This question appeared in an old math contest, and I seem to remember that the answer I had back when I first saw it was unsatisfactory.
You have a deck of cards numbered from $1$ to $60$. First, you shuffle them, and deal them out into six rows of ten cards each. Then, you do the following two things:
You rearrange the cards in each row so that they are in increasing order from left to right.
Then, you rearrange the cards in each column so that they are in increasing order from top to bottom.
Are the cards still in increasing order from left to right after the second step?