Two players $A$ and $B$ play a game with a ball and $n$ boxes placed onto the vertices of a regular $n$-gon where $n$ is a positive integer. Initially, the ball is hidden in a box by player $A$. At each step, $B$ chooses a box, then player $A$ says the distance of the ball to the selected box to player $B$ and moves the ball to an adjacent box. If $B$ finds the ball, then $B$ wins. Find the least number of steps for which $B$ can guarantee to win.
Problem
Source: Turkey JBMO Team Selection Test 2013, P3
Tags: geometry, geometric transformation, combinatorics proposed, combinatorics
31.05.2013 23:52
There is something lost in the translation. Let say that after $B$ chooses a box, not only player $A$ says the distance of the ball to the selected box to player $B$, but even shows it to $B$, before moving the ball to an adjacent box, of course without $B$ knowing where. So $B$ knows the two places where the ball can be, but how can he decide?
01.06.2013 00:11
mavropnevma wrote: There is something lost in the translation. Let say that after $B$ chooses a box, not only player $A$ says the distance of the ball to the selected box to player $B$, but even shows it to $B$, before moving the ball to an adjacent box, of course without $B$ knowing where. So $B$ knows the two places where the ball can be, but how can he decide? I think $B$ can guess just after $A$ says the distance.
01.06.2013 02:06
How can $A$ say the distance before $B$ takes a guess? Then $A$ only knows where the ball is, not where $B$ will choose, so he cannot say the distance.
01.06.2013 15:07
It's obviously wrong for $n=3$
01.06.2013 15:26
mavropnevma wrote: How can $A$ say the distance before $B$ takes a guess? Then $A$ only knows where the ball is, not where $B$ will choose, so he cannot say the distance. I am also unsure about the statement, it is originally not clear, however I suppose that B can guess any time, I mean at each step B can guess more than one times. Because otherwise it is clear that B can never find the ball.
01.06.2013 15:31
crazyfehmy wrote: mavropnevma wrote: How can $A$ say the distance before $B$ takes a guess? Then $A$ only knows where the ball is, not where $B$ will choose, so he cannot say the distance. I am also unsure about the statement, it is originally not clear, however I suppose that B can guess any time, I mean at each step B can guess more than one times. Because otherwise it is clear that B can never find the ball. If this is a original problem and $B$ can guess more than one time in each step , so $B$ can ask and after $A$ answer , $B$ guess the two boxes and he can win !!!
07.06.2013 12:38
The question can be salvaged by requiring that the ball always move in the same direction. If $B$ knows the direction, he can obviously find the ball in no more than three guesses. If $B$ does not know the direction beforehand, it may take four.
12.11.2013 09:28
This is the official English translation of the problem statement: There are $n$ chests places on the vertices of a regular $n-$gon and a bead. Alice and Bob play a game. At the beginning Alice hides the bead in one of the chests. Each move consists of three stages: Alice if wishes can secretly move the bead from the located chest to the any of two neighboring chest if the neighboring chest is not chosen by Bob at the previous move Bob chooses one of the chests Alice tells the distance from the chosen chest the chest where the bean is located If Bob can determine the chest containing bean at the end of some move he wins. For all values of $n\geq 3$ determine the minimal number of moves necessary for Bob to guarantee winning.