Question

Puzzle #101

Answer 1

Question 2. What if Randall goes first?
Question 3. What if it's an m×n grid?
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Source: http://spikedmath.com/puzzle-006.html

Answer 2

can mike darken only the squares he has written numbers into or all the squares with the largest number in any row?

Answer 3

The ones with largest number I think.

Answer 4

For question 1 and 2, the second player has the winning strategy. For any turn, if Player 1 puts a number in row i, Player 2 needs to put a number in the same row. In each row, make the first three spots group 1, and the second 3 spots group 2. Whichever group Player 1 puts the number in, Player 2 puts his number in the other group. Now, in the odd rows, Player 2 can force the black square to end up in Group 1 since:

1) If player 1 puts numbers in group 1, player 2 responds with smaller numbers in group 2
2) If player 1 puts numbers in group 2, player 2 responds with larger numbers in group 1.

In the even rows, do the opposite (switch the words smaller and larger), and this will result in the black squares ending up in group 2.

If player 2 plays like this, the only possible way player 1 can win is if the grid looks like this:So to counter this, all player 2 has to do is change his strategy in the sixth row, and make sure the black square doesnt end up in column 2, 3, or 4. to accomplish this task, make group 1 columns 2, 3, and 4, and group 2 columns 1, 5, and 6. If player 2 plays like he did in the other even rows, this will result in his victory.