## The Coin Collection

A PuzzleCritic Original:

The Museum of Mathematical Mysteries houses a peculiar collection of coins, each shaped like a polygon, with the two largest proper factors of n inscribed on either face, where n is the number of edges the given coin possesses. For example, there is a hexagonal coin on which the numbers 2 and 3 are inscribed. The museum’s curator examines one of the coins and sees the number 15 inscribed on one face. Determine all numbers that might appear on the other face.

## The Vault: Robbers

Here I present one of my favourites; a problem of rare invention. From the 1998 Tournament of the Towns:

(Two-person case) Two robbers stole a bag of coins from a merchant. Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the two robbers (that is, they both get coins with the same total value in pennies).

Prove that after one coin is removed, the number of coins remaining is even.

(General case) A gang of robbers stole a bag of coins from a merchant. Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the robbers (that is, they all get coins with the same total value in pennies).

Prove that after one coin is removed, the number of coins remaining is divisible by the number of robbers.

## Numbers on a board

Here is a gentle thinker from the 2014 Grey Kangaroo:

Dean’s teacher asks him to write several different positive integers on the board. Exactly two of them are to be divisible by 2 and exactly 13 of them are to be divisible by 13. M is the greatest of these numbers.

What is the least possible value of M?

## The 1137 Shuffle

From the 2003 South African Maths Olympiad:

The first four digits of a positive integer n are 1137. Prove that the digits of n can be shuffled in such a way that the new number is divisible by seven.