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An aged King, after losing his only son in battle, determines to divide his kingdom amongst his wisest advisors.

He presents two solid cubes, one made of gold, the other made of silver, and a wooden box into which can fit either of the cubes perfectly.

He explains that he will divide his kingdom among those who are able to find all the ways to fill the box with pieces cut from the gold and silver cubes.

The rules are as follows:

  1. The pieces may not be curved. They must be formed by planes.

  2. The corners of each piece must meet one of the corners of the box.

  3. Also, the same exact configuration of gold and silver pieces, rotated differently, will count as a unique way to fill the box if that rotation/configuration is not identical to one you previously counted.

The king does not know the answer, but is looking for those who can solve the riddle and prove it correct, in a way he can understand.

What answer would you give him?

Note:

I've researched this for quite some time but have been unable to come up with a solution. Any ideas are welcome!

This image is an example of a cube being divided in different legal ways:

cube cut

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  • $\begingroup$ I don't quite understand 1. or 2. $\endgroup$
    – Ben Frankel
    Apr 20, 2015 at 15:13
  • $\begingroup$ @BenFrankel In order to cut a piece out of one of the cubes, you would have to start at one corner, and cut directly to another two corners forming a triangle plane inside the cube. $\endgroup$
    – ridthyself
    Apr 20, 2015 at 15:18
  • $\begingroup$ @ghosts_in_the_code noted, I adjusted my comment to remove the word "line" and include "plane" as it was misleading. $\endgroup$
    – ridthyself
    Apr 20, 2015 at 15:22
  • $\begingroup$ Do different arrangements/geometry of the same ratio count as different solutions? $\endgroup$
    – Mark N
    Apr 20, 2015 at 15:26
  • $\begingroup$ @etothepowerofx Please add this to the post itself. Comments are not meant to contain part of the question. $\endgroup$
    – ghosts_in_the_code
    Apr 20, 2015 at 15:26

2 Answers 2

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Here is my answer

Is it can easily be observed the only way this can be solved is by dividing the two cubes into two prisms and and taking one prism from each cube and fitting it into the box in as many different configurations as possible, i.e., by rottating them so that each corner of the container meets all the corners of the new cube formed (by new cube I mean the one formed by the two prisms). This logicaly is the only possible answer because other pieces cut out won't satisfy the first and second condition

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As far as I can tell you can at most make

5 cuts.
The cuts are non-intersecting. One goes through opposite faces across the diagonal. With the remaining prisms, you can make up to 2 more cuts on each one: near point to opposite edge, near point with one near corner + one far corner, combo of the two, second with its mirror cut.

Ignoring rotation, we have no cuts, main diagonal cut, 1 side cut, 1 side cutout, main diag + 1 side cut (2x), 1 + 1 side cut (5x), 1/2 rotated (5x), 2 side (1x, 6x), main diag + 1 side + 1 side (9x), main diag + 2 side (3x), main diag + 2 side + 1 side (9x), 1/2 rotated (9x), main diag + 2 side + 2 side (9x). Those give the number of pieces at $1, 2, 2, 3, 3, 3, (2,3), 4, 4, 5, 5, 6$.

For each piece, it can either be gold or silver, and each combination would be different.
For each of our arrangements, you have $2 ^ n$ (where n = pieces) choices. So,

$2 + 4 + 4 + 8(x2) + 8(x5) + 8(x5) + (4+8(x6)) + 16(x9) + 32(x9) + 32(x3)+ 32(x9) + 64(x9) = 1550$

Edit
etothepowerofx has noticed 2 more configurations, that add 32 and 8 options respectively, putting the total to $1590$

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  • $\begingroup$ Don't forget about rotation $\endgroup$
    – Mark N
    Apr 20, 2015 at 15:39
  • $\begingroup$ That's a good start, but I think the number is much higher. After you make your initial diagonal cut, dividing the cube into halves, each half can then be cut twice more, 5 cuts total. $\endgroup$
    – ridthyself
    Apr 20, 2015 at 15:44
  • $\begingroup$ But that would add corners that don't touch the original corners. $\endgroup$
    – JonTheMon
    Apr 20, 2015 at 15:47
  • $\begingroup$ @JonTheMon Your first cut divided the cube in half, with a plane the shape of a rectangle. Your next cut on one of the halves drew a plane the shape of an equilateral triangle which cut off one of the corners of the cube. What remains is a skewed pyramid with a rectangle base, 3 right triangle faces (which are exterior to the cube) and one equilateral triangle which was the plane formed when you cut off the corner. This remaining pyramid-like shape can be divided in two different ways along it's 5 corners. $\endgroup$
    – ridthyself
    Apr 20, 2015 at 15:59
  • $\begingroup$ @JonTheMon I added an image to my question to give you an idea of what I mean... $\endgroup$
    – ridthyself
    Apr 20, 2015 at 16:30

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