VicKmath7 wrote: A teacher and her 30 students play a game on an infinite cell grid. The teacher starts first, then each of the 30 students makes a move, then the teacher and so on. On one move the person can color one unit segment on the grid. A segment cannot be colored twice. The teacher wins if, after 31 games, there exists a rectangle $1$x$2$ or $2$x$1$, such that each segment from it's border is colored, but the segment between the two adjacent squares isn't colored. Prove that the teacher can win. I believe there might be a slight inaccuracy in translation - the condition where the teacher wins should read "the teacher wins if, after the move of one of the 31 players, there is a $1\times 2$ or $2\times 1$ rectangle, ..." The solution to this problem is fairly simple. The teacher just draws a $100\times 100$ square within $400$ cycles, which means that there are at most $400\times 30$ segments coloured in the square. This is less than the $2\times 100\times 99$ total segments inside, which means that there are still some empty segments. Then the teacher and students can colour in the segments until the square is fully coloured. But this means that just before the square is fully coloured, there exists exactly one empty segment inside, which fulfils the desired condition, and the teacher wins.

Pretty Nice problem. Call a $n\times n$ square colorful if it's border is colored. We try to notice some observation. If teacher tries to select a $k\times k$ square coloring it's border, students will try to make all the inner small squares colorful. If teacher make the $k\times k$ square colorful before making all the inner small squares colorful(by the students), after some finite step their will be at least one $1\times 2$ or $2\times 1$ rectangle in the $k\times k$ square. So, teacher jave to assure that, student can't make all the the inner small squares colorful. To make $k\times k$ colorful teacher have to make $4k$ moves. Students have to do $2k(k-1)$ moves to make all the inners small squares colorful. But, students can make $30\times(4k-1)$ moves since teacher made 4k moves. Then teacher will win if the inequality satisfies, \[30\times (4k-1)<2k(k-1)\]Some sort of calculation will give, $k\geq 61$. So, the teacher will win.

Elegant solutions!

color 200*200 border square with 4*200 then 2*200*199>400*30 so the all squares is not colored when only one left is anwer of this question