Maths Competition Question - Algebra/Number Theory Twelve cowboys sit in a circle around a bonfire. Each observes that his age (viewed as an integer) is the average of the ages of his left and right neighbours. Which of the following could be the sum of their ages? A. 224 B. 226 C. 227 D. 228

They must all be the same age, for if someone A is sitting next to two people of diferent ages, then one of those must be older, and on the other side of the person that is older than A must be someone that is older yet, etc. But with a circle, ages can't keep rising, for eventually you get back to A.

So, the sum must be a multiple of 12, which is 228.