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# The Classic Moat Problem

## Question

this is a classic problem so i was surprised that i could not find it in the forum searches but of course that is not to say that it isn't there .

Problem:
Suppose you have two unit-length boards. What is the widest moat you can cross if you have no means to nail or otherwise attach them together?

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Assuming there's a 90-degree corner, then the widest moat that's possible to cross is slightly smaller than sqrt(5) / 2.

Reasoning:
Lay one board at a 45-degree angle across the outer corner of the moat, and then rest the other board perpendicular to the first such that it lands on the corner on the inside of the moat.

This gives you a triangle with legs 1 and 1/2, which has a hypotenuse (corresponding to the moat width) of sqrt(5) / 2. The actual width has to be a little less than that, since you lose some length of the board in the process of bearing on the ground and on the other board.

I'm not really sure how to post a quick MS Paint image to make it clearer...

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Assuming there's a 90-degree corner, then the widest moat that's possible to cross is slightly smaller than sqrt(5) / 2.

Reasoning:

Lay one board at a 45-degree angle across the outer corner of the moat, and then rest the other board perpendicular to the first such that it lands on the corner on the inside of the moat.

This gives you a triangle with legs 1 and 1/2, which has a hypotenuse (corresponding to the moat width) of sqrt(5) / 2. The actual width has to be a little less than that, since you lose some length of the board in the process of bearing on the ground and on the other board.

I'm not really sure how to post a quick MS Paint image to make it clearer...

Here's what you're describing, I believe...

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With that setup, I don't think you can call the width of the moat equal to the hypotenuse of the triangle formed by half of the first board plus all of the second board. A line from one end of the first board to the end of the second board that's touching the opposite shore would not be perpendicular to the moat. You would need to solve for the length that's perpendicular to the moat in order to find out how wide of a moat you could cross.

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What shape is the moat? Does it have to be a straight line? Can it vary in width?

If it can be any shape, just choose a moat in the shape of a very pointy arrow and cross over at the point using 1 board.

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What shape is the moat? Does it have to be a straight line? Can it vary in width?

If it can be any shape, just choose a moat in the shape of a very pointy arrow and cross over at the point using 1 board.

What shape is the moat? Does it have to be a straight line? Can it vary in width?

If it can be any shape, just choose a moat in the shape of a very pointy arrow and cross over at the point using 1 board.

What shape is the moat? Does it have to be a straight line? Can it vary in width?

If it can be any shape, just choose a moat in the shape of a very pointy arrow and cross over at the point using 1 board.

The question isn't whether you CAN cross the moat. The question is, what is the widest the moat can be to make it across.

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Rookie's classic (with some assumptions not made in the OP).

If the moat has a square corner,

then two planks of length x

will allow you to cross a moat of width 1.5x/sqrt(2).

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