Hackenbush for Kids

Hackenbush for Kids

This game is presented in Berlycamp, Conway and Guy's book Winning Ways and is mathematically interesting as a game with positive or negative values (it's partesan) that may be infinitesimal, in contrast with the ordinary Hackenbush.

Draw a horizontal line as the “floor” e complete a figure using strokes of two colours, say blue and red. We'll say player A is blue (positive) and B is red (negative). So a figure that favours A will have a positive value, and one that favours B negative. If the figure favours whoever plays first it will have a fuzzy value, and if it favours the second to play it's value is zero.

As in Hackenbush, A and B alternate erasing one trace of their colour. A trace is considered connected to the floor if there is a sequence of strokes linking it to the floor. The rule of Hackenbush for Kids that distinguishes it from common Hackenbush is that no move may disconnect other parts of the figure from the floor: it's forbidden to create a disconnected body.

We'll call the value of a single isolated blue stroke +! and that of a red stroke -1. To attribute a value to a figure we do as done before in other sections. Consider the various options that each player may have and calculate the value of the result of that move. Between braces we put on the left the possible results of A and on the right those of B. The value of the figure will be the most simple “number” (real or surreal) contained between the highest left value and the smallest right value. The figure below, very simple indeed, will permit B to move leaving a figure of value 3 and A leaving one with value 2 – 1 = 1. So the configuration has value { 1 | 3 } = 2.

The figure below values zero. In fact, whovever begins the match will lose – confirming our definition of “value zero”.

{ -1 | 1} = 0

It won't always be that easy. The figure below allows only A de play, so the method says the value will be { -1 | }. What value is that? Zero – as whover begins will lose again!

{ -1 | }

For similar motives, the configuration below on the left is worth zero and the one to the right is worth – 1.

{ -3 | } = 0 { - 2 | 0, 1 } = - 1

More interesting are the following configurations.

{ -1, 0 | 1 } = ½ {-1, 0 | 0, 1} = *

On the right we have aconfiguration with a fuzzy value – something that does not exist in common Hackenbush. That means that the first player to play can win if she plays correctly. But its value is also such that the sum of it will any positive value whatsoever, for small as it may be, will be positive, and the sum with an arbitrarily small negative value is negative.