# NCTM Circle Problem

If you’ve been here before, you probably know I really like the NCTM problems they post on their Twitter and Facebook pages.  Well they posted one last week that I just couldn’t seem to stop thinking about (doesn’t that usually mean you’ve got a good question on your hands!?!)

In case you missed it, here was the question:

What’s the value of the question mark?

The first time I saw the picture I immediately noticed that 4 and 8 were on the circle and that they were only one wedge apart.  I really, really wanted the question mark to be worth 16, but then that would mean that the 13 should have been a 32 and that the 23 should have been a 64 and that the 92 should have been a 128 . . . clearly I was wrong.  (P.S. can you tell what I was going in order to fill in the wedges of the circle?).

Then, I noticed that there were a few prime numbers on the circle (remember how to check to see if a number is prime or composite (via Math Warriors)?).

Here’s my tip . . . anytime you’re looking for a pattern in a situation prime numbers should make you a little suspicious.  You should be suspicious because they don’t mix well with other numbers, when considering “typical” pattern generators such as multiplication, division, or power properties.

Given my prime number predicament I felt like I had two choices . . . I could look for a pattern generated by adding or subtracting the numbers or I could look at the prime factors of the numbers in the circle and see what happened.  I wasn’t really compelled to try the whole add/subtract thing because from looking at the entries, that didn’t seem very likely.

Instead rewrote the numbers on the circle as a product of their prime factors and I came up with something that looked like this:

Based on my little diagram, I was able to fill in the missing value.

If you happened to come across this question via Twitter or Facebook last week, you know there were two common answers floating around.  The first common answer was 120.  The second was 79.

1. How did people arrive at both of these answers?

2. Do you think that one is “right” and one is “wrong”?  If so, why?  Take a side and convince me the other answer is incorrect.

Another point that was brought up online was that if you started at different points on the circle, you could get a different question mark value.

3. What do you think about that argument?