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- Larsson, Urban, 1965, et al.
(author)
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From heaps of matches to the limits of computability
- 2013
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In: Electronic Journal of Combinatorics. - : The Electronic Journal of Combinatorics. - 1077-8926 .- 1097-1440. ; 20:3
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Journal article (peer-reviewed)abstract
- We study so-called invariant games played with a fixed number d of heaps of matches. A game is described by a finite list M of integer vectors of length d specifying the legal moves. A move consists in changing the current game-state by adding one of the vectors in M, provided all elements of the resulting vector are nonnegative. For instance, in a two-heap game, the vector (1, -2) would mean adding one match to the first heap and removing two matches from the second heap. If (1, -2) is an element of M, such a move would be permitted provided there are at least two matches in the second heap. Two players take turns, and a player unable to make a move loses. We show that these games embrace computational universality, and that therefore a number of basic questions about them are algorithmically undecidable. In particular, we prove that there is no algorithm that takes two games M and M' (with the same number of heaps) as input, and determines whether or not they are equivalent in the sense that every starting-position which is a first player win in one of the games is a first player win in the other.
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| 3. |
- Larsson, Urban, 1965, et al.
(author)
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Maharaja Nim : Wythoff's Queen meets the Knight
- 2013
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Other publication (other academic/artistic)abstract
- We relax the hypothesis of a recent result of A. S. Fraenkel and U. Peled on certain complementary sequences of positive integers. The motivation is to understand to asymptotic behavior of the impartial game of \emph{Maharaja Nim}, an extension of the classical game of Wythoff Nim. In the latter game, two players take turn in moving a single Queen of Chess on a large board, attempting to be the first to put her in the lower left corner, position $(0,0)$. Here, in addition to the classical rules, a player may also move the Queen as the Knight of Chess moves, still taking into consideration that, by moving no coordinate increases. We prove that the second player's winning positions are close to those of Wythoff Nim, namely they are within a bounded distance to the half-lines, starting at the origin, of slope $\frac{\sqrt{5}+1}{2}$ and $\frac{\sqrt{5}-1}{2}$ respectively. We encode the patterns of the P-positions by means of a certain \emph{dictionary process}, thus introducing a new method for analyzing games related to Wythoff Nim. Via Post's Tag productions, we also prove that, in general, such dictionary processes are algorithmically undecidable.
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| 4. |
- Larsson, Urban, 1965, et al.
(author)
-
Maharaja Nim
- 2013
-
Journal article (other academic/artistic)abstract
- We relax the hypothesis of a recent result of A. S. Fraenkel and U. Peled on certain complementary sequences of positive integers. The motivation is to understand to asymptotic behavior of the impartial game of \emph{Maharaja Nim}, an extension of the classical game of Wythoff Nim. In the latter game, two players take turn in moving a single Queen of Chess on a large board, attempting to be the first to put her in the lower left corner, position $(0,0)$. Here, in addition to the classical rules, a player may also move the Queen as the Knight of Chess moves, still taking into consideration that, by moving no coordinate increases. We prove that the second player's winning positions are close to those of Wythoff Nim, namely they are within a bounded distance to the half-lines, starting at the origin, of slope $\frac{\sqrt{5}+1}{2}$ and $\frac{\sqrt{5}-1}{2}$ respectively. We encode the patterns of the P-positions by means of a certain \emph{dictionary process}, thus introducing a new method for analyzing games related to Wythoff Nim. Via Post's Tag productions, we also prove that, in general, such dictionary processes are algorithmically undecidable.
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