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JLee
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Here is a pretty long path that does not visit any square twice, but misses misses a few spots:

enter image description here
The path visits 96 squares.

To turn this into a solution, we have to add a few extra moves to mow those missed spots:

The five squares inside of the P will take 8 extra moves to mow.
The other five loose unmowed squares can be done individually using 2 moves each.

That gives a total number of moves of:

96+8+10=114 moves

I don't know if this is optimal

Here is a pretty long path that does not visit any square twice, but misses misses a few spots:

enter image description here
The path visits 96 squares.

To turn this into a solution, we have to add a few extra moves to mow those missed spots:

The five squares inside of the P will take 8 extra moves to mow.
The other five loose unmowed squares can be done individually using 2 moves each.

That gives a total number of moves of:

96+8+10=114 moves

I don't know if this is optimal

Here is a pretty long path that does not visit any square twice, but misses a few spots:

enter image description here
The path visits 96 squares.

To turn this into a solution, we have to add a few extra moves to mow those missed spots:

The five squares inside of the P will take 8 extra moves to mow.
The other five loose unmowed squares can be done individually using 2 moves each.

That gives a total number of moves of:

96+8+10=114 moves

I don't know if this is optimal

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Jaap Scherphuis
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Here is a pretty long path that does not visit any square twice, but misses misses a few spots:

enter image description here
The path visits 96 squares.

To turn this into a solution, we have to add a few extra moves to mow those missed spots:

The five squares inside of the P will take 8 extra moves to mow.
The other five loose unmowed squares can be done individually using 2 moves each.

That gives a total number of moves of:

96+8+10=114 moves

I don't know if this is optimal