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King's Maths School

Challenge 465: Arithmetic Acquisition

You might be familiar with Nim-style games - here's a gentle twist!

There is a pile of N counters. Player 1 starts by removing 1 counter. Player 2 then removes either 1 or 2 counters. Player 1 continues by removing 1, 2 or 3 counters. Player 2 continues by removing from 1-4 counters. The game continues until one player removes the last counter; they are the winner.

For what values of N does Player 1 win, assuming both players are perfect at the game?

Submit your solution

Please do send in your solution to this problem to weeklymaths@kcl.ac.uk You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide feedback on your solution, assuming that you are in year 11 or below. If you are older than this, we hope you enjoy trying the problems and reviewing your solutions against those we publish on the website.