1st toss bet $14.29. If lose, bet 14.29 again. If win bet zero. If win 2nd toss, bet zero again on 3rd toss. If lose bet $28.58. IOW, make sure you bet zero twice if you keep winning.
Very sorry about that. I seemed to have fat fingered that (very crucial) point. Hopefully the example I gave of a loss (which was correct) and the incorrect explanation at least gave people a pause to look for clarifications
How does the suggestion not meet that criterion? You bet nothing until two heads are already out. If two heads never come out, you lose the game, but you keep your money. If two heads come out, you bet $100. On the next toss you either get heads, win the game AND meet the $200 goal, or you get tails, lose the game and lose your money - it's 50/50. Is it really possible to do better than that?
At first glance, 50 25 25 50 100. Bet those amounts until you've won the 100 or the game is over. Then bet 0.
Again, see my counter example: first flip, heads. you betted nothing. fine. Second flip, heads, great, you bet $100. third flip: tail. You lose the $100. You have nothing. Fourth flip: head. You won the game. You have nothing. You didn't meet the payoff you desired.
Explain this. You can bet per flip so what's the point of the 3 heads? That ends the game and doubles your bank (which you called stake).?
No dice. Suppose the flip outcome is tail tail head tail head head? you lose 50, lose 25, win 50 (so now your stack is $75). You can't bet $100 on the last cause you don't have it
Winning the game is different than meeting your goal. It's a game within a game. You need to win the game AND double your money while doing it. You can win the game and lose your money. You can double your money and lose the game. You need to win AND double. That's the brainteaser part.