The Koebe function is defined on the unit circle of the complex plane by: \[f(x) = \frac{x}{(1 - x)^2}.\] Its Maclaurin series expansion tells that \[f(x) = \sum_{n = 1}^{\infty} n x^n,\] which allows us to interpret the Koebe function as the expected value of the following random variable \(Y\).

Assume that \(X_i\) is an infinite Bernoulli process with the probability that \(X_i = 1\) is \(p.\) Regarding its realization as a sequence of 0s and 1s, group the input in non-overwrapping consective bits of length \(1, 2, 3,\) etc. in this order. For example, if the input sequence is `011010111101100...`

, then grouping looks like `(0)(11)(010)(1111)(01100)...`

. Then discard the groups containing 0; e.g. the final output of the above example is `(11)(1111)...`

. Call the length of the remaining 1s \(Y\). Then \[\mathbb{E}[Y] = f(p).\]

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GNU Emacsにはgomokuという組み込みの五目並べゲームがありますが、そのコンピュータ側プレーヤーのメッセージが6歳児を泣かせるという事態が起こりました。

保護者がバグ報告したおかげで、よりmotivationalなメッセージに修正されました。その差分を見ると、確かに従来のコンピュータ側プレーヤーとはかなり性格が違っています。

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