Antonio went to play miniature golf on Monday, when it cost $16 to rent the club and ball, plus $2 per game. Clayton went Thursday, paying $4 per game, plus rental fees of $4. By coincidence, they played the same number of games for the same total cost. How much did each one spend? How many games did each one play?
Let's say they both played x games.
Antonio spent 16 + 2x dollars.
Clayton spent 4x + 4 = 2x + 20 dollars.
If they spent the same amount of money, then
16 + 2x = 2x + 20
16 = 20, which is not true.
Let's try to find how much they would have spent if they each played one more game.
Antonio would have spent 16 + 2(x+1) = 16 + 2x + 2 = 18 + 2x dollars.
Clayton would have spent 4(x+1) + 4 = 4x + 8 + 4 = 4x + 12 dollars.
If they spent the same amount of money, then
18 + 2x = 4x + 12
18 - 12 = 4x - 2x
6 = 2x
x = 3
So they each played 3 games.
Antonio spent 16 + 2(3) = 16 + 6 = <<16+2*3=22>>22 dollars.
Clayton spent 4(3) + 4 = 12 + 4 = <<4*3+4=16>>16 dollars.
Therefore, Antonio spent 22 dollars and played 3 games, and Clayton spent 16 dollars and played 3 games. Answer: \boxed{22, 3 \quad\text{and}\quad 16, 3}.