Let $d$ be a positive real number. The scorpion tries to catch the flea on a $10\times 10$ chessboard. The length of the side of each small square of the chessboard is $1$. In this game, the flea and the scorpion move alternately. The flea is always on one of the $121$ vertexes of the chessboard and, in each turn, can jump from the vertex where it is to one of the adjacent vertexes. The scorpion moves on the boundary line of the chessboard, and, in each turn, it can walk along any path of length less than $d$. At the beginning, the flea is at the center of the chessboard and the scorpion is at a point that he chooses on the boundary line. The flea is the first one to play. The flea is said to escape if it reaches a point of the boundary line, which the scorpion can't reach in the next turn. Obviously, for big values of $d$, the scorpion has a strategy to prevent the flea's escape. For what values of $d$ can the flea escape? Justify your answer.
Problem
Source: 2011 Lusophon Mathematical Olympiad - Problem 6
Tags: combinatorics proposed, combinatorics
25.09.2011 21:26
$d=5$ If it is less, the flea can win, he does sqaures of $1*1$ and the scorpion will do a bit less than the whole boundary and you can easily see the best strategy for the socorpion doesn't go. But when $d\ge 5$ he will win with that one.
26.09.2011 05:08
The winning strategy for flea is to achieve to the boundary line in less moves as possible, ie, in 5 moves. As a flea starts, the scorpion must to reach the flea in him fifth move, otherwise, the flea will escape. The flea will run away from the scorpion by opposite side when the scorpion was placed. Now, we have choose the best position to put the scorpion. If we choose the middle of the side, so he will have to walk 20 sides of squares in 5 moves. Then for d = 4, the scorpion still wins. But if we choose one of chessboard's corner, for simetry, the flea will have two equals options. For example, if we choose the upper left corner, the flea will run to the right side or to the underside. Now, the scorpion will have to walk 15 sides of squares in 5 moves. Then for d = 3, the scorpions wins. So for d < 3, the flea can escape.
26.09.2011 08:55
False, a simple thing to prove the flea can win if $d<4:$ Name the square $ABCD$ and $E,F,G,H$ are the middles of $AB,BC\cdots.$ By symmetrie , we can say the scorpion starts in $[AE]$ The flea goes away of the scorpion, if this one goes to the side of $B$ he will lose, by the flea going to $G.$ So the scorpion is at the other way, then the flea can go to side $BC$ and see what scorpion does and win.
27.09.2011 19:28
SCP wrote: False, a simple thing to prove the flea can win if $d<4:$ Name the square $ABCD$ and $E,F,G,H$ are the middles of $AB,BC\cdots.$ By symmetrie , we can say the scorpion starts in $[AE]$ The flea goes away of the scorpion, if this one goes to the side of $B$ he will lose, by the flea going to $G.$ So the scorpion is at the other way, then the flea can go to side $BC$ and see what scorpion does and win. gaga_t14 wrote: The winning strategy for flea is to achieve to the boundary line in less moves as possible, ie, in 5 moves. As a flea starts, the scorpion must to reach the flea in him fifth move, otherwise, the flea will escape. The flea will run away from the scorpion by opposite side when the scorpion was placed. Now, we have choose the best position to put the scorpion. If we choose the middle of the side, so he will have to walk 20 sides of squares in 5 moves. Then for d = 4, the scorpion still wins. But if we choose one of chessboard's corner, for simetry, the flea will have two equals options. For example, if we choose the upper left corner, the flea will run to the right side or to the underside. Now, the scorpion will have to walk 15 sides of squares in 5 moves. Then for d = 3, the scorpions wins. So for d < 3, the flea can escape. Your solutions are wrong. $d$ doesn't have to be integer... it can assume any real value.
27.09.2011 20:17
Sth to answer: if we look to $d<5$ it consist lots of reals, and otherwise $d\ge5$ also, he had mistake in proving for $d=4$ as ex.
30.09.2011 19:45
Well, as nunoarala said, d can assume any real value. You were right when you said that if $d\ge 5$ the scorpion wins. Keep trying
07.06.2012 01:24
Has anyone already solved this problem? Please post your solutions.
06.12.2018 13:18
Any ideas, pls?
27.12.2018 15:51
Someone??