Posts Tagged ‘combinatorics’  One, Two, Three, Four, Six(?)
October 15, 2008  How many necklaces can you make if you have
two different color beads (like purple and gold) at your
disposal?  If you put 0 beads on your necklace, there’s only one
way to do so and it’s mighty boring.  Suppose you put 1 bead on
your necklace. It might be purple or it might be gold, giving
you two possible necklaces.  If you want to put 2 beads on your
necklace, you can do so in three ways. And if you want to use 3
beads, there are four possibilities.  What about with 4 beads?
It looks like the answer should be five, right? Right? You know
you want to say that. ALAS, there are actually six different
ways to do it.  Once the pattern of consecutive integers is
broken, it stays broken. There are eight ways to make a necklace
out of 5 beads, and either thirteen or fourteen ways to make it
out of 6 beads: thirteen if you allow reflexive symmetry — that
is, the necklace is “free” and you can flip it over in addition
to rotating it — and