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Cards, permutations, and sums

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Abstract:

We are given four cards, each containing four nonnegative real numbers, written one below the other, so that the sum of the numbers on each card is 1. We are allowed to put the cards in any order we like, then we write down the first number from the first card, the second number from the second card, the third number from the third card, and the fourth number from the fourth card, and we add these four numbers together. We wish to find real intervals $[a,b]$ with the following property: there is always an ordering of the four cards so that the above sum lies in $[a,b]$. We prove that the intervals $[1,2]$ and $[2/3,5/3]$ are solutions to this problem. It follows that $$[0,1], [1/3,4/3], [2/3,5/3], [1,2]$$ are the only minimal intervals which are solutions. We also discuss a generalization to $n$ cards, and give an equivalent formulation of our results in matrix terms.

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