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        riddles >> easy >> 24 play...
        
(Message started by: Noke Lieu on Apr 28th, 2013, 11:26pm)
Title: 24 play...
Post by Noke Lieu on Apr 28th, 2013, 11:26pm
You're 1 of 24 people in a room.
24 envelopes, each with a number 1-24.
Each person gets to have 1 chance to choose an envelope, and guess the number inside.

If they're right, they win the jackpot.  The game ends.

If they're wrong, the envelope is discarded, the jackpot goes up by $1.

When do you choose to guess your number?
Title: Re: 24 play...
Post by rmsgrey on Apr 29th, 2013, 5:14am
Assuming that each person guesses a number which wasn't in any of the previously opened envelopes, but doesn't mind when they guess a number, I'd guess mine [hide]last[/hide].

Reasoning:
[hideb]
Since, by assumption, the previous number guessed has never been revealed to be in an opened envelope before the last, and, by the rules of the game, if the game continues, can't have been in the last either, it's always possible that the next envelope contains the last number chosen.

By symmetry, each number that could be in the next envelope has an equal chance, so the common-strategy where everyone chooses the same number - 1, say - is an optimal one.

Obviously, there's an equal, 1/24 chance of any given envelope containing the number 1, so you maximise your expected winnings (at $1) by picking the envelope with the highest jackpot.
[/hideb]
Title: Re: 24 play...
Post by Grimbal on Apr 29th, 2013, 8:41am
While we are at it, who can expand the following polynomial

   (x-a)(x-b)(x-c)...(x-z)

and express it in the canonic form

   a26 x26 + a25 x25 + ... + a1 x + a0

?

Computers are not allowed.


PS: his somehow got in a wrong thread.
Title: Re: 24 play...
Post by jollytall on Apr 29th, 2013, 10:08am
Do you mean the old riddle where the result is [hideb]zero[/hideb]?
Title: Re: 24 play...
Post by Noke Lieu on Apr 29th, 2013, 6:07pm
Perhaps, rmsgrey, you like the 100% success rate of going last...
It's just that you don't get to go that often.

It was something that seemed pretty counterintuitive- that it didn't matter when you choose- it all works out the same...
Title: Re: 24 play...
Post by rmsgrey on Apr 30th, 2013, 5:03am

on 04/29/13 at 18:07:24, Noke Lieu wrote:
Perhaps, rmsgrey, you like the 100% success rate of going last...
It's just that you don't get to go that often.

It was something that seemed pretty counterintuitive- that it didn't matter when you choose- it all works out the same...


Actually, I like the fact that the jackpot is biggest then - if I go first, I have a 1/24 chance of winning $1; if I go last, I have a 1/24 chance of winning $24...

Same chance of winning, whenever I guess, but the prize gets bigger the longer I wait...
Title: Re: 24 play...
Post by towr on Apr 30th, 2013, 8:03am
Then how about we use 300 numbers/envelopes, and the first person can guess 24 times, the second 23 times, etc. Otherwise same setup.
Title: Re: 24 play...
Post by Noke Lieu on May 1st, 2013, 4:25pm
...or you have to pay $1 to be the first guesser, $2 to be second... Everyone pays up front.
Title: Re: 24 play...
Post by towr on May 1st, 2013, 10:27pm

on 05/01/13 at 16:25:56, Noke Lieu wrote:
...or you have to pay $1 to be the first guesser, $2 to be second... Everyone pays up front.
Up front as in: before the game starts, or before you have your turn?
Title: Re: 24 play...
Post by Noke Lieu on May 2nd, 2013, 1:50am
pay before the game starts. Otherwise why wouldn't you go last?
Title: Re: 24 play...
Post by rmsgrey on May 3rd, 2013, 6:25am

on 05/01/13 at 16:25:56, Noke Lieu wrote:
...or you have to pay $1 to be the first guesser, $2 to be second... Everyone pays up front.


A strange game, Professor. The only way to win is not to play...
Title: Re: 24 play...
Post by rmsgrey on May 3rd, 2013, 6:27am

on 04/30/13 at 08:03:34, towr wrote:
Then how about we use 300 numbers/envelopes, and the first person can guess 24 times, the second 23 times, etc. Otherwise same setup.


Then I'd go for 12 chances at $12 rather than 1 chance at $24 or 24 chances at $1...
Title: Re: 24 play...
Post by towr on May 3rd, 2013, 6:58am
Perhaps I wasn't clear enough in that by "same setup" I still meant $1 is added to the jackpot after each discarded envelope. So the first player guesses for $1,$2,$3,..,$24, and the last guesses for $300.
Title: Re: 24 play...
Post by Noke Lieu on May 3rd, 2013, 4:10pm

on 05/03/13 at 06:25:52, rmsgrey wrote:
A strange game, Professor. The only way to win is not to play...

um, yeah... that way the jackpot is the $300. Doesn't change the game enough to be good though.
Title: Re: 24 play...
Post by rmsgrey on May 6th, 2013, 5:38am

on 05/03/13 at 06:58:12, towr wrote:
Perhaps I wasn't clear enough in that by "same setup" I still meant $1 is added to the jackpot after each discarded envelope. So the first player guesses for $1,$2,$3,..,$24, and the last guesses for $300.

In that case, I'd have to do some real calculation to figure out where the maximum expected value is (which is the point at which the sum of the jackpots for "your" envelopes is greatest).

For the variant where you pay $1 before opening each envelope, you'll want to be playing a turn or two later than when the jackpot comes from the bank - you break even when the jackpot is won before your turn, win the total contributions of everyone before you when it's one of your envelopes, and lose $1 per envelope you opened when the winning envelope is after yours.
Title: Re: 24 play...
Post by whizen on May 29th, 2013, 6:47pm

on 04/29/13 at 08:41:51, Grimbal wrote:
While we are at it, who can expand the following polynomial

   (x-a)(x-b)(x-c)...(x-z)

and express it in the canonic form

   a26 x26 + a25 x25 + ... + a1 x + a0

?

Computers are not allowed.


PS: his somehow got in a wrong thread.


I know you forbid bringing computers into this... but I could not resist.

I fed this to a polynomial expander at... http://www.mathportal.org/calculators/polynomials-solvers/polynomials-expanding-calculator.php
and the program has probably crashed.
Its been 20 minutes and its still calculating... I fear the worst.

Then I went and did this to 5 other polynomial solving servers online... they all reeled under weight of this perplexing riddle. Even wolfram alpha cracked under pressure.

This is obviously a ridiculously tough problem, and as you remarked, misplaced in the easy section. Is it by chance one of "the ones which must not be computed"?
Title: Re: 24 play...
Post by towr on May 29th, 2013, 10:24pm

on 05/29/13 at 18:47:36, whizen wrote:
This is obviously a ridiculously tough problem, and as you remarked, misplaced in the easy section. Is it by chance one of "the ones which must not be computed"?
It's a lot easier to do by hand; as long as you're a bit smarter about it than a computer.
Title: Re: 24 play...
Post by whizen on May 29th, 2013, 11:29pm

on 05/29/13 at 22:24:51, towr wrote:
It's a lot easier to do by hand; as long as you're a bit smarter about it than a computer.


Apologies if I conveyed that I didn't know the solution... I did make an attempt at humour and got carried away in the last statement in my post.

However, I did feed the equation to various polynomial solvers to see if they could figure out the result. They could handle it as long as the equations were small. However, all of them croaked on the 26 term product.
Title: Re: 24 play...
Post by towr on May 30th, 2013, 8:56am
Ah :)
Title: Re: 24 play...
Post by Annettagiles on Oct 14th, 2014, 4:30am
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