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A Misère-Play \(\star \)-Operator

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 220)


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).


  • Combinatorial game
  • Game convergence
  • Game creation operator
  • Impartial game
  • Misére play
  • Star operator
  • Sum-free set

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  1. 1.

    He also restricted the set of terminal positions to contain only 0, a definition not used in connection with the \(\star \)-operator.

  2. 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. 3.

    Note that the \(\star \)-operator under misère rules is the same as the \(\star \)-operator in normal-play [5, 6]. However, since in misère-play \({0}\) is never a P-position, the definition simplifies in this case.

  4. 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. 5.

    An example of this case is \(x=12\in X(1)\) in Fig. 3.

  6. 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.


  1. 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)

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  2. C. Bloomfield, M. Dufour, S. Heubach, U. Larsson, Properties for the \(\star \)-operator of vector subtraction games (in preparation)

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  3. E. Duchêne, M. Rigo, Invariant games. Theoret. Comput. Sci. 411(34–36), 3169–3180 (2010)

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  4. S.W. Golomb, A mathematical investigation of games of “take-away”. J. Combin. Theory 1, 443–458 (1966)

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  5. 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)

    CrossRef  MATH  Google Scholar 

  6. U. Larsson, The \(\star \)-operator and invariant subtraction games. Theoret. Comput. Sci. 422, 52–58 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. U. Larsson, J. Wästlund, From heaps of matches to the limits of computability. Electron. J. Combin. 20, 41 (2013)

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Thanks to Lydia Ievins and Michale Bergman for making the trip to CANT 2016 possible.

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Correspondence to Urban Larsson .

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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.

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