Cynthia Lanius

Algebra - More Fun with Calendars

Another fun mathematical puzzle to play with your friends.
(Or teachers, with your class.)

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The Trick: Take any calendar. Choose 9 days that form a square like the ones to the right. Add all nine numbers and then divide by the number in the middle.

The Answer: You always get nine.

How It Works: You may be thinking that this is one of those special properties of the number nine, but it's not. No, this is another excellent example of the use of algebra. Here's what I mean.

Let's call the number in the center n. Then the nine numbers would be

n-8 n-7 n-6
n-1 n n+1
n+6 n+7 n+8

To do the trick, I had you add the numbers, so let's add these numbers.

n - 8 + n - 7 + n - 6 + n - 1 + n + n + 1 + n + 6 + n + 7 + n + 8

Then we divided by the number in the middle, which in this case is n, but first, let's simplify the expression by adding like terms.


Now let's divide by the number in the middle, (n) and we get 9. Always.

Why It Works: What we are really doing is averaging. Look back at this step which is the simplified sum of all nine numbers.


If I had divided that by nine, I would have got the average of all nine numbers, which is n, the number in the middle. Divide by the number in the middle, you get 9.

Explore: Where else might this work? Could you tweak the trick for the 3 X 5 calendar pattern below?

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Thanks to Joseph J. Castanza, Ph.D. for sending me this calendar "trick".

Other Math Lessons by Cynthia Lanius

Copyright 2008 by Cynthia Lanius.