Question 11
Initially, there is one beetle on each square in the set $S$. Suddenly each beetle flies to a new square, subject to the following conditions:
(1) the new square may be the same as the old or different;
(2) more than one beetle may choose the same new square;
(3) if two beetles are initially in squares with a common vertex, then after the flight they are either in the same square or in squares with a common vertex.
Suppose $S$ is the set of all squares in the middle row and column of a $99 \times 99$ chess board, is it true that there must always be a beetle whose new square shares a vertex with its old square (or is identical with it)?
What if $S$ also includes all the border squares (so $S$ is rows $1$, $50$ and $99$ and columns $1$, $50$ and $99$)?
What if $S$ is all squares of the board?
(1) the new square may be the same as the old or different;
(2) more than one beetle may choose the same new square;
(3) if two beetles are initially in squares with a common vertex, then after the flight they are either in the same square or in squares with a common vertex.
Suppose $S$ is the set of all squares in the middle row and column of a $99 \times 99$ chess board, is it true that there must always be a beetle whose new square shares a vertex with its old square (or is identical with it)?
What if $S$ also includes all the border squares (so $S$ is rows $1$, $50$ and $99$ and columns $1$, $50$ and $99$)?
What if $S$ is all squares of the board?