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281.—­PAINTING A PYRAMID.

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.”

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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.

282.—­THE ANTIQUARY’S CHAIN.

An antiquary possessed a number of curious old links, which he took to a blacksmith, and told him to join together to form one straight piece of chain, with the sole condition that the two circular links were not to be together.  The following illustration shows the appearance of the chain and the form of each link.  Now, supposing the owner should separate the links again, and then take them to another smith and repeat his former instructions exactly, what are the chances against the links being put together exactly as they were by the first man?  Remember that every successive link can be joined on to another in one of two ways, just as you can put a ring on your finger in two ways, or link your forefingers and thumbs in two ways.

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283.—­THE FIFTEEN DOMINOES.

In this case we do not use the complete set of twenty-eight dominoes to be found in the ordinary box.  We dispense with all those dominoes that have a five or a six on them and limit ourselves to the fifteen that remain, where the double-four is the highest.

In how many different ways may the fifteen dominoes be arranged in a straight line in accordance with the simple rule of the game that a number must always be placed against a similar number—­that is, a four against a four, a blank against a blank, and so on?  Left to right and right to left of the same arrangement are to be counted as two different ways.