Reach 150

Logic Level 4

Dan, Sam and their new friend Dimitri (the one who plays chess), play a game in which the first to start says the number 1, the next says 2, and the one who's next must say an integer number between the number previously said and its double (but not including).

For example, Dan begins saying 1, then Sam says 2, and then Dimitri can say whichever number he wants between 2 and 4; as the only integer between 2 and 4 is 3, he must say 3 3 . Then, Dan can choose any number between 3 and 6; that is, he can say either 4 or 5.

The game finishes when anyone reaches 150 (who is the winner). If Dan begins and then goes Sam, who will win? This means, who has a winning strategy?

This is the sixth problem of the set Winning Strategies .
All of them Nobody Angie Sam Dan Dimitri

Mateo Matijasevick by Mateo Matijasevick , 22, Colombia

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Let's suppose that somebody has a winning strategy, and call him X. Let's analyze the numbers that X must have told in order to win.

The one that was before him (Y) should have said any number between 76 and 149, including the endpoints. In fact, Y was forced to say a number in that interval, because X has a winning strategy. By the same reason, this means that Z was forced to say 75... but how is this possible? In the previous turn, X said k k , so Z could have replied any number between k + 1 k+1 and 2 k − 1 2k-1 (including endpoints); k k could have been at most 38, because if it was a greater number, then Z could have said 76, so Y would have win. But, Z could have said any number between 39 and 75... actually, Z was not forced to say 75.

We have a contradiction, because this means that X does not have a winning strategy, contradicting our hypothesis that X was the optimally winner. Therefore, nobody has a winning strategy.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Hmmmm , I just think you are not being completely articulate about what makes a player not have a winning strategy that is that you do not point out in an explicit manner it and let it implicit. So it might be useful to think at what makes the game be undecided by any player and by thinking that see articulately enough what that reason is so maybe it is firstly important to analyze how they by the play do determine the game because in this way of determining seems to be the reason.

A A - 5 years, 1 month ago
A A
Apr 27, 2016

If one of them would have a wining strategy , say Dan , it would mean that that player will be able to determine in such a way the game (and by this make optimal choices) that no matter how the other 2 players combined would play they would never win that is there will never be one of them that will get the number 76 (150/2 + 1) while that player also can be assured too that he will get a number of 76 or greater. Observe that while Dan can choose his numbers so that he can affect Sam's choices in such a way that Dimitri would never be able to get 76 he won't be able to affect the choices Dimitri makes so that Dimitri can't get the next player have a number which is losing therefore being that his control over the combined play of Dimitri and Sam is limited and ineffective. By this it can be said that the game is not determined in such a way that the choices of only one player would determine a result in such a way that it is necessary that none of the other 2 players would get the 76 being because it can be said that the winner of the game is innately determined by 2 players. However it seems that in this game it was assumed that the combined play of Dimitri and Sam is arbitrary just for the purpose of verifying if the combined play of them in any of it's instances gives or not results where Dan wouldn't win regardless of it's choices and therefore that it didn't consider further the choices of Dimitri and Sam as strategically rational. Consider therefore that they will choose to prevent any player before them win until it is clear that they are in a losing position. Then by considering things this way it can be said that it is necessary one of them loses and choose which of the other wins.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...