Sunday, June 8, 2008

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.

2 comments:

kayla said...

This trick reinforces the concept of a number as a quantity, rather than as the series of digits by which we represent it. By doing a "magic trick" and using a "secret code machine," children actually learn how to convert back and forth between decimal and binary expansions of numbers. Here, binary expansions are disguised as secret codes. This trick reinforces the concept of a number as a quantity, rather than as the series of digits by which we represent it. By doing a "magic trick" and using a "secret code machine," children actually learn how to convert back and forth between decimal and binary expansions of numbers. Here, binary expansions are disguised as secret codes.
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Ricky

Holistic Rehab

robertinho said...

Excellent I've always wanted to get mathematical tricks, by the way why the man in the article wearing that ridiculous shirt.