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Yellow-Brown Hackenbush
Published online by Cambridge University Press: 28 February 2011
Summary
This game is played on a sum of strings. In its “restricted” form, Left, at her turn, picks a bichromatic string and removes its highest yeLLow branch. Right, at his turn, picks a bichromatic string and removes its highest bRown branch. As in the well-known game of bLue-Red Hacken-bush, all higher branches, being disconnected, also disappear. But in yellow-brown Hackenbush, unlike blue-red Hackenbush, all moves on monochromatic strings are illegal. This makes all values of yellow-brown Hackenbush all-small.
This paper presents an explicit solution of restricted yellow-brown Hacken-bush. The values are sums of basic infinitesimals that have appeared in many other games found in Winning Ways.
Yellow-Brown (YB) Hackenbush is played on sums of strings. Each mixed string is played analogously to LR Hackenbush. Left can remove a yeLLow branch, and all other branches (if any) above it; Right can remove a bRown branch and all other branches (if any) above it. But yellow-brown strings differ from blue-red strings in an important respect:
Neither player is allowed to move on any monochromatic string.
That rule ensures that all stopping positions are 0, and that all YB values are infinitesimal.
There are (at least) two variations of YB Hackenbush: restricted and unrestricted. In the restricted variation, each player is allowed at most a single option on any string, namely, his branch which is highest above the ground. Although this restriction would have no effect on the values of LR Hackenbush, it has a major effect on YB Hackenbush. In the unrestricted YB Hackenbush, either player can move to 0 by playing his lowest branch, so all nonzero values are confused with 0.
We now present a complete solution for restricted YB strings.
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- Games of No Chance 3 , pp. 413 - 418Publisher: Cambridge University PressPrint publication year: 2009
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