We analyze the classic board game of Mastermind with n holes and a constant number of colors. The classic result of Chvatal (Combinatorica 3 (1983), 325-329) states that the codebreaker can find the secret code with Theta(n / log n) questions. We show that this bound remains valid if the codebreaker may only store a constant number of guesses and answers. In addition to an intrinsic interest in this question, our result also disproves a conjecture of Droste, Jansen, and Wegener (Theory of Computing Systems 39 (2006), 525-544) on the memory-restricted black-box complexity of the OneMax function class.
@InProceedings{doerr_et_al:LIPIcs.STACS.2012.441, author = {Doerr, Benjamin and Winzen, Carola}, title = {{Playing Mastermind With Constant-Size Memory}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {441--452}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.441}, URN = {urn:nbn:de:0030-drops-34112}, doi = {10.4230/LIPIcs.STACS.2012.441}, annote = {Keywords: Algorithms, Mastermind, black-box complexity, memory-restricted algorithms, query complexity} }
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