Cover the surface of a cube with five congruent Roman crosses.
geometry/tiling/roman.crosses New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
Cover the surface of a cube with five congruent Roman crosses. A Greek cross consists of five squares joined at edges so that one square is central and each of the other four is attached along one edge of the central one. It is easy to cover a cube with six such congruent crosses of the appropriate size, pasting them on without overlap and folding them over the edges: put four corners of a cross at the four corners of each face, and a triangular piece of each arm folds over an edge to fill exactly a space not filled by the cross on the adjacent face. A simple Roman cross is composed of six squares, five arranged as for the Greek cross, with the sixth square adjoined to one arm to make it twice as long as any of the other three. Cover the cube with five congruent Roman crosses. Is there a short-cut way to a solution? Is there more than one distinct solution? Which other hexominoes can similarly cover a cube?
Show solution
A solution is shown below (as it appears when the cube is flattened).
It can best be seen by reproducing it on cardboard, colouring each
section with a different colour according to the number with which it
is labelled (1 through 5), and folding it to form a cube.
+--------------------+
|^--_ 5 / /|
| ^--_/ 3 / |
| /^--_ / |
| 5 / ^--_/ 3 |
| / 5 /^--_|
|^--_/ / |
| 5 /^--_ / 3 |
| / ^--_/ |
| / 2 /^--_ |
|/ / 3 ^--_|
+--------------------+--------------------+--------------------+
|^--_ 5 / /|^--_ 2 / /|^--_ 3 / /|
| ^--_/ 5 / | ^--_/ 3 / | ^--_/ 3 / |
| /^--_ / | /^--_ / | /^--_ / |
| 5 / ^--_/ 2 | 2 / ^--_/ 1 | 1 / ^--_/ 3 |
| / 2 /^--_| / 1 /^--_| / 3 /^--_|
|^--_/ / |^--_/ / |^--_/ / |
| 5 /^--_ / 2 | 2 /^--_ / 1 | 1 /^--_ / 4 |
| / ^--_/ | / ^--_/ | / ^--_/ |
| / 4 /^--_ | / 2 /^--_ | / 1 /^--_ |
|/ / 2 ^--_|/ / 1 ^--_|/ / 4 ^--_|
+--------------------+--------------------+--------------------+
|^--_ 2 / /|
| ^--_/ 1 / |
| /^--_ / |
| 2 / ^--_/ 1 |
| / 1 /^--_|
|^--_/ / |
| 4 /^--_ / 4 |
| / ^--_/ |
| / 4 /^--_ |
|/ / 4 ^--_|
+--------------------+
|^--_ 4 / /|
| ^--_/ 4 / |
| /^--_ / |
| 5 / ^--_/ 4 |
| / 4 /^--_|
|^--_/ / |
| 5 /^--_ / 3 |
| / ^--_/ |
| / 5 /^--_ |
|/ / 3 ^--_|
+--------------------+
-- Ilan Mayer <ilan@isgtec.com>
[ Editor: the puzzle was posed by Mary Krimmel <mary.krimmel@sdcs.org>
in geometry.puzzles (December 1996), who thought it might come from a
Martin Gardner column, and reposted to rec.puzzles by Barry Wolk
<wolk@ccm.umanitoba.ca>. Her further questions were not answered in
the collected postings. ]