## PAINTING A PYRAMID.

(

Combination and Group Problems)

This puzzle concerns the painting of the four sides of a tetrahedron, or

triangular pyramid. If you cut out a piece of cardboard of the

triangular shape shown in Fig. 1, and then cut half through along the

dotted lines, it will fold up and form a perfect triangular pyramid. And

I would first remind my readers that the primary colours of the solar

spectrum are seven--violet, indigo, blue, green, yellow, orange, and

red. When I was a child I was taught to remember these by the ungainly

word formed by the initials of the colours, "Vibgyor."

In how many different ways may the triangular pyramid be coloured, using

in every case one, two, three, or four colours of the solar spectrum? Of

course a side can only receive a single colour, and no side can be left

uncoloured. But there is one point that I must make quite clear. The

four sides are not to be regarded as individually distinct. That is to

say, if you paint your pyramid as shown in Fig. 2 (where the bottom side

is green and the other side that is out of view is yellow), and then

paint another in the order shown in Fig. 3, these are really both the

same and count as one way. For if you tilt over No. 2 to the right it

will so fall as to represent No. 3. The avoidance of repetitions of this

kind is the real puzzle of the thing. If a coloured pyramid cannot be

placed so that it exactly resembles in its colours and their relative

order another pyramid, then they are different. Remember that one way

would be to colour all the four sides red, another to colour two sides

green, and the remaining sides yellow and blue; and so on.

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THE ANTIQUARY'S CHAIN.
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BUILDING THE TETRAHEDRON.