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.
Answer:
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 /
/ / /
' ' '
]
