The following is a simple activity for learning a simple number theory concept. This is based on the procedure described by Martin Gardner in Time Travel and Other Mathematical Bewilderments (1988).
All you need for this is a pack of cards. First, remove all the cards of one suit (e.g. all the spades). Afterwards, arrange the cards face down in a row in an ascending numerical order (Ace = 1, Jack = 11, Queen = 12 and King = 13), from left to right. Now, do the following steps:
- Turn each of the cards over.
- Turn over every second card (2, 4, 6, etc.).
- Turn over every third card (3, 6, 9, 12), every fourth card (4, 8, 12), then every fifth card (5, 10).
- Continue in this fashion until you have turned over the last card (the thirteenth card in this case).
After the procedure, all the cards except for the ace, four and nine would be face down.
Do you see a pattern here? All the numerical values of the cards that are face up are squares (1, 4 and 9).
You might think that this is a coincidence but it’s not. You can try to use 30 blank cards and number them from 1 to 30 and do the procedure described above. After turning over the 30th card, you would see that the card numbered 1, 4, 9, 16 and 25 are the only cards face up.
So, What’s Going On?
I told you earlier that this activity will teach you a simple number theorem, right? The concept that this activity shows is that every positive integer has an even number of divisors (the divisors include 1 and the number itself) except for square numbers. For example, the divisors of number 10 (a non-square number) are 1, 2, 5 and 10, a total of four divisors while the number 16 (a square number) has five divisors, namely, 1, 2, 4, 8 and 16. Every prime number also has an even number of divisors since it is divisible by 1 and by itself. This means that all non-square numbers have an even number of divisors while square numbers have an odd number of divisors.
So, how was the activity related to the concept? When you are turning over the cards, the cards are turned over equivalent to the number of their numerical values’ divisors. For example, the 7 of Spades was turned two times because 7 has two divisors, 1 and 7. Since all the cards were face down in the beginning, when you turned the cards an even number of times, the cards would be face down at the end. However, if you turned the cards an odd number of times (e.g. the four of Spades was turned three times since 4 has three divisors, 1, 2, and 4), the cards would end up being face up. Hence, all the cards representing the square numbers were face up after the procedure.