{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T09:36:58Z","timestamp":1773740218504,"version":"3.50.1"},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2023,11,8]],"date-time":"2023-11-08T00:00:00Z","timestamp":1699401600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Since the 1960s Mastermind has been studied for the combinatorial and information-theoretical interest the game has to offer. Many results have been discovered starting with Erd\u0151s and R\u00e9nyi determining the optimal number of queries needed for two colours. For <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000366_inline1.png\"\/><jats:tex-math>\n$k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> colours and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000366_inline2.png\"\/><jats:tex-math>\n$n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> positions, Chv\u00e1tal found asymptotically optimal bounds when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000366_inline3.png\"\/><jats:tex-math>\n$k \\le n^{1-\\varepsilon }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Following a sequence of gradual improvements for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000366_inline4.png\"\/><jats:tex-math>\n$k\\geq n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> colours, the central open question is to resolve the gap between <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000366_inline5.png\"\/><jats:tex-math>\n$\\Omega (n)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000366_inline6.png\"\/><jats:tex-math>\n$\\mathcal{O}(n\\log \\log n)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000366_inline7.png\"\/><jats:tex-math>\n$k=n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In this paper, we resolve this gap by presenting the first algorithm for solving <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000366_inline8.png\"\/><jats:tex-math>\n$k=n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> Mastermind with a linear number of queries. As a consequence, we are able to determine the query complexity of Mastermind for any parameters <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000366_inline9.png\"\/><jats:tex-math>\n$k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000366_inline10.png\"\/><jats:tex-math>\n$n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1017\/s0963548323000366","type":"journal-article","created":{"date-parts":[[2023,11,8]],"date-time":"2023-11-08T01:27:55Z","timestamp":1699406875000},"page":"143-156","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["Mastermind with a linear number of queries"],"prefix":"10.1017","volume":"33","author":[{"given":"Anders","family":"Martinsson","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5614-962X","authenticated-orcid":false,"given":"Pascal","family":"Su","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,11,8]]},"reference":[{"key":"S0963548323000366_ref7","doi-asserted-by":"publisher","DOI":"10.1109\/SP.2009.4"},{"key":"S0963548323000366_ref13","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1965-034-2"},{"key":"S0963548323000366_ref14","doi-asserted-by":"publisher","DOI":"10.2307\/2310126"},{"key":"S0963548323000366_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2009.02.021"},{"key":"S0963548323000366_ref1","doi-asserted-by":"crossref","first-page":"42","DOI":"10.4153\/CJM-1966-007-2","article-title":"Determination of a subset from certain combinatorial properties","volume":"18","author":"Cantor","year":"1966","journal-title":"Can. J. Math."},{"key":"S0963548323000366_ref15","first-page":"25","article-title":"Mastermind is NP-complete","volume":"5","author":"Stuckman","year":"2006","journal-title":"INFOCOMP J. Comput. Sci."},{"key":"S0963548323000366_ref5","first-page":"229","article-title":"On two problems of information theory","volume":"8","author":"Erd\u0151s","year":"1963","journal-title":"Magyar Tud. Akad. Mat. Kutat\u00f3 Int. K\u00f6zl."},{"key":"S0963548323000366_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-30347-0_36"},{"key":"S0963548323000366_ref12","first-page":"195","article-title":"On a combinatory detection problem I","volume":"9","author":"Lindstr\u00f6m","year":"1964","journal-title":"I. Magyar Tud. Akad. Mat. Kutat\u00f3 Int. K\u00f6zl."},{"key":"S0963548323000366_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579188"},{"key":"S0963548323000366_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-13122-6_21"},{"key":"S0963548323000366_ref10","first-page":"1","article-title":"The computer as master mind","volume":"9","author":"Knuth","year":"1976\/77","journal-title":"J. Recreational Math."},{"key":"S0963548323000366_ref4","doi-asserted-by":"crossref","first-page":"ArticleNo:42","DOI":"10.1145\/2987372","article-title":"Playing Mastermind with many colors","volume":"63","author":"Doerr","year":"2016","journal-title":"J. ACM"},{"key":"S0963548323000366_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-61332-3_138"},{"key":"S0963548323000366_ref11","first-page":"251","article-title":"An optimal Mastermind strategy","volume":"25","author":"Koyama","year":"1993","journal-title":"J. Recreational Math."},{"key":"S0963548323000366_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2011.06.009"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548323000366","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T11:23:08Z","timestamp":1707132188000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548323000366\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,8]]},"references-count":16,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,3]]}},"alternative-id":["S0963548323000366"],"URL":"https:\/\/doi.org\/10.1017\/s0963548323000366","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,8]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}