I think Karl is just saying that if you've bet on HHH, the only way you can possibly win against THH is to get your three H's in a row. If a T occurs at any time, your opponent is certain to win eventually: he either gets his two H's and wins immediately, or a T occurs in which case he's got the first of "THH" and you need 3 H's and he only needs two to win.

So we can certainly say (as a strategy) that:

THH > HHH

and for the same reason:

HTT > TTT

In both of the above examples, the person with the non-constant triple wins,
on average, 7 times out of 8.

So it would be crazy to pick HHH or TTT, but how about THH or HTT? Can they
be beaten? And can the pattern that beats them be beaten? That's the essence
of the problem.

The rule seems to be to take the pattern of your opponent, say it's ABC, where A, B and C
are either H or T, and pick as your rule B'AB, where B' is the opposite of B.

So HHT -> THH, et cetera.

To convince yourself that THH beats HHT, draw a state diagram. It's hard to draw them
here, but here's what mine would look like for the contest above:

I've got 7 states: S = start, A, B, C, D, E and F, where C is a win for HHT and F is a
win for THH.

From each state other than the ones that indicate a win, draw two arrows: one for flipping an H from that state, and one for flipping T, so label the arrows with H or T.

For my diagram, A, B and C represent states on the way to a win for HHT and D, E and F on the way to a win for THH.

Here are the "H" arrows: S->A, A->B, B->B, D->E, E->F

Here are the "T" arrows: S->D, D->D, E->D B->C, A->D

Note that once you get to D, it's a certain win for THH and once you get to B, it's a certain win for HHT.

The first two flips can be HH, HT, TH, TT. HH puts you at B; all the others put you at D.

I think I got that right...