BUILDING THE TETRAHEDRON.
(
Combination and Group Problems)
I possess a tetrahedron, or triangular pyramid, formed of six sticks
glued together, as shown in the illustration. Can you count correctly
the number of different ways in which these six sticks might have been
stuck together so as to form the pyramid?
Some friends worked at it together one evening, each person providing
himself with six lucifer matches to aid his thoughts; but it was found
that no two results were the same. You see, if we remove one of the
sticks and turn it round the other way, that will be a different
pyramid. If we make two of the sticks change places the result will
again be different. But remember that every pyramid may be made to stand
on either of its four sides without being a different one. How many ways
are there altogether?
Answer:
Take your constructed pyramid and hold it so that one stick only lies on
the table. Now, four sticks must branch off from it in different
directions--two at each end. Any one of the five sticks may be left out
of this connection; therefore the four may be selected in 5 different
ways. But these four matches may be placed in 24 different orders. And
as any match may be joined at either of its ends, they may further be
varied (after their situations are settled for any particular
arrangement) in 16 different ways. In every arrangement the sixth stick
may be added in 2 different ways. Now multiply these results together,
and we get 5 x 24 x 16 x 2 = 3,840 as the exact number of ways in which
the pyramid may be constructed. This method excludes all possibility of
error.
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!
Informational
