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In terms of mathematical analysis and combinatorial game theory, the essence of any game is captured by its game tree, the tree whose nodes represent the current game state, and to make a move in the …

My question concerns the (completely deterministic) card game known as War, played by seven-year-olds everywhere, such as my son Horatio, and sometimes also by others, such as their fathers. The ques …

Consider the following simplified example of the same phenomenon, which many students find clarifying. Let $f(n)=1$, if there are $n$ consecutive $7$s in the decimal expansion of $\pi$, and otherwise …

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 …

This phenomenon extends beyond the traveling salesman problem, and even beyond NP, for there are even some undecidable problems with the feature that most instances can be solved very quickly. There …

It's a great problem! Theorem. The mathematicians have a winning strategy in the game for every ordinal $\alpha$. Proof. Let's prove the theorem by transfinite induction. Suppose that the mathematic …

This question arises from my recent visit to my daughter's second-grade class, where I led some discussion and activities on graph coloring (see Math for seven-year-olds). In one such activity, each c …

A partial order $\mathbb{B}$ is universal for a class $\cal{P}$ of partial orders if every order in $\cal{P}$ embeds order-preservingly into $\mathbb{B}$. For example, every partial order $\langle\ma …

Many chess positions that one may easily set up on a chess board are impossible to achieve in a game of legal moves. For example, among the impossible situations would be: A position in which both k …

Apart from your specific example, the idea of truth-by-accident has been studied in the context of formal first-order languages, which includes the language of graph theory, and in his dissertation, K …

This question spins off of Gil Kalai's recent question on Conway's game of life for a random initial configuration. There are numerous configurations in the game of life that are known to be stable …

Update. A research collaboration growing out of this question and some of its answers has now resulted in the following article, providing an account of the rearrangement number: A. Blass, J. Bren …

This is too long for a comment, so I am placing it here. In this article of the Notices of the AMS, Gonthier describes a full formal proof of the four-color theorem, which makes explicit every logica …

My example is the classical proof that sqrt(2) is irrational. More generally, many proofs that proceed by showing that there are no minimal counterexamples exemplify your phenomenon. The method of no …

Update. I've improved the argument to use only the consistency of $T$. (2/7/12): I corrected some over-statements previously made about Robinson's Q. I claim that for every statement $\varphi$, the …