Prime

Does anyone else experience this problem? When in IE9 I cannot sign out unless I manually delete all cookies for BD Domain. In Chrome it appears to sign me out. But after I close the browser, then open it again and go to BD Forums, I find myself signed on.

Meanwhile, here are 13 different words

And then you must watch out for traps

I can’t find any simple formal description for the strategy. It appears, for an even “n” - you win; for an odd - she does. Interesting question is: with an even n, can she have a winning strategy in the modified version, where she can jump the King to any available square?

Let me offer the following perspective:

In this case... As for the question in the OP, I think I came up with a closed from equation for the losing probability, but haven't solved it.

Hello Jake. Welcome to the Den. Bushindo's answer has the right logic. His numbers are slightly off. I'll put my solution inside Spoiler, to let other people tackle this problem on their own.

There is one thing that is not quite clear to me. The one minute delay is it between the rounds or between the individual rolls? Say, after the first roll I got $3. A minute later, do I get 3 rolls (one for each $) in quick succession, or is there a minute delay before each individual roll?

Am I allowed to stop betting and leave at any time? If so, will they exchange my dollars for larger denominations, or write a check?

Funny. I felt uncomfortable about using such complex math as Lagrange Interpolating Polynomial while not remembering how it actually worked. I set out to invent a simpler switching function, which would select one value out of 11 based on an integer variable N and then convert that value into an integer of my choice. And I have invented such formula! Thereafter, I consulted a respectable mathematical reference website and discovered that my simple and beautiful switching function was the Lagrange Interpolating Polynomial! Now I understand how it works! It appears 18th century mathematicians (Waring and Euler) invented it just for this puzzle. I shall give a better explanation the puzzle solution and Lagrange Polynomial. It could be useful to some math students and, perhaps, parents of high school students who are about to get bogged down in those dreaded AP math classes. Lagrange Polynomial: http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html

Here's a winning strategy In this example, doesn't 2nd player win on his 3rd move having (2,7,6)?

No. We have exactly 11 pairs of points (xk,yk). Where xk is one of the tan(...) values and yk is one of the integer values 0 through 10. It should be big and messy, but a regular polynomial of 10th degree. And another correction to start output with zeros instead of ones, just subtract another 1 from that whole thing.

Oops, messed up a little...

But they don't need to be integers, those 11 periodic values.

I am assuming N must be an integer. As for how to slice two pi into 11 pieces, which would yield the required progression, I am stumped. I just want to throw a different thought into this subject:

That is a good solve! Exact and exhaustive proof. Complexity could be reduced just a bit, though.

A note of clarification: BrainDen Casino table games are set in a typical way. When a guest wins, he keeps his bet too. E.g., in this single die roll game, if you bet $1 and your number is rolled, you get $4 and keep your $1 bet.

Mr. Gamble and Mr. Prude went to the Brain Den Casino, which offered the following game: A customer bets on a number on a die (1 through 6). The dealer rolls a single die. If the customer’s number comes up, he is paid 4 times his bet, if not - customer loses his bet. Without delay, Mr. G makes his mind to play 5 rounds betting $1 on number “1” each time. Mr. P, believing the game is in favor of the house offers a side bet, whereby if after 5 rolls Mr. G ends up behind he must pay $5 to Mr. P, otherwise Mr. P pays $5 to Mr. G. After 5 rounds, what is the average payoff to: 1) BrainDen Casino? 2) Mr. Gamble? 3) Mr. Prude? Results may be rounded to the nearest cent. Calculators are allowed. Answers obtained by way of computer simulations may not claim the “best answer” spot.

The statement of the problem is unambiguous. It requires finding the smallest number of dice for the winning bet on average (not the average). Bonanova is a Grandmaster of stating problems. If he wanted us to find the average number of rolls before you get all 6 numbers, he would say so (in a better way.)

That's the OP. A computer, or real life simulation of this simply and economically stated probability problem must be simple and straightforward. I look at it as a gambling game, where I can bet $1 on the outcome.

I haven't read the entire thread, not sure if someone has not suggested that already.

The bottom line, the ultimate truth for calculation of a probability is the number of all variations that meet your criteria divided by the number of all possible variations.