Grades 11-12 Video Solutions 2025-2025_11-12

Video Transcription

This is problem 16 on the grade 11 and 12 2025 math kangaroo contest. On a giant 4x4 chessboard, there are 16 kangaroos, one in each square.  On each turn, each of the kangaroos jumps to a neighboring square, up, down, left, or right, but not diagonally, this is important.  All kangaroos can stay on the board, or all kangaroos must stay on the board. There are multiple kangaroos on any square.  After 100 turns, what is the largest possible number of empty squares? Our answer choices are A, 15, B, 14, C, 12, D, 10, and E, 8.  Take your time, try to go about doing this one, maybe try not to play 100 rounds of this, but if it's fun, I suppose you can.  So this problem sort of introduces this notion of what are called, at large, invariants. So let's say I am this kangaroo right here, in the corner.  I'm currently on a black square, at turn N, so I guess turn 0 for now. And in two turns, I must jump to a white square, and then now that I'm on a white square at  turn 1, on turn 2, I must jump to a black square. And I won't draw all of them in, but I have to end up on a black square.  So at turn N, if I'm on either black or white, N plus 2 means I need to be on that same black or white tile. So on turn 0, kangaroos are either on black or white tiles.  So that means after 100 turns, that's an even number, so we can do 0, then turn 2, then turn 4, then turn 6, all the way up to turn 100, there must still be kangaroos on white  and black squares. So at minimum, at least two squares must be occupied. One white square and one black square. So that eliminates A, 15.  And then it's really just a game of, well, can I fit every kangaroo on a square? And it turns out the answer is yes. So let's just try to keep some target squares in mind.  Maybe let's call this bottom right square the target white square, and this top right square the target black square. Still I can just sort of outline an algorithm.  If at turn 0 I am a kangaroo on black, then I should try to go to the target square. And if I'm on the target, then I should try to just bounce on and off the target.  So go on and off. And this algorithm actually works for both black and white. You notice that they have to go to the target. So let's take this one here.  So after four turns, it gets to the target, and then it sort of infinitely loops back and forth and back and forth and back and forth up until turn 100.  And we can be confident that it ends up on a black square by our parity argument, by this invariant.  And for any white kangaroo, well, it just takes some even number of turns to get to this target white square. And then on every even turn, you end up back at this target square.  So that means that at the end of turn 100, and actually much earlier than that, every single kangaroo will either be on this square or it will be on this square.  And that's our answer. It can't be 15, but we've devised a method to get 14. So 14 is our answer.

Video Summary

In a 4x4 chessboard, 16 kangaroos each occupy one square, and they must jump to a neighboring square every turn without leaving the board. After 100 turns, the goal is to find the maximum number of empty squares. The problem introduces invariants, explaining how kangaroos starting on a black square will return to a black square after an even number of turns, and the same applies to kangaroos starting on white squares. To maintain coloring parity after 100 turns, at minimum, one black and one white square must be occupied. By strategically jumping, all remaining kangaroos can be gathered on these two squares, leaving 14 others empty. Thus, the largest number of possible empty squares is 14, option B.

Keywords

4x4 chessboard

kangaroo jumps

maximum empty squares

coloring parity

strategic jumping