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A cube with an edge length of 10 is divided into two cuboids with integer edge lengths by a flat cut. Afterwards, one of those cuboids is again being divided into two smaller cuboids with integer edge lengths by a second flat cut.

What is the smallest possible volume of the biggest of the three cuboids?

I´m pretty sure the result is 350 but I couldn´t figure out a way to prove it (probably an extremum problem).

cosmo5
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Martin Müller
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    Please [edit] the question to tell us why you are "pretty sure", even though you don't have a proof. – Ethan Bolker Feb 07 '21 at 15:27
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    Wait another month. Deadline is 8th Mar 2021 for this problem from ongoing contest : https://math.meta.stackexchange.com/questions/32947/german-contest-bundeswettbewerb-mathematik-2021 – cosmo5 Feb 07 '21 at 15:32
  • You can get this result with an elimination process: For the first cut, eventually there are only 5 possibilities (10x10x9 & 10x10x1 , 10x10x8 & 10x10x2 , 10x10x7 & 10x10x3 , 10x10x6 & 10x10x4 , 10x10x5 & 10x10x5). If you take another cut of one of the two 10x10x5 cuboids, the volume of the biggest of the three cuboids will be 500. Thats why you need to examine the bigger cuboids in the other pairs and try to cut them in two exact halves in order to make them as sall as possible. Soon you will find out that this is only possible when dividing the 10x10x7 on a side with the length 10 – Martin Müller Feb 07 '21 at 15:39

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