The fact of the matter is that numbers ending with 5 are particularly easy to square, but there is a more general rule which works for all numbers. Here is how you do it…
Take any number between 1-100… 67, ok, lets square 67. First you find a number close to 67 that is easy to multiply such as 70. Next you move the same number of steps (there are 3 steps from 67 to 70) in the opposite direction giving you 64. Multiply 70 and 64 (4200 + 280= 4480). Then add the square of the number of steps that you took in either direction i.e. 3, giving you 9. Now simply add these two numbers to get 4489, which is the square of 67.
Ok sure, multiplying 64 and 70 does require some mathematical skull but is not that hard and most people can do it with a little bit of training I think. The trick is much easier for smaller numbers such as 18. For 18 you would multiply 20 and 16, giving you 320, plus the square of 2, which is 4, giving you a total of 324. Here is an illustration of the method taken from the teaching company course "The joy of mathematics".
The very easiest cases are numbers ending with 5. Take the number 45 and apply the previous rule. You go five steps in either direction giving you 40 and 50. Multiplying these gives you 2000. Add the square of the number of steps that you took i.e. 5, giving you 25. So the square of 45 is 2000 + 25 = 2025. For numbers ending with five you can simply multiply the number below and above the first number and then write 25 behind that number i.e. 4*5 = 20, and with 25 behind that gives you 2025. For 65 it would be 6*7 = 42 and then 25 behind that giving you 4225, for 15 it would be 1*2 = 2, and then 25 behind giving you 225.
This trick seems to work well with higher numbers as well. The only limitation is your multiplication skills. So go out and impress your friends. Next time I write about math I will explain how you can pull of the almost autistic feat of telling what day of the week it was or will be at any date in history or future