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On a conjecture of Butler and Graham

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Abstract

Motivated by a hat guessing problem proposed by Iwasawa, Butler and Graham made the following conjecture on the existence of a certain way of marking the coordinate lines in [k]n: there exists a way to mark one point on each coordinate line in [k]n, so that every point in [k]n is marked exactly a or b times as long as the parameters (a, b, n, k) satisfies that there are nonnegative integers s and t such that s + t = k n and as + bt = nk n−1. In this paper we prove this conjecture for any prime number k. Moreover, we prove the conjecture for the case when a = 0 for general k.

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Authors and Affiliations

  1. Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China

    Tengyu Ma & Huacheng Yu

  2. Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China

    Xiaoming Sun

Authors
  1. Tengyu Ma
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  2. Xiaoming Sun
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  3. Huacheng Yu
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Corresponding author

Correspondence to Xiaoming Sun.

Additional information

Communicated by C. J. Colbourn.

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Ma, T., Sun, X. & Yu, H. On a conjecture of Butler and Graham. Des. Codes Cryptogr. 69, 265–274 (2013). https://doi.org/10.1007/s10623-012-9656-8

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  • DOI: https://doi.org/10.1007/s10623-012-9656-8

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