Some Remarks on End-Nim

We reexamine Albert and Nowakowski's variation on the game of Nim, called End-Nim, in which the players may only remove coins from the leftmost or rightmost piles. We reformulate Albert and Nowakowski's solution to this game. We examine its misère version and a further variant where the wi...

Full description

Saved in:
Bibliographic Details
Published in:International journal of combinatorics 2011-12, Vol.2011 (2011), p.1-9
Main Authors: Cairns, Grant, Ho, Nhan Bao
Format: Article
Language:English
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by
cites cdi_FETCH-LOGICAL-c1302-8b0e1c29c07e7c5adb846f6e814a630ffb462e621c97a44f54503ffbf2052c9b3
container_end_page 9
container_issue 2011
container_start_page 1
container_title International journal of combinatorics
container_volume 2011
creator Cairns, Grant
Ho, Nhan Bao
description We reexamine Albert and Nowakowski's variation on the game of Nim, called End-Nim, in which the players may only remove coins from the leftmost or rightmost piles. We reformulate Albert and Nowakowski's solution to this game. We examine its misère version and a further variant where the winner is the player who reduces the game to a single pile; we call this Loop-End-Nim. We show that the three games, End-Nim, misère-End-Nim, and Loop-End-Nim, all have the same losing positions, except for the positions where all the piles are of equal size. We also give some partial results concerning the higher Sprague-Grundy values of the three games.
doi_str_mv 10.1155/2011/824742
format article
fullrecord <record><control><sourceid>emarefa_cross</sourceid><recordid>TN_cdi_crossref_primary_10_1155_2011_824742</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>501088</sourcerecordid><originalsourceid>FETCH-LOGICAL-c1302-8b0e1c29c07e7c5adb846f6e814a630ffb462e621c97a44f54503ffbf2052c9b3</originalsourceid><addsrcrecordid>eNqFjz1LxEAQhhdR8DivsrASUivrzcx-ppTj_IBDwY86bDa7GDWJZAvx37shcq3TzDA8vLwPY6cIV4hKrQkQ15akkXTAFqit4SUaPNzfWhyzVUrvkMdmmMyCnT0PXSieQufGj1QMfbHtG_7QdifsKLrPFFZ_e8leb7Yvmzu-e7y931zvuEcBxG0NAT2VHkwwXrmmtlJHHSxKpwXEWEtNQRP60jgpo5IKRP5GAkW-rMWSXc65fhxSGkOsvsY2l_mpEKpJq5q0qlkr0xcz_db2jftu_4HPZ3iSC9HtYQUI1opfOzVSxA</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Some Remarks on End-Nim</title><source>Wiley-Blackwell Open Access Collection</source><source>Publicly Available Content Database</source><creator>Cairns, Grant ; Ho, Nhan Bao</creator><contributor>Hattingh, Johannes</contributor><creatorcontrib>Cairns, Grant ; Ho, Nhan Bao ; Hattingh, Johannes</creatorcontrib><description>We reexamine Albert and Nowakowski's variation on the game of Nim, called End-Nim, in which the players may only remove coins from the leftmost or rightmost piles. We reformulate Albert and Nowakowski's solution to this game. We examine its misère version and a further variant where the winner is the player who reduces the game to a single pile; we call this Loop-End-Nim. We show that the three games, End-Nim, misère-End-Nim, and Loop-End-Nim, all have the same losing positions, except for the positions where all the piles are of equal size. We also give some partial results concerning the higher Sprague-Grundy values of the three games.</description><identifier>ISSN: 1687-9163</identifier><identifier>EISSN: 1687-9171</identifier><identifier>DOI: 10.1155/2011/824742</identifier><language>eng</language><publisher>Cairo, Egypt: Hindawi Puplishing Corporation</publisher><ispartof>International journal of combinatorics, 2011-12, Vol.2011 (2011), p.1-9</ispartof><rights>Copyright © 2011 Grant Cairns and Nhan Bao Ho.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c1302-8b0e1c29c07e7c5adb846f6e814a630ffb462e621c97a44f54503ffbf2052c9b3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27900,27901</link.rule.ids></links><search><contributor>Hattingh, Johannes</contributor><creatorcontrib>Cairns, Grant</creatorcontrib><creatorcontrib>Ho, Nhan Bao</creatorcontrib><title>Some Remarks on End-Nim</title><title>International journal of combinatorics</title><description>We reexamine Albert and Nowakowski's variation on the game of Nim, called End-Nim, in which the players may only remove coins from the leftmost or rightmost piles. We reformulate Albert and Nowakowski's solution to this game. We examine its misère version and a further variant where the winner is the player who reduces the game to a single pile; we call this Loop-End-Nim. We show that the three games, End-Nim, misère-End-Nim, and Loop-End-Nim, all have the same losing positions, except for the positions where all the piles are of equal size. We also give some partial results concerning the higher Sprague-Grundy values of the three games.</description><issn>1687-9163</issn><issn>1687-9171</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2011</creationdate><recordtype>article</recordtype><recordid>eNqFjz1LxEAQhhdR8DivsrASUivrzcx-ppTj_IBDwY86bDa7GDWJZAvx37shcq3TzDA8vLwPY6cIV4hKrQkQ15akkXTAFqit4SUaPNzfWhyzVUrvkMdmmMyCnT0PXSieQufGj1QMfbHtG_7QdifsKLrPFFZ_e8leb7Yvmzu-e7y931zvuEcBxG0NAT2VHkwwXrmmtlJHHSxKpwXEWEtNQRP60jgpo5IKRP5GAkW-rMWSXc65fhxSGkOsvsY2l_mpEKpJq5q0qlkr0xcz_db2jftu_4HPZ3iSC9HtYQUI1opfOzVSxA</recordid><startdate>20111229</startdate><enddate>20111229</enddate><creator>Cairns, Grant</creator><creator>Ho, Nhan Bao</creator><general>Hindawi Puplishing Corporation</general><general>Hindawi Publishing Corporation</general><scope>ADJCN</scope><scope>AHFXO</scope><scope>RHU</scope><scope>RHW</scope><scope>RHX</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20111229</creationdate><title>Some Remarks on End-Nim</title><author>Cairns, Grant ; Ho, Nhan Bao</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c1302-8b0e1c29c07e7c5adb846f6e814a630ffb462e621c97a44f54503ffbf2052c9b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2011</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Cairns, Grant</creatorcontrib><creatorcontrib>Ho, Nhan Bao</creatorcontrib><collection>الدوريات العلمية والإحصائية - e-Marefa Academic and Statistical Periodicals</collection><collection>معرفة - المحتوى العربي الأكاديمي المتكامل - e-Marefa Academic Complete</collection><collection>Hindawi Publishing Complete</collection><collection>Hindawi Publishing Subscription Journals</collection><collection>Hindawi Publishing Open Access</collection><collection>CrossRef</collection><jtitle>International journal of combinatorics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Cairns, Grant</au><au>Ho, Nhan Bao</au><au>Hattingh, Johannes</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Some Remarks on End-Nim</atitle><jtitle>International journal of combinatorics</jtitle><date>2011-12-29</date><risdate>2011</risdate><volume>2011</volume><issue>2011</issue><spage>1</spage><epage>9</epage><pages>1-9</pages><issn>1687-9163</issn><eissn>1687-9171</eissn><abstract>We reexamine Albert and Nowakowski's variation on the game of Nim, called End-Nim, in which the players may only remove coins from the leftmost or rightmost piles. We reformulate Albert and Nowakowski's solution to this game. We examine its misère version and a further variant where the winner is the player who reduces the game to a single pile; we call this Loop-End-Nim. We show that the three games, End-Nim, misère-End-Nim, and Loop-End-Nim, all have the same losing positions, except for the positions where all the piles are of equal size. We also give some partial results concerning the higher Sprague-Grundy values of the three games.</abstract><cop>Cairo, Egypt</cop><pub>Hindawi Puplishing Corporation</pub><doi>10.1155/2011/824742</doi><tpages>9</tpages><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier ISSN: 1687-9163
ispartof International journal of combinatorics, 2011-12, Vol.2011 (2011), p.1-9
issn 1687-9163
1687-9171
language eng
recordid cdi_crossref_primary_10_1155_2011_824742
source Wiley-Blackwell Open Access Collection; Publicly Available Content Database
title Some Remarks on End-Nim
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-02-24T21%3A18%3A12IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-emarefa_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Some%20Remarks%20on%20End-Nim&rft.jtitle=International%20journal%20of%20combinatorics&rft.au=Cairns,%20Grant&rft.date=2011-12-29&rft.volume=2011&rft.issue=2011&rft.spage=1&rft.epage=9&rft.pages=1-9&rft.issn=1687-9163&rft.eissn=1687-9171&rft_id=info:doi/10.1155/2011/824742&rft_dat=%3Cemarefa_cross%3E501088%3C/emarefa_cross%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c1302-8b0e1c29c07e7c5adb846f6e814a630ffb462e621c97a44f54503ffbf2052c9b3%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_id=info:pmid/&rfr_iscdi=true