Problem 227

[noitues]

I would like some clarification on Problem 227.

"The game ends when one player has both dice after they have been rolled and passed, that player has then lost."
- Does this mean that as the dice go around, if at the end on one turn any player has both dice he is out of the circle?
- If so, does the game continue with the other 99 players? Again, if so, how do they start opposite each other?
(Side Note: If this is not it, it would be an interesting twist to the problem, "Average number of turns till only one person left")

- Thinking of another interpretation, Is it just the straight forward 100 players are playing and when one person gets both dice, the game is over. The average number of turns to reach that point is the answer?

I appreciate the help here. I love to try out the new problems, though don't usually get the answer. Problem makers, keep up the great work!

Noitues

[Robert_Gerbicz]

Another question: how to enter the number? With point or comma?

[harryh]

when one person gets both dice, the game is over. The average number of turns to reach that point is the answer?

Yes.

Another question: how to enter the number? With point or comma?

Decimal point as is the norm with all problem answers.

[Robert_Gerbicz]

when one person gets both dice, the game is over. The average number of turns to reach that point is the answer?

Yes.

Another question: how to enter the number? With point or comma?

Decimal point as is the norm with all problem answers.

Thanks. Now solved the problem, in fact for me the problem was the definition with the significant digits. Who doesn't know: http://en.wikipedia.org/wiki/Significant_figures.

Solving about 2000 problems on different sites none of them asked significant digits, even the oldish acm contests problems.

[karlo]

Do we need to submit the answer with 10 digits after the point? Or we just round it and if, after that, some last digits are zero we can let them out?

[Robert_Gerbicz]

Do we need to submit the answer with 10 digits after the point? Or we just round it and if, after that, some last digits are zero we can let them out?

No, read the wikipedia's article! For this problem, because the answer will be in [1000,10000) the submitted answer should be in the form: abcd.efghij
(and use rounding to get the last digit).

[karlo]

Thanks Robert_Gerbicz. Also I am not sure what the statement below implies:
'The game ends when one player has both dice after they have been rolled and passed, that player has then lost.'
Does it mean that at the moment one player has both dice the game ends. Or does it imply that the player rolls them once more, and that counts as a turn? In my opinion, the comma makes the statement a little ambiguous.

[DavidF]

Another thing I thought was unclear.

Suppose the two people with dice are next to each other: D1 D2

Then D1 throws a 6, D2 throws a 1. Does the game end (because one of them must in practice throw first and give the dice to the other one), or do we assume the transitions occur simultaneously giving D2 D1 without ending the game.

(It wasn't difficult to try both possibilities, so it wasn't a big issue, but possibly this could be made clearer).

[Robert_Gerbicz]

Does it mean that at the moment one player has both dice the game ends. Or does it imply that the player rolls them once more, and that counts as a turn? In my opinion, the comma makes the statement a little ambiguous.

The true is the following: "the moment one player has both dice the game ends"

[karlo]

Another thing I thought was unclear.

Suppose the two people with dice are next to each other: D1 D2

Then D1 throws a 6, D2 throws a 1. Does the game end (because one of them must in practice throw first and give the dice to the other one), or do we assume the transitions occur simultaneously giving D2 D1 without ending the game.

Same question.

[hk]

Another thing I thought was unclear.

Suppose the two people with dice are next to each other: D1 D2

Then D1 throws a 6, D2 throws a 1. Does the game end (because one of them must in practice throw first and give the dice to the other one), or do we assume the transitions occur simultaneously giving D2 D1 without ending the game.

(It wasn't difficult to try both possibilities, so it wasn't a big issue, but possibly this could be made clearer).

So discuss it in the problem forum and not HERE.

BTW the second sentence reads "On each turn, the two players with a die roll it." Isn't that clear enough??

[David F]

So discuss it in the problem forum and not HERE.

Seeing as anyone in the problem forum will have had to correctly interpret the question, I'm not sure discussing it there is helpful compared with discussing it HERE.

BTW the second sentence reads "On each turn, the two players with a die roll it." Isn't that clear enough??

If I thought it was clear enough, I wouldn't have posted. I actually made the other interpretation first, but as I was aware there were 2 possibilities, it was easy enough for me to try the other one and see it gave the correct answer.

[hk]

So discuss it in the problem forum and not HERE.

Seeing as anyone in the problem forum will have had to correctly interpret the question, I'm not sure discussing it there is helpful compared with discussing it HERE.

BTW the second sentence reads "On each turn, the two players with a die roll it." Isn't that clear enough??

If I thought it was clear enough, I wouldn't have posted. I actually made the other interpretation first, but as I was aware there were 2 possibilities, it was easy enough for me to try the other one and see it gave the correct answer.

Then you saw it wrong.

[daniel.is.fischer]

Odd. We specifically wrote "The game ends when one player has both dice after they have been rolled and passed" to remove the possible ambiguity.

[TripleM]

Another thing I thought was unclear.

Suppose the two people with dice are next to each other: D1 D2

Then D1 throws a 6, D2 throws a 1. Does the game end (because one of them must in practice throw first and give the dice to the other one), or do we assume the transitions occur simultaneously giving D2 D1 without ending the game.

(It wasn't difficult to try both possibilities, so it wasn't a big issue, but possibly this could be made clearer).

Perhaps this wasn't in the original problem and was added after you posted this, but

The game ends when one player has both dice after they have been rolled and passed,

covers this exact issue.

[Sunhill]

This is a great question, and I am nowhere near even thinking about planning a solution.

I am however writing examination questions for a final year Mathematics paper, and I have used this question as inspiration.

HOWEVER ... at first glance, I thought a "turn" was the same as a dice throw. So, in a two-player game, Player 1 throws first. If s/he gets 1 or 6, the die is passed to Player 2 and the game is over. Otherwise the game continues. Turn 2 is Player 2's turn, turn 3 is Player 1's turn, and so on.

I can now see that the two turns are really one simultaneous turn, but I am in sympathy with DavidF's earlier comment. Most dice games I've ever played have one throw at a time.

[hk]

Another thing I thought was unclear.

Suppose the two people with dice are next to each other: D1 D2

Then D1 throws a 6, D2 throws a 1. Does the game end (because one of them must in practice throw first and give the dice to the other one), or do we assume the transitions occur simultaneously giving D2 D1 without ending the game.

(It wasn't difficult to try both possibilities, so it wasn't a big issue, but possibly this could be made clearer).

Perhaps this wasn't in the original problem and was added after you posted this, but

The game ends when one player has both dice after they have been rolled and passed,

covers this exact issue.

It has been there all the time but I think people overlook its importance.
Such things happen, especially with this kind of problem.
That is why I said: discuss it in the problems forum.

[sfabriz]

Ok, this problem is driving me a little crazy.

What about at some point player in k position has one die and his neghbour, player k+1 has the other one.
They both roll: player k makes a 3 so he keeps the die, player k+1 makes a 1, so he passes the die to player k.
Now player k would have both dice, they would have been "rolled", but not both of them would have been "passed".
Would the game end at this point?

[hk]

Yes.

[sfabriz]

Thank you!
Now I am ready NOT to solve this one...