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 / \ / \ / \ / ' ' '
]
