Actual interview brainteaser at a top flight shop

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?
 
Quote from Bogwaluth:

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?
More...

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.
 
Quote from dtrader98:

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.
More...

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.
 
Quote from MustPlayOptions:

Format is Result of first 2 flips and then bet for each flip

<pre>
Bet 1 2 3 4 5
WW 37.5 37.5 25 50 100
11001 137.5 175 150 100 200
1101 137.5 175 150 200
111 137.5 175 200


WL 37.5 37.5 50 50 100
10101 137.5 100 150 100 200
1011 137.5 100 150 200
10011 137.5 100 50 100 200



LW 37.5 37.5 50 50 100
0111 62.5 100 150 200
01101 62.5 100 150 100 200
01011 62.5 100 50 100 200

LL 37.5 37.5 25 50 100
00111 62.5 25 50 100 200

</pre>
More...

I think this is correct. Could someone else double check to confirm?
 
Quote from Kovacs:

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.
More...

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.:D
 
Quote from nitro:

Mapping the betting to the Fibonacci ratios can't be right either, as betting 1 / 1 has already been shown to not work...But What about starting with 1/2 2/3 etc? You have to have a choice function, otherwise you are not keeping track of the heads and tails.
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
 
Quote from dtrader98:

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.:D
More...

Come up with a three heads scenario where you don't think it'll win. I'll give you the betting sequence.
 
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