COMMENT ⊗   VALID 00002 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	The problem is to construct a 5x5x5 cube out of the following blocks:
C00006 ENDMK
C⊗;
The problem is to construct a 5x5x5 cube out of the following blocks:
	A. 3 1x1x3
	B. 1 1x2x2
	C. 1 2x2x2
	D. 13 1x2x4

1. Consider the 5x5x5 as 5 stacked 5x5 planes.  Every block type except A
occupies covers an even number of blocks in a plane, so every plane must
intersect a block of type A or contain it fully.

2. Looking at the stack from any of the three directions, we see that at
least one A block must have its long axis in that direction since 5
planes must be covered by only three blocks.  Therefore, the three A's
must have mutually perpendicular axes.

3. An A intersects 5 different planes; it lies in two planes and crosses
three others.  Since there are 15 planes in all, the three A's can barely
cover all of them, so that no plane can intersect more than one A.

4. Now color the 1x1 cubes in the stack so that adjacent cubes have
opposite colors - the usual checkerboard coloring extended to three
dimensions.  Suppose that the corners of the stack are colored white.

5. Each non A covers the same number of squares of each color, so
two of the A's must have the color sequence white-black-white and one
must have the sequence black-white-black, since there is one more white
cube than black in the whole stack.

6. Any plane in which a black-white-black A lies must have one more
black cube than white, so it must be the next plane to a face.  Up to
symmetry, this precisely locates the black-white black.
If we label the co-ordinates from 1 through 5 in each direction, we
may assume that the black-white-black occupies (2,2,2), (2,2,3) and
(2,2,4).

7. Now consider the projection of the A's in the x-y plane.  They
must be mutually perpendicular, intersect all lines parallel to
the x and y axes and intersect no line twice.  This can only be done
if one of them occupies (3,1), (4,1) and (5,1) and the other occupies
(1,3), (1,4) and (1,5).  Since these blocks must be in the z=1 and
z=5 planes, this locates the A's precisely up to symmetry.

Here is a map of the projection in the x-y plane.

X
X
X
  X
    X X X

8. The rest is an exercise for the reader.