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Cheese Cut Calculation (Posted on 2016-03-03) Difficulty: 3 of 5
I have another 10cm x 13cm x 14cm block of cheese just like in Cheese Cut Conclusion. Again I am slicing ten 1cm slices off of the block, all parallel to the faces like the original problem.

How many different sizes can the remaining block be after all ten slices are removed?

  Submitted by Brian Smith    
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The sum of the edge lengths of the original block is 10+13+14 = 37cm. This total is reduced by 10 after the cuts which means that the edges of the remaining block sum to 37-10 = 27 cm.

The largest dimension of the remaining block must be at most 14cm. The smallest dimension can be any amount not exceeding the largest dimension. To see this consider removing ten 13x14 slices. That removes all of the block leaving nothing, or a 0x13x14 degenerate block. Since 0 cm is an option then all thicknesses are viable.

If the smallest edge is 1cm then the remaining edges are 12x14 or 13x13, for two possibilities. 2cm yields 14x11 or 13x12 for the other two dimensions, again 2 possibilities. Similarly 3cm has 3 possibilities from 14x10 to 12x12, 4cm has 3 possibilities 14x9 to 12x11, 5cm has 4 possibilities from 14x8 to 11x11, 6 cm has 4 possibilities 14x7 to 11x10.

At this point a 14cm edge is no longer possible now that the smallest edge is getting fairly thick. 7cm has 4 possibilities 13x7 to 10x10, 8 cm has 2 possibilities 11x8 and 10x9, and 9cm has one possibility 9x9.

The total number of possibilities is 2+2+3+3+4+4+4+2+1 = 25 possible sizes for the remaining block (ignoring a degenarate block of zero volume).

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  Subject Author Date
Solutioncomputer solutionCharlie2016-03-03 11:31:38
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