Hi, Just looked over this cool teaser. Just to clarify a few things: 1) All cases of three 1s are considered winners, yes? And A case such as TTHHT is therefore a winner, implying you must have made a rigid rule to bet and taken it on the first two tosses, correct? Or if you chose not to bet on toss one, it must be consistent across all trials. 2) You mentioned somewhere to double on every winning trial (big factor). Since that is a restriction, I assume you may do a completely different action on non-wins, but that rule must be consistent from non-win to non-win, yes? 3)37.5 would not work on every trial. You would be out on the first two tosses of tail tail. ie. 100 toss 1 = tail => 62.5 toss 2 tail =>25, you don't have enough for 3rd toss. The optimal rule set will have staggered and conditional bet sizing. 4) With the optimial rule set, you are guaranteed to have a +100 gain, in every possible case of three ones showing up in a 5 toss trial. Very important criteria here. 5) there were some examples where you showed 6 outcomes, but it will always only be 5 or less, correct? I.e. you can stop if 1st three tosses were H, and you reached +100. Not required to continue. 6) I was almost about to declare the objective impossible, until I realized that you were allowed to have a different strategy on wins and losses. Very nice teaser. thanks for sharing.
Hmmmmm, I seem to have gotten 37.50 for the first 2 bets. Then the following strategies: If you go WW or LL you bet 25,50,100 for bets 3,4,5. If you go LW or WL you bet 50,50,100 for bets 3,4,5. I had the following 10 scenarios: W W W - - W W L W - W W L L W L W W W - L W L W W L W W L W L L W W W W L W W - W L W L W W L L W W Did I miss something?
here is your answer ... in the orcam language http://ecx.images-amazon.com/images/I/51vnxMWiIpL._SL500_AA240_.jpg tell the "boys" to open their eyes and see the whole picture as traded by the best of the best whom do not give a shit if the whole world goes to hell for their little bonus using OPM sad isn't it ?
I show a loss on 25,50,100 scenario: T,H,H,T,H 62.5,100,125,75... you are out on the 4th toss as you don't have enough left to bet 100 on the 5th.
There are only 120 possible HT combinations. The hard part is the bet choice, otherwise it would be easy to write a backtracking computer program. Still, it seems that by induction, we should be able to work backwards at each point towards the right bet. So for example, $100 as the first bet is shown to be false. Can we eliminate other factors of 100? Maybe the solution is not independent of the initial conditions, since if in fact it is a problem specific to integers, then the prime decomposition of 100 (or whatever amt of money we had left) would be important, e.g., 5^2 * 2^2, or maybe only prime decompositions of the amount of money left is right. If so, only 2, 4, 5, 25, (100 ?)would be initial possible bets. Now, construct a tree at each HT using prime bets as parent nodes.
You still double your initial stake in the TTHHH scenario if you start the third toss with only $25. Third toss you bet 25 and double your money to 50 if it's heads. Fourth toss you bet 50 and double your money to 100 if it's heads. Fifth toss you double your money to 200 if it's heads. If at any time you get a tails, you go broke, which is fine because the sequence is a loser as well since three heads at that point is impossible.
I'll let OP address that. But it sounds like a violation, as you did not win in every case with three heads. If your rule is to stand aside on 1st two bets, and you only counted three ones on the tail end, 50,25,25 would guarantee +100 for the only two all three Head cases (of XXHHH). However, I don't think that will fly, as it's too easy, and I believe violates rules. I was gravitating towards 0 being the optimal 1st bet however.. but will investigate a bit more.
This is beginning to feel right. Since the Fibonacci series can be found in Pascals triangle... Code: Row 0 1 Row 1 1 1 Row 2 1 2 1 Row 3 1 3 3 1 Row 4 1 4 6 4 1 Row 5 1 5 10 10 5 1 Row 6 1 6 15 20 15 6 1
Come up with a three heads scenario where you don't think it'll win. I'll give you the betting sequence.