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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.