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

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.