Is it possible to write a given positive integer as an arithmetic expression in which each of the numbers 1 through k appears exactly once? For instance, using the numbers 1 through 4 one may write 17=13+24. What is the smallest number not obtainable as such an expression? How many ways can a given number be expressed as such an arithmetic combination? The British television show Countdown has a numbers game in which players try to construct a 3-digit target number as an arithmetic combination of some randomly chosen smaller numbers. What strategies are best? In this note we explore some methods for investigating such matters.
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