Exact-Bound analyzes and optimal strategies for mastermind with a lie

  • Authors:

  • Li-Te Huang;Shan-Tai Chen;Shun-Shii Lin

  • Affiliations:

  • Graduate Institute of Computer Science and Information Engineering, National Taiwan Normal University, Taipei, Taiwan, R.O.C.;Department of Computer Science, Chung Cheng Institute of Technology, National Defense University, Tao-Yuan, Taiwan, R.O.C.;Graduate Institute of Computer Science and Information Engineering, National Taiwan Normal University, Taipei, Taiwan, R.O.C.

  • Venue:
  • ACG'05 Proceedings of the 11th international conference on Advances in Computer Games
  • Year:
  • 2006

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Abstract

This paper presents novel and systematic algorithms to solve a variant of the Mastermind game, which is called “Mastermind with a Lie”. Firstly, we use the k-way-branching(KWB) algorithm to get an upper bound of the number of guesses for the problem. With the help of clustering technique, the KWB algorithm is able to obtain near-optimal results effectively and efficiently. Secondly, we propose a fast backtracking(PPBFB) algorithm based on the pigeonhole principle to get the lower bounds of the number of guesses. That is a computer-aided approach, which is able to estimate the depth of the game tree and to backtrack when the depth is larger than a predefined value. Moreover, we also develop two novel methods, named “volume-renewing” and “preprocessing”. They can improve the precision in the estimation of the lower bound and speed up the game tree search. As a result of applying the KWB algorithm and the PPBFB algorithm, we are able to show that the upper bound is 7 and that is also the lower bound. Thus, the problem is solved completely and the exact bound of the number of guesses for the problem is 7.