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        riddles >> easy >> Another hat riddle
        
(Message started by: BNC on Mar 3^rd, 2003, 2:00am)

Title: Another hat riddle
Post by BNC on Mar 3^rd, 2003, 2:00am

Three uberpuzzlers participated in the riddle Olympics, and naturally got to the finals. As the three puzzlers are equally smart, and follow the exact same thinking process, they all got the same (highest possible) score at the final stage.
To decide which would win, they were blindfolded, and a hat was placed on each of their heads. They were placed in such a way so that each could see the hats of the other two contestants, but not his own. They were asked the same question:
“On each of the hats you’re wearing there in a natural number. The numbers form an algebraic series. The first to guess the number on his hat would win the Olympics.”

The blindfolds were then removed. One of the contestants saw the numbers 4 and 6 on the other hats. After a while he stood and said: “The number on my hat is ____”

Title: Re: Another hat riddle
Post by wolfgang on Mar 3^rd, 2003, 11:59am

I'm not completely sure, but I guess [hide] 8.
The reasoning being
He could have a 2, 5, or 8.
If he had a 2, 6 would have seen 2,4 and known he couldn't have a 3 because then 4 would have seen 2,3 and known he couldn't have a 1 because 6 would have seen 1,2 and known he had to have a 3.
If he had a 5, 6 would have seen 4,5 and known he couldn't have a 3 because then 5 would have seen 3, 4 and known he couldn't have a 2 because 4 would have seen a 2,3 and known he couldn't have a 1.
But if he had an 8, 4 would have seen 6,8 and quickly ruled out 7 for similar reasons, but would have taken a longer time to rule out 10; and 6 would have seen 4,8, and ruling out 12 would probably take a long time to rule out (it's all making me too dizzy to actually work through it all the way). [/hide]

Title: Re: Another hat riddle
Post by Chronos on Mar 5^th, 2003, 9:09pm

It's a shame that I didn't make it to the finals, because I have a faster answer. I would stand up and say "My number is 2. My number is 5. My number is 8." as fast as I could. I can probably talk faster than the other two can metareason, and one of those is, in fact, the correct guess.

Title: Re: Another hat riddle
Post by BNC on Mar 5^th, 2003, 11:27pm

Yeah, I thought at first to state the question's final sentence as "The first to guess the number on his hat would win the Olympics. Guessing wrong would result in eternal shame"

But then I figured it might confuse, and be considered a "hint".

Oh well. Can't win them all.

Title: Re: Another hat riddle
Post by TimMann on Mar 6^th, 2003, 10:26pm

I was stumped on this one when I got to the term "algebraic series". From the answers, it looks like you mean what I was taught to call an "arithmetic sequence." Is the term "algebraic series" common?

Title: Re: Another hat riddle
Post by Icarus on Mar 7^th, 2003, 3:27pm

I too was confused by "algebraic series". This was not a term I had heard of before. A search with Google turned up several references to algebraic series. But it turns out they are much more complicated beasties than the simple arithmetic sequences desired here.

Title: Re: Another hat riddle
Post by Jeremiah Smith on Mar 7^th, 2003, 3:50pm

I interpreted "algebraic" as meaning "either arithmetic or geometric".