It will be convenient to imagine that we are painting our pyramids on
the flat cardboard, as in the diagrams, before folding up. Now, if we
take any _four_ colours (say red, blue, green, and yellow), they may be
applied in only 2 distinctive ways, as shown in Figs, 1 and 2. Any other
way will only result in one of these when the pyramids are folded up. If
we take any _three_ colours, they may be applied in the 3 ways shown in
Figs. 3, 4, and 5. If we take any _two_ colours, they may be applied in
the 3 ways shown in Figs. 6, 7, and 8. If we take any _single_ colour,
it may obviously be applied in only 1 way. But four colours may be
selected in 35 ways out of seven; three in 35 ways; two in 21 ways; and
one colour in 7 ways. Therefore 35 applied in 2 ways = 70; 35 in 3 ways
= 105; 21 in 3 ways = 63; and 7 in 1 way = 7. Consequently the pyramid
may be painted in 245 different ways (70 + 105 + 63 + 7), using the
seven colours of the solar spectrum in accordance with the conditions of
the puzzle.
[Illustration:
1 2
+---------------+ +---------------+
\ R / \ B / \ B / \ R /
\ / \ / \ / \ /
\ / G \ / \ / G \ /
\-------/ \-------/
\ / \ /
\ Y / \ Y /
\ / \ /
' '
3 4 5
+---------------+ +---------------+ +---------------+
\ R / \ R / \ R / \ G / \ Y / \ R /
\ / \ / \ / \ / \ / \ /
\ / G \ / \ / G \ / \ / G \ /
\-------/ \-------/ \-------/
\ / \ / \ /
\ Y / \ Y / \ Y /
\ / \ / \ /
' ' '
6 7 8
+---------------+ +---------------+ +---------------+
\ G / \ Y / \ Y / \ Y / \ G / \ G /
\ / \ / \ / \ / \ / \ /
\ / G \ / \ / G \ / \ / G \ /
\-------/ \-------/ \-------/
\ / \ / \ /
\ Y / \ Y / \ Y /
\ / \ / \ /
' ' '
]