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I claim that Player A has a winning strategy in your game, and furthermore, it is a winning strategy for her simply to play the smallest available number. Let me consider the game along with several …

Consider the following open knight's tour on a $5\times 5$ board, starting at position $1$ and then touring the $5\times 5$ board in the indicated move order. The final position is $25$, from which th …

Update. (Oct 28, 2015) See below, for a position with game value $\omega^4$. This is a great question, which I have been pondering for some time. I have just completed a joint article Transfinit …

There is a positive solution for the decidability of the mate-in-$n$ version of the problem. Many of us are familiar with the White to mate in 3 variety of chess problems, and we may consider the na …

This is a really great question! Previous attempts to make sense of infinite Go have sometimes had problems because it wasn't clear how to define the winner of a game of Go after transfinite play. T …

I have a question about the Chocolatier's game, which I had introduced in my recent answer to a question of Richard Stanley. To recap the game quickly, the Chocolatier offers up at each stage a finite …

Consider the game of infinite Hex, where two players Red and Blue alternately place their stones on the infinite hex grid, each aiming to create a winning configuration. Red wins after infinite play, …

I don't know about the game attributed to Specker, but here is a simple game with your desired features. Let us call it the Chocolatier's game. There are two players, the Chocolatier and the Glutton. …

Consider the arithmetic progression game, a two-player game of perfect information, in which the players take turns playing natural numbers, or finite sets of natural numbers, all distinct, and the fi …

The game of infinite Hex, proceeding from an arbitrary position, is a good example with all the features you seek. The game was the subject of my Oxford student Davide Leonessi's masters MFoCS dissert …

This is a great question! I've now managed to eliminate the use of countable choice. Theorem. Without using any choice principle, it follows that player I can have no winning strategy in the game. …

In ZFC, the player aiming for the empty set has a winning strategy in the game played on any infinite set, including the reals. Using the axiom of choice, we can well-order the set and thereby pretend …

The answer to the topological version of Banach-Mazur games is negative, proved by Gabriel Debs in 1985: Debs, Gabriel, Stratégies gagnantes dans certains jeux topologiques (Winning strategies in cer …

Consider the Sudoku Solver-Spoiler game, a natural variation of the Sudoku game recently appearing in the question Who wins two-player Sudoku? posted by user PyRulez. In that game, the players attempt …

I explain in my answer to Richard Stanley's question on the decidability of infinite chess that this is precisely how Dan Brumleve, Philip Schlicht and I proved that the mate-in-$n$ problem of infinit …