math/logics puzzle

[Kooldino]

I have some ideas...
1-instead of the 10th man saying BLACK or WHITE based on parity, base it on MAJORITY (duh). That way it has a better chance of helping out the fellow man. THere is an odd # of hats in front of him, so there must be a majority (5/4 at worst). Then things could continue with one of the above algorithms, but in the event that a guess must be made, they could keep track of the colors of hats behind them, and base their guess on probability.

Take it a step further...

10 - starts by making a call on the majority...as stated above.

9 - looks at the 3 guys in front of him, and says "black" if the majority of 8,7,6 is "black", and "white" if the majority is white. #9 himself has a 50/50 chance of survival.

8-looks at 7 and 6 and makes his decision on majority...

EXAMPLE: If 9 says "black", then he is looking at 2 possibilities of the 2 guys in front of him. Both black (which he would know he was white) or mixed (which he knows he's black).

Either way, 8 definitely knows which he is.

7-Sees #6. If 9 said "black", and let's say 8 said "black", and #6 (whe he can see) is white, then he knows he's black. If #6 is black, then he knows he's white.

6-Knows his color based on the answers of 8 and 7.

5-He's gotta be a real man, and take one for the team. He has to have the same task as man #9, and make his call on the majority of the 3 in front of him. Let's say that this time the majority is white, for example. So he says "white". There's also a 50/50 chance of him surviving just based on random chance.

4-same logic as #8
3-same logic as #7
2-same logic as #6
1-Bases his decision on #10's initial majority call, and is definitely saved.

So here's how it comes down...

10=50/50
9=50/50
8=saved
7=saved
6=saved
5=50/50
4=saved
3=saved
2=saved
1=saved (because he kept track of everyone behind him, and knows the majority based on 10's call. Keep in mind, he's not to take #10's color into account, because #10 didn't know his color when he called it).

The above method GUARANTEES us 7/10 survivors. And Chances are that 8 or 9 of them will survive.

:-)

(Edited for spelling error, that ended up not being an error...lol)

[Kooldino]

Crap, I figured out a hole in my logic. If EACH the clusters of 3 (8,7,6 and 4,3,2) are all the same color (within their own clusters), it could cause 1 person from each cluster to POSSIBLY make the wrong call. But the chances of 3 guys matching are 1 in 8...and the chances of BOTH clusters of 3 guys having all the same color hat (amongst themselves) are quite slim, but if it were the case, it would drop my WORST case scenario down to 50% of them getting saved, but it will almost always be better then 50% savior rate, usually be 70 or 80% savior rate, and 1 in 5 times or so, everyone should be saved.

[Gurm]

Kooldino,

Thank you. That's a good alternative strategy!

I have spoken.

- Gurm

[Kooldino]

Ok, just got back from lunch and figured out another inefficiency about my algorithm. I have the last guy saying the color of the majority of the hat colors. This always helps the first guy make a decision, but doesn't always guarantee the first guy to make the correct decision. And the only other person it COULD help is one of the guys in the "2,3,4" group, but that's ONLY if they're all the same color, AND there was the right ratio of black to white hats. (You guys following my logic here, I know it's hard to explain).

ANYWAY, I would just like to tweak it so that the #10 guy just states the color of the #1 guy, so now it does guarantee that the #1 guy is spared (but before it only guaranteed it if there was a certain range of black:white ratios). Everyone with me still?

This still works out to a 50% savior rate, WORST case scenario, but the case where you'd get 50% is less than 2%, and your chances are better of saving 100% than they are 50%. And you'll still get 70-80% savior rate on average.

[Kooldino]

PS-But keep in mind that pretty much NONE of these strategies will work unless they are allowed to associate BEFORE hand so they know what to look out for.