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Cutting Edge (Posted on 2007-09-18) Difficulty: 3 of 5
Start with a rectangular solid, two ends of which are squares. Label the vertices of one square A, B, C, D. The square opposite this is labeled A', B', C', D', with A' connected to A by an edge of the solid, and so forth.

Make a cut through plane ACD', and another cut through A'C'D. This results in the block being divided into four pieces. Discard the smallest three pieces. The volume of the remaining piece is a two-digit number of cubic centimeters. Each of the two digits happens to be one of the dimensions of the original rectangular solid, in centimeters.

What is that volume of that largest piece?

  Submitted by Charlie    
Rating: 4.0000 (2 votes)
Solution: (Hide)
Let the length be L and the width W. Each, we are told, is a one-digit number.

The volume of the whole block is W^2 * L.

The volume cut off by the first cut is that of a pyramid whose base is half the square face and whose height is L, and is thus L*W^2 / 6.

The volume cut off by the second cut would be the same as the first cut except for the portion occupied by both pyramids.

The portion occupied by both pyramids is a tetrahedron whose vertices are D, D', M, M', where M is the center of rectangle DAA'D', and M' is the center of rectangle DCC'D'. Considered as a pyramid, the base can be considered as triangle DMA', with area L*W/4 and height W/2, for a volume of L*W^2/24.

The volume of the largest remaining piece is therefore, by inclusion/exclusion:

W^2 * L - 2 * L*W^2 / 6 + L*W^2/24

Tabulated for single-digit dimensions, this comes out to:

L \ W  1       2       3       4       5       6       7       8       9
1     0.708   2.833   6.375  11.333  17.708  25.500  34.708  45.333  57.375
2     1.417   5.667  12.750  22.667  35.417  51.000  69.417  90.667 114.750
3     2.125   8.500  19.125  34.000  53.125  76.500 104.125 136.000 172.125
4     2.833  11.333  25.500  45.333  70.833 102.000 138.833 181.333 229.500
5     3.542  14.167  31.875  56.667  88.542 127.500 173.542 226.667 286.875
6     4.250  17.000  38.250  68.000 106.250 153.000 208.250 272.000 344.250
7     4.958  19.833  44.625  79.333 123.958 178.500 242.958 317.333 401.625
8     5.667  22.667  51.000  90.667 141.667 204.000 277.667 362.667 459.000
9     6.375  25.500  57.375 102.000 159.375 229.500 312.375 408.000 516.375

Only L=3, W=4 shows the volume as a 2-digit integer whose digits represent the dimensions of the original solid. The remaining piece therefore has volume 34 cm^3.

From Enigma No. 1456, by Susan Denham, New Scientist 18 August 2007.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
AnswerK Sengupta2009-01-09 15:57:11
SolutionSolutionhoodat2007-09-18 16:01:53
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