An Unusual Sequence

From the 2003 Tournament of the Towns:

Each term in a sequence of positive integers is obtained from the previous term by adding it to its largest digit. What is the greatest possible number of successive odd terms in such a sequence?

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.

The Vault: Powers of two

My new book of maths puzzles is on its way! It’s packed full of interesting problems to sink your teeth into. I’ll post an update as the launch approaches.

In the meantime, here is a fantastic problem from the USA:

It is given that $2^{2004}$ is a 604-digit number beginning with a 1. How many of the numbers $2^0, 2^1, 2^2, 2^3, ..., 2^{2003}$ begin with a 4?

A Game of Stones

A PuzzleCritic Original:

There are 999 stones in a pile. Amisi and Boaz take it in turns removing either 3 or 5 stones from the pile, with Amisi going first, until no more moves are possible. The last player to make a move wins. Which player can guarantee victory?

L’s on a square

Happy New Year! From the 2004 Georg Mohr Contest:

Find all positive integers n such that a 2n x 2n chessboard can be covered by non-overlapping L-pieces, each covering 4 squares. Rotations and reflections are allowed.