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Martin Kutz über "Angels With Broken Wings" Abstract. We play a game with a chess king on an infinite checker board. The king moves across the board, according to the usual chess rules and we want to trap him by destroying squares of the board. Precisely, in each move the king steps onto an undamaged square adjacent to his current position and we in turn delete an arbitrary square from the board (except the one under the king, of course). Can you trap the king, that is, do you have a strategy such that at some point the king cannot move any more? Or is he able to run on forever? Berlekamp showed that you can trap him. In fact, the strategy is not very complicated. But the problem gets amazingly difficult if you increase the power of the king. Conway defined a k-angel to be a "king" that can "fly" in one move to any untouched square at distance at most k from his current position. He then asked whether the opponent--whom he figuratively calls "the devil"--can catch any angel of arbitrary power. This problem remains unsolved for at least two decades now. In this talk we will revisit some well-known, yet rather counterintuitive facts about possible escape strategies. Then we introduce a slight variant of the problem which turns out asymptotically equivalent. Under these refined conditions we are able to prove slight improvements upon Berlekamp's devil result revealing further peculiarities and possible fallacies of this "everlasting game." Exercise. As a preparation for the talk, everybody is invited to solve the original problem, i.e., catch the ordinary chess king! |