Dissect a 6x6x6 cube into as few pieces as possible that can be reassembled into three cubes 3x3x3, 4x4x4...
geometry/dissections/cube.3.4.5.6 New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
Dissect a 6x6x6 cube into as few pieces as possible that can be reassembled into three cubes 3x3x3, 4x4x4 and 5x5x5, illustrating 3^3 + 4^3 + 5^3 = 6^3. [ Editor: problem statement reconstructed from the postings; the pieces are made of unit cubes. ]
Show solution
Correct solutions were sent in by Kevin Mundstock, Keith Ramsay and
Michael Hartley. Michael Hartley did it with 10 pieces. Here is his
solution:
Our pieces are:
0,1,2: three 3x3x1 rectangles
3: a 5x5x5 cube (guess where that goes!)
4: a really weird shape consisting of 3 units thick L shape
(with the arms of the L being four units each) glued onto
a 3x3x1 rectangle - or, in other words, a 3x3x1 rectangle
glued (upright) to the end of the large face of a 3x4x1
rectangle (to form the L), this being then glued onto
the large face of a 3x3x1 rectangle so that the corner
of the L is attached to the corner of the 3x3x1 rectangle.
5: a 4x2x1 rectangle with an extra cube glued to the end of 5555
one of its long edges. 5555
6,9: two 3x2x1 rectangle 5
7: a 4x1x1 rectangle with an extra cube glued to the end of 7
one of its edges (so it looks like a thin L) 7777
8: a two units thick L shape with the arms being 3 and 2 units
long respectively - that is, a 3x2x1 rectangle with a 2x1x1
rectangle glued across the end of one of its large faces.
If any of these directions seem confusing, you should be able to
tell the shapes from the solution (below) anyway...
000 111 222
000 111 222
000 111 222
7777 7884 9994 9994
5444 5884 5884 5884
5444 5664 5664 5664
5444 4444 4444 4444
33333 33333 33333 33333 33333
33333 33333 33333 33333 33333
33333 33333 33333 33333 33333
33333 33333 33333 33333 33333
33333 33333 33333 33333 33333
equals
000111 333336 333336 333336 333339 333339
000111 333336 333336 333336 333339 333339
000111 333334 333334 333334 333339 333339
222444 333334 333334 333334 333338 333338
222444 333334 333334 333334 333338 333338
222444 554444 554444 554444 555788 777788
-- Michael Hartley <hartley@maths.uwa.edu.au>
However, Geoff Bailey (Fred the Wonder Worm) managed to locate an
appearance of this problem in print, with an 8-piece solution:
"Mathematical Models", by H. M. Cundy and A. R. Rollett, Oxford
University Press 1961; recent re-issue: ISBN 0 906212 20 0.
C & R say...
A very interesting dissection puzzle takes its origin from the
arithmetical fact that 3^3 + 4^3 + 5^3 [Editor: "+ ^3" in the
posting] = 6^3. The problem of dissecting the 6x6x6 cube to display
this was solved some years ago and the solution as published in
Eureka, the magazine of the Cambridge Archimedeans (an undergraduate
mathematical society). It makes a good puzzle. There are eight
pieces; the 3x3x3 cube is uncut; there is a 2x2x2 cube, a block
2x1x1, another 3x2x1 and four others. One of these is a 4x4x4 with
a 2x1x1 block cut from its corner; the others are shown in Fig. 253.
The assembly of the 4x4x4 cube is obvious; five pieces make the
5x5x5 cube, and the whole set can be assembled to make the 6x6x6
cube.
Geoff Bailey has made a model by glueing Cuisenaire [Editor: "cuisinaire" in the posting] rods together, and
suggests that interested puzzlers try the same. Geoff also typed up
ascii diagrams from the book, reproduced here:
------------------
Piece A is a 4x3x2 block with a 2x3x1 block on top of one of the 4x3 sides,
so that two of the length 3 edges coincide.
TOP: SIDE: FRONT:
+---+---+---+---+ +---+---+ +---+---+---+
| | | | | | |
+ + + + +---+---+ +---+---+---+
| | | | | | |
+ + + + + + +
| | | | | | |
+---+---+---+---+ +---+---+---+---+ +---+---+---+
PERSPECTIVE:
+---+---+
/ /|
/ / +---+---+
/ / / /|
+---+---+ / / +
| |/ / |
+ +---+---+ +
| | /
+ + /
| |/
+---+---+---+---+
Piece B is a 5x3x2 block with a 3x2x1 block on one of the 5x3 sides (as in
the previous block) and a 2x2x1 block on top of that, on a 2x1 face.
TOP: SIDE: FRONT:
+---+---+---+---+---+ +---+---+ +---+
| | | | | | |
+---+---+ + + + + +
| | | | | | |
+ + + +---+---+ +---+---+ +
| | | | | | |
+---+---+---+---+---+ + +---+---+---+ +---+---+---+
| | | |
+ + + +
| | | |
+---+---+---+---+---+ +---+---+---+
PERSPECTIVE:
+---+---+
/---+---/|
| |+
+ +|
| |+
+---+---+|
/ / +---+---+---+
/ / / /|
+---+---+ / / +
| |/ / |
+ +---+---+---+ +
| | /
+ + /
| |/
+---+---+---+---+---+
Piece C is a 5x3x2 block with a 2x2x2 cube on top of one of the 5x3 faces in
a corner, and a 3x1x1 block next to it along the length 3 edge.
TOP: SIDE: FRONT:
+---+---+---+---+---+ +---+---+ +---+---+
| | | | | | |
+ + + +---+---+---+ +---+ +
| | | | | | | |
+---+---+---+ | + +---+---+ +---+---+---+
| | | | | | |
+---+---+---+---+---+ + + + +
| | | |
+---+---+---+---+---+ +---+---+---+
PERSPECTIVE:
+---+---+
/ /|
/ / +
+---+---+ |
| | /---+---+---+
+---+---+-'-+ /|
/---+---+---/| / |
| |/ / |
+ +---+---+ +
| | /
+ + /
| |/
+---+---+---+---+---+
----------------------------
Have fun trying to find the 5^3 and 6^3 assemblies! (Not given in the
book.)
-- Bill Taylor <wft@math.canterbury.ac.nz>