{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T10:00:19Z","timestamp":1780480819174,"version":"3.54.1"},"reference-count":19,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T00:00:00Z","timestamp":1731456000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>Given integer n and k such that 0&lt;k\u2264n and n piles of stones, two players alternate turns. On each move, a player is allowed to choose any k piles and remove exactly one stone from each. The player who has to move but cannot is the loser in the normal version of the game and (s)he is the winner in the mis\u00e8re version. Cases k=1 and k=n are trivial. For k=2, the game was solved for n\u22646. For n\u22644, the Sprague\u2013Grundy function was efficiently computed (for both versions). For n=5,6, a polynomial algorithm computing P-positions was obtained for the normal version. Then, for the case k=n\u22121, a very simple explicit rule that determines the Smith remoteness function was found for the normal version of the game: the player who has to move keeps a pile with the minimum even number of stones; if all piles have an odd number of stones, then (s)he keeps a maximum one, while the n\u22121 remaining piles are reduced by one stone each in accordance with the rules of the game. Computations show that the same rule works efficiently for the mis\u00e8re version too. The exceptions are sparse. We list some. Denote a position by x=(x1,\u2026,xn). Due to symmetry, we can assume wlog that x1\u2264\u2026\u2264xn. Our computations partition all exceptions into the following three families: x1 is even, x1=1, and odd x1\u22653. In all three cases, we suggest formulas covering all found exceptions, but it is not proven that there are no others.<\/jats:p>","DOI":"10.3390\/g15060037","type":"journal-article","created":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T06:23:16Z","timestamp":1731478996000},"page":"37","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On Remoteness Functions of k-NIM with k + 1 Piles in Normal and in Mis\u00e8re Versions"],"prefix":"10.3390","volume":"15","author":[{"given":"Vladimir","family":"Gurvich","sequence":"first","affiliation":[{"name":"Higher School of Economics, National Research University, 101978 Moscow, Russia"},{"name":"Rutgers Center for Operations Research (RUTCOR), Rutgers University, Piscataway, NJ 08854, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vladislav","family":"Maximchuk","sequence":"additional","affiliation":[{"name":"Higher School of Economics, National Research University, 101978 Moscow, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Georgy","family":"Miheenkov","sequence":"additional","affiliation":[{"name":"Higher School of Economics, National Research University, 101978 Moscow, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mariya","family":"Naumova","sequence":"additional","affiliation":[{"name":"Rutgers Business School, Rutgers University, Piscataway, NJ 08854, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Albert, M.H., Nowakowski, R.J., and Wolfe, D. (2007). Lessons in Play: An Introduction to Combinatorial Game Theory, A. K. Peters Ltd.. [2nd ed.].","DOI":"10.1201\/b10691"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Berlekamp, E.R., Conway, J.H., and Guy, R.K. (2001\u20132004). Winning Ways for Your Mathematical Plays, A.K. Peters. [2nd ed.].","DOI":"10.1201\/9780429487309"},{"key":"ref_3","unstructured":"Conway, J.H. (1976). On Numbers and Games, Academic Press."},{"key":"ref_4","unstructured":"Siegel, A.N. (2023). Combinatorial Game Theory, American Mathematical Society."},{"key":"ref_5","unstructured":"Gurvich, V., Martynov, D., Maximchuk, V., and Vyalyi, M. (2024). On remoteness functions of exact slow k-NIM with k + 1 piles. Integers."},{"key":"ref_6","first-page":"5777","article-title":"Slow k-Nim","volume":"3","author":"Gurvich","year":"2015","journal-title":"RUTCOR Res. 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Game Theory"}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/15\/6\/37\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:31:22Z","timestamp":1760113882000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/15\/6\/37"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,13]]},"references-count":19,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["g15060037"],"URL":"https:\/\/doi.org\/10.3390\/g15060037","relation":{},"ISSN":["2073-4336"],"issn-type":[{"value":"2073-4336","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,13]]}}}