## 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.

## Shares

From the 2004 Tournament of Towns:

Each day, the price of the shares of the corporation “Soap Bubble, Limited” either increases or decreases by n%, where n is an integer such that 0<n<100. The price is calculated with unlimited precision. Does there exist an n for which the price can take the same value twice?

## The Mathlon

A, B and C are the participants in a Mathlon – an athletics competition made up of several events. In each event x points are awarded for finishing 1st, y points for finishing 2nd, and z points for finishing 3rd, where x>y>z are positive integers. In the end, A scored 22 points, and B and C each scored 9 points. B won the 100 metres. How many events were there in total, and who finished second in the high jump?

## Queuing

From the 2011 South African Junior Olympiad:

Several people line up in single file. A solitary latecomer wishes to join the queue. Prove that it is always possible for them to join the line so that the number of men in front of them is equal to the number of women behind them.

## Binary strings

From the 2010 Australian Maths Challenge:

There are sixteen different ways of writing four-digit strings using 1s and 0s. Three of these strings are 1010, 0100 and 1001. These three can be found as substrings of 101001. There is a string of nineteen 1s and 0s which contains all sixteen strings of length 4 exactly once. If this string starts with 1111, what are the last four digits?