How to square any number between 1 and 100

In my post on magic mathematical tricks I claimed to be able to easily square any number between 1-100. Later I got a comment enquiring about this claim, according to my reader there is only a calculation "fast track" for numbers ending with 5, i.e. 5, 15, 25, 35, 45, 55 etc, and not for all the other numbers between 1-100.

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

Magical mathematical tricks

Parallell to working on forthcoming articles I have been listening to the Teaching Company series called "The joy of mathematics", taught by professor Arthur T. Benjamin (see picture).

It is a truly great series of lectures which has allowed me to fresh up my math as well as to learn som impressive new skills. For example I can now square any number between 1-100 very fast (perhaps I will reveal the trick in a subsequent post). The lecturer has also perfomed two "magical tricks". The first one I think most people will be familiar with, I recall hearing it in 3rd grade, it goes as follows:

1. Think of a number between 1-10
2. Double that number
3. Add 10
4. Divide that number by 2
5. Subtract the number that you first thought about

Ok, now concentrate on the number that you are left with, a little bit more... I think I am getting... wait for it... yes, I almost have it... 5! Was that the number that you were left with? If not, it is because you did an error. Some simple algebra will prove why you are always left with 5.

x*2 = 2x,
2x+10 = 2x+10,
(2x+10)/2 = x+5,
x+5-x = 5

If you want to end up with a different number you can merely swap out the 10 with something else. Your answer will always be half of the number that is added in the second step.

The second magical trick is somewhat more complicated and also more impressive if you ask me. It goes as follows:

1. Think of a number between 1-10
2. Triple that number
3. Add 6
4. Triple the number again
5. Now add the individual digits in your numer
6. If you still have a two digit number, add the individual digits again

Now, concentrate again. Unless you have done a mistake you are thinking about the number 9. The following algebra plus some explanation shows why this trick works.

x*3 = 3x
3x+6 = 3x + 6
(3x+6)*3= 9x+18

Now 18 is a multiple of 9 and adding x number of nines to that will always result in a multiple of 9. Multiples of nine always consists of digits that add up to 9, or to another multiple of 9 which will then add up to 9. You can check this out for yourself. The multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126 etc

Adding the individual digits of any of these number will give 9 except 99, which gives 18 which you then add again and get 9.