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.
This research was funded by a grant NSC 93-2213-E-003-001 from National Science Council, R.O.C.
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© 2006 Springer-Verlag Berlin Heidelberg
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Huang, LT., Chen, ST., Lin, SS. (2006). Exact-Bound Analyzes and Optimal Strategies for Mastermind with a Lie. In: van den Herik, H.J., Hsu, SC., Hsu, Ts., Donkers, H.H.L.M.(. (eds) Advances in Computer Games. ACG 2005. Lecture Notes in Computer Science, vol 4250. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11922155_15
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DOI: https://doi.org/10.1007/11922155_15
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