Odd Scoring

The applet below serves a playboard for a problem (Kvant, n 2, 1970, M8, p. 47) that I paraphrase the following way:

A chip is placed at the end of a grid band with N cells. On a move the chip is shifted leftwards 1, 2, or 3 steps. When it reaches the last cell, the total numbers of steps made by you and the computer are counted and the player who made an even number of steps is declared a winner.

The game is played against the computer. You move first.

To avoid a draw, the number of available steps is always odd. The idea is of course to devise a winning strategy.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

A few words

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The applet does implement a certain strategy which up to a certain point is exactly that of Scoring. However, cell #9 from the left is break point. From then on the number of steps you have accumulated so far - or rather its parity - is important.

It's not hard to determine the right strategy - see if you can beat the applet. Start from the left. A step away from the end gives you no choice. If you are there you have to make a single step and, with it, the game ends and the step count starts. Two steps away from the left end, you may make 1 or 2 steps. Choose the number that will make your total even. Etc.

The game can be played with a pile of odd objects from which the players remove 1, 2, or 3 items at a time. When all the items have been removed, each player counts the number of items in his/hers/its possession. The winner is the player with an even number of items.

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