My roommate challenged me with the following riddle, claiming he had a solution to the problem. After two weeks of being puzzled, I asked for a hint, and later for his solution. Unfortunately, I was able to point out a flaw in his inductive reasoning, and now neither of us have a solution. Presented is the (mathematically informal) riddle.

There are 15 buckets and 20 bottles. Each of the 15 buckets contains between 1 and 20 marbles (inclusive). Each of the 20 bottles contains between 1 and 15 marbles (inclusive). Prove (or counterexample), that no matter what the distribution of marbles happens to be, you can always select a non-empty subset of buckets and non-empty subset of bottles such that the sum of marbles in both of these subsets is equal.

Feel free to substitute (15,20) with the arbitrary positive integers (A,B). I have proven up to (5,N), but am searching for a more elegant proof in one exists.

  • $\begingroup$ Did you consider induction on A+B ? Gerhard "Sometimes Induction Makes It Easy" Paseman, 2015.06.01 $\endgroup$ Jun 2 '15 at 0:50
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    $\begingroup$ This is essentially Problem A-4 from the 1993 William Lowell Putnam Mathematical Competition (and a hard Putnam problem at that - only 5 of the top 200 solved it). You can find a solution by Kiran Kedlaya posted here written shortly after he took that test. $\endgroup$ Jun 2 '15 at 1:19
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    $\begingroup$ Seen before on math.SE math.stackexchange.com/questions/307517 $\endgroup$ Jun 2 '15 at 1:29

While I won't repeat the proof here, this question has been answered sufficiently. It was 1993 Putnam problem A-4, answered by 5 of the top 207 participants. I no longer feel bad about having spent more than 6 hours and a 30-minute lunch break working on this.


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