A common cause of error is this. If you calculate your combinations by working upwards from a basic triangle lying on the table, you will get half the correct number of ways, because you overlook the fact that an equal number of pyramids may be built on that triangle downwards, so to speak, through the table. They are, in fact, reflections of the others, and examples from the two sets of pyramids cannot be set up to resemble one another--except under fourth dimensional conditions! 281.--PAINTING A PYRAMID. 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 / \ / \ / \ / ' ' '