zzle 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."

[Illustration]

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.

[Illustration]

283.--THE FIFTEEN DOMINOES.

In this case we do not use