Abstract
We study the \(\star \)-operator (Larsson et al. in Theoret. Comp. Sci. 412:8–10, 729–735, 2011) of impartial vector subtraction games (Golomb in J. Combin. Theory 1:443–458, 1965). Here, we extend the operator to the misère-play convention and prove convergence and other properties; notably, more structure is obtained under misère-play as compared with the normal-play convention (Larsson in Theoret. Comput. Sci. 422:52–58, 2012).
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Notes
- 1.
He also restricted the set of terminal positions to contain only 0, a definition not used in connection with the \(\star \)-operator.
- 2.
In one dimension, \(\min ({{\mathscr {M}}})\) consists of a single value and we sometimes abuse notation and write the minimal number instead of the set. If \({\mathscr {M}}=\varnothing \) then we define \(\min {\mathscr {M}}=\varnothing \).
- 3.
- 4.
The \(\star \)-operator is in fact an infinite class of operators, one operator for each dimension. However, we will refer to ‘the’ \(\star \)-operator because the operator acts in the same way in each dimension.
- 5.
An example of this case is \(x=12\in X(1)\) in Fig. 3.
- 6.
Examples of this case are \(x=5\in X(0)\), \(x=16\in X(2)\), and \(x=48\in X(3)\) in Fig. 3.
References
E.R. Berlekamp, J.H. Conway, R.K. Guy, Winning Ways, 2nd edn. (1–2 Academic Press, London 1982). 1–4. A. K. Peters, Wellesley/MA (2001/03/03/04)
C. Bloomfield, M. Dufour, S. Heubach, U. Larsson, Properties for the \(\star \)-operator of vector subtraction games (in preparation)
E. Duchêne, M. Rigo, Invariant games. Theoret. Comput. Sci. 411(34–36), 3169–3180 (2010)
S.W. Golomb, A mathematical investigation of games of “take-away”. J. Combin. Theory 1, 443–458 (1966)
U. Larsson, P. Hegarty, A.S. Fraenkel, Invariant and dual subtraction games resolving the Duchêne-Rigo Conjecture. Theoret. Comp. Sci. 412(8–10), 729–735 (2011)
U. Larsson, The \(\star \)-operator and invariant subtraction games. Theoret. Comput. Sci. 422, 52–58 (2012)
U. Larsson, J. Wästlund, From heaps of matches to the limits of computability. Electron. J. Combin. 20, 41 (2013)
Acknowledgements
Thanks to Lydia Ievins and Michale Bergman for making the trip to CANT 2016 possible.
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Dufour, M., Heubach, S., Larsson, U. (2017). A Misère-Play \(\star \)-Operator. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_12
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DOI: https://doi.org/10.1007/978-3-319-68032-3_12
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