Home > Combinatorics, puzzles > El País’ weekly challenge

## El País’ weekly challenge

For a while now, the Spanish newspaper “El País” has been posing a weekly mathematical challenge to promote a collection of books, and celebrate a hundred years of their country’s Royal Mathematical Society.

The latest of these challenges—the fourth—is a beautiful problem in combinatorics:

Consider a clock, with its twelve numbers around a circle: $1, 2, \dotsc, 12.$ Color each of the twelve numbers in either blue or red, in such a way that there are exactly six in red, and six in blue. Proof that, independently of the order chosen to color the numbers, there always exists a line that divides the circle in two perfect halves, and on each half there will be exactly three numbers in red, and three numbers in blue.

While there are many different ways to solve this challenge, I would like to propose here one that is solely based upon purely counting techniques.