**From New Scientist #2617, 18th August 2007**

I started with a small rectangular block with one end a square, ABCD, and with a correspondingly labelled square at the other end, A*B*C*D*. I made two cuts through the block, one through the plane ACD* and the other through the plane A*C*D. This cut the block into four pieces, and I discarded the smallest three of those.

The volume of the remaining piece is a two-figure number of cubic centimetres. By coincidence, in that number each of the two digits was the length of one of the sides of the original rectangular block.

What, in cubic centimetres, is the volume of this remaining piece?

[enigma1456]

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Suppose the dimensions of the block are: AB = AD = BC = BD = A*B* = A*D* = B*C* = B*D* = a and AA* = BB* = CC* = DD* = b.

Then the volume of the complete block is v = a²b.

The volume of the pyramid removed by the cut ACD* is ⅓ × ½ a²b = v/6.

Similarly the volume of the pyramid removed by the cut A*C*D is v/6.

And the intersection of these two pyramids is a third pyramid DM

_{1}D*M_{2}, where M_{1}is the centre point of face AA*D*D and M_{2}is the centre point of face CC*D*D.It has a volume of ⅓ × (½ × ½ ab) × ½ a = v/24.

So the volume of the remaining piece of the block is v – 2 × v/6 + v/24 = (17/24) v.

The following Python program finds the dimensions on the block such that the dimensions form the digits of the volume of the remaining piece of the block. It runs in 39ms.

Solution:The remaining piece has a volume of 34 cu cm.