyle

17^2 = 289

$

Think:

$\displaystyle

1^2 = 1

$

$\displaystyle

7+7 = 14

$

$\displaystyle

7^2 = 49

$

If you remember the addition you need to do, it shouldn't take much longer.

This metod works excellently for myself, and it will usually take me between 2-5 seconds to calculate the product of any number between 10-19 multiplicated with itself.

A bit more mathematically we can write:

$\displaystyle

f(x) = x^2 \Leftrightarrow 100\alpha^2 + 20\beta + \beta^2, x \epsilon [10, 19]

$

Where alpha denotes the first digit of the product, and beta the second.