# A A strategy better than blind chance

1. Jul 21, 2016

### asmani

This is an interesting riddle from here: http://www.brand.site.co.il/riddles/201607q.htm [Broken]

I'm having difficulty understanding the problem. If each hat is black/white with 50-50 probability, independent of the colors of other hats, then the probability of winning for n=2 is always 1/4, no matter what strategy they plan. Can you please provide an example strategy, in which the probability is not 1/4?

Last edited by a moderator: May 8, 2017
2. Jul 21, 2016

### RUber

I am having difficulty accessing your link. Could you copy some of the pertinent text?
Usually these sorts of problems involve grouping to increase your odds, but I would have to see the riddle to know for sure.

3. Jul 21, 2016

### FactChecker

4. Jul 22, 2016

### asmani

Yeah, sorry I accidentally omitted the "l" in "html".

5. Jul 22, 2016

### RUber

Got it, thank you.

One possible strategy I can think of is for everyone to assume they are in the majority.
If the true proportion is 60:40 in favor of white, then most people would see that white is more common. If everyone says white, then 60% of the people are correct.
Unfortunately, this strategy also guarantees failure, unless everyone is wearing the same color hat, since success is defined as everyone guessing correctly.

So, what if everyone were to form a circle? Then you could look left and just guess that your hat is the opposite color compared to the person on your left.
For n=2, you have the following cases:
white, white = fail
white, black = success
black, black = fail
black, white = success.
This would at least give you 50/50 odds of winning for n=2.

As far as working out the large number options, I have not gone that far yet.

6. Jul 22, 2016

### asmani

Hey we have an infinite number of hats...

7. Jul 22, 2016

### RUber

Sorry, I was looking at the guess your color variation with one hat.
You are talking about the "pick a white hat" option?
That seems to be about the same in theory, but a lot more complicated to draw out the specific cases.

8. Jul 25, 2016

### Citan Uzuki

Sure. Both players will almost surely be wearing at least one white hat. For $i=1,2$, let $n_i$ be the position of the first white hat on player $i$'s head. Then have player 1 point at hat $n_2$ and player 2 point at hat $n_1$. It is easy to see that both players win iff $n_1 = n_2$, which happens with probability $\sum_{n=1}^{\infty}\left(\frac{1}{4}\right)^n = \frac{1}{3}$.

9. Jul 29, 2016

### Zafa Pi

An easier statement is to adopt post #8 strategy and notice that there 3 equally likely possible outcomes (b,w), (w,b), and (w,w). (b,b) is excluded. Thus 1/3.

I don't think your post should be labeled A, but rather B

Last edited by a moderator: May 8, 2017
10. Aug 1, 2016

### haruspex

That's not quite right. BB is possible, e.g. the stacks are BWB... and BBW...
With this ingenious algorithm, the probabilities are equally likely WW, BB, BW. If it's not WW then one of them has the earlier first white. That player will equally likely pick a B or W from his own stack, but the other player is guaranteed to pick a B from her own.

11. Aug 1, 2016

### Zafa Pi

12. Aug 1, 2016

### Staff: Mentor

And that is a key point. Each player still has 1/2 probability to pick a white hat, but the choices are now correlated.

With 1/3 probability they both pick a white hat, with 1/3 probability they get black+white (or white+black), with 1/3 probability they both pick a black hat.

The generalization to N participants gives 1/(N+1) probability to win. The 1/log(N) in the puzzle is interesting - it looks specific enough to suggest a solution, but it needs some better strategy.

Edit: Indeed. And there is one.

13. Aug 1, 2016

### Zafa Pi

OK, I get the 1/(N+1). Are you going to share the 1/log(N)? I find it amazing.

14. Aug 1, 2016

### haruspex

My guess is that it involves picking the kth height at which all the visible hats are white, k tending to infinity. But I've not attempted the algebra yet.

15. Aug 2, 2016

### Zafa Pi

Is the kth height the same as the minimum height where you see all white? If so then you get probability 1/(N+1).

16. Aug 2, 2016

### haruspex

No, it's the kth height at which you see all white.
I.e., of all the heights at which you see all white, the kth instance.

17. Aug 2, 2016

### Staff: Mentor

The webpage has a link to the solution.

That would not give a result better than 1/(N+1) I think.

18. Aug 2, 2016

### haruspex

I don't understand the description. Where does L come from? Have they agreed this in advance too?

19. Aug 2, 2016

### Staff: Mentor

Yes, they agree on one L in advance. Ideally the value that is the most probable - but as there are just k options, there is certainly a value of L that gives at least 1/k probability (with the caveat discussed before).