We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 27 June 2025
The game of Top Entails introduces the curious theory of entailing moves. In Winning Ways, simple positions are analysed and stacks of one and three coins are shown to be loony positions, since any move leaving such a stack is a losing move, regardless of other stacks that might also be present. We analyse all stacks of up to 600,000 coins and find three more loony stacks: those having 2403, 2505 and 33,243 coins.
The game of Top Entails is introduced in Winning Ways [Berlekamp et al. 1982, p. 376]. It is a nim-like, impartial game played with stacks of coins and the following two types of move. A player may either (1) split just one stack into two strictly smaller stacks; or (2) remove just one coin from just one stack; in this case the opponent is entailed to reply with a move (of either type) in the resulting stack.
An unentailed stack of zero coins can never really occur in this game, since neither type of move can create one. Nevertheless, we can posit one as a hypothetical object; indeed, it is clear that it has value *0 = 0, since it hardly matters whether there is such a stack on the table or not!
What about an entailed stack of zero coins? Such a position can certainly occur, by an entailing (type 2) move from any stack containing just one coin. Since the next player, N, is now forced to move in an empty stack, the game ends instantly with a win for the previous player, P.
To save this book to your Kindle, first ensure no-reply@cambridge-org.demo.remotlog.com is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
