Reach 90
Dan and Sam play a game in which the first to start says the number and the one who's next must say an integer number between the number previously said and its triple (but not including).
For example, Dan begins saying 1, then Sam can say whichever number he wants between 1 and 3; as the only integer between 1 and 3 is 2, he must say 2. Then, Dan can choose any number between 2 and 6; that is, he can say either 3, 4 or 5.
The game finishes when someone reaches 90 (who is the winner). If Dan begins, who will win? This means, who has a winning strategy?
This is the third problem of the set Winning Strategies .
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.
1 solution
For one of the players to be assured that he loses if the adversary plays optimally it must mean that that player has the number 90/3 = 30 since for that number whichever number he will say afterwards will give the adversary the possibility to reach 90 but for which it is not possible to reach more than a value that when is tripled is less than 90. This means that whoever can make the other player reach 30 will have a winning strategy and observe further that this can be considered generally for the case of each step of the game which for this particular case of the problem implies that whoever achieves at the other 2 steps 10 and 3 will lose being therefore implied that if the first player plays optimally , that is Dan , Dan will win.