Mowing a Chessboard
Imagine that you're mowing an
chessboard pattern into your lawn. Here's how you plan to do it:
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Your mower is wide enough to only mow one row or column at a time.
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You proceed by choosing a row or column and mowing the entire thing. Once you start mowing a row or column you must continue mowing it until you reach the other side of the board.
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You can mow over squares that you've already mowed over as long as at least one square in that row/column has yet to be mowed.
If is the minimum number of times you must cross the board and is the maximum number of times, what is ?
The answer is 23.
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1 solution
The minimum amount of mows is 8. This is obtained by mowing 8 rows or 8 columns in succession. The minimum cannot be less than 8 because that would imply that a row or column remains un-mowed.
The maximum amount of mows is 15. This can be obtained by mowing the 1st column, then the 1st row, then the 2nd column, then the 2nd row, and so on. You will notice that if you continue this pattern, you will not be able to mow the 8th row after you have mowed the 8th column. This is because the lawn will be completely mowed after you have mowed the 8th column. Thus, the maximum is 8 (columns) + 7 (rows) = 15.
The maximum cannot be more than 15 because a single row or column cannot be mowed more than once, and after 8 rows or 8 columns are mowed, the lawn is completely mowed. If 8 columns are mowed, then the maximum number of rows that can be mowed is 7. Similarly, if 8 rows are mowed, then the maximum number of columns that can be mowed is 7.
Thus, the maximum 15 plus the minimum 8 is 23.