Mark:
SMALL RECTANGULAR BLOCK
S = length of side of square end
L = third dimension of rectangular box
With a little bit of calculus, we get:
V = (17 * L * S^2) / 24
To eliminate the 24 in the denominator, we need L * S^2 to be a multiple of 24. If L = 3 and S = 4, then V = 34.
Mark, you are indeed a chip off the tree of knowledge. For the rest of us:
First, let the length of a square side be s, and the other length be r, so the block's volume is V = rs^2. Also let the midpoint of DD* be E.
Making JUST the first cut would remove a triangular based pyramid ACDD* whose volume is V/6 (volume of pyramid is base-area * height / 3).
Making JUST the second cut would remove an identical pyramid A*C*D*D.
However, when making both cuts, the cutting planes intersect, say at FG, giving a triangle EFG, halfway along the block, defining two pyramids, EFGD and EFGD*, each of which is similar to ACDD*, but half-size in each dimension, hence with volume V/48. These two small pyramids are contained in both the large pyramids.
The four pieces are thus:
DFGD* consisting of the two small pyramids, with volume V/24.
ACDFG being the pyramid ACDD* with DFGD* removed and with volume V/6-V/24 = V/8.
A*C*D*FG, being the pyramid A*C*D*D with DFGD* removed and also with volume V/8.
ABCFGA*B*C*, "the remaining piece", with volume V - 2V/8 - V/24 = 17V/24.
For 17/24 rs^2 to be an integer, rs^2 must be a multiple of 24, so at least one of r and s must be a multiple of 3, and at least one must be a multiple of 2.
17/24 rs^2 must therefore be a two-digit multiple of 17 with at least one digit being a multiple of 3 and at least one a multiple of 2.
34 and 68 are possibilities, and by inspection, r=3, s=4, volume=34 is the solution.
Thank you Miss mi for the update on Valter the pregnant Pole - sweet dreams.
Ksanfo conundrums:
GUM PRESIDENT = Toothless President?
Today I would like to remember one of our number who has been missing far too long. I am referring to Gustavratzenhofer; or the ?'whipper snapper' as he is referred to in my village. I therefore re-print his one and only contribution to this forum:
"
Ok, there's a guy in a straw hat standing, legs apart, both fists clenched tightly. Inside of each hand there is a bee. They are alive; you can clearly hear them buzzing."
What does the guy have in his eye
UU
OPUER TAICMEE
that whole blurb there was just way too much effort - bet it goes over like a pregnant pole vaulter...k. Going to bed...I am way too chatty. 
(ouch)
