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Posted by Paul R. Martin on September 9, 2002 22:07:14 UTC

Hi Mike and Richard,

Overlap is not the problem; the problem is a firm grass bank. But let's suppose, in the California style, that the bank is rigid concrete painted to look like grass.

To show that you don't have to worry about overlap or how wide the boards are (as long as you are able to walk on them), mark a little dot at each end of the first board at the exact center of the edge on the bottom. For example, if the board is 10 inches wide, the dots will be on the bottom of the board 5 inches from the two corners at each end of the board.

When you place the first board at 45 degrees, have the dots down and place the board so the dots are touching the exact edge of the concrete. You can still walk on the board as long as you are on the correct side of the center line. Otherwise the board will pivot on the concrete edge and dump you into the moat. So far, so good.

Now, when you place that second board you do have to overlap the second board over the centerline of the first board a little so it doesn't tip the first board into the drink. Similarly, the other end of the second board must overlap the hard, sharp 90 degree concrete corner that will be its abutment. Make each of these overlaps in the amount of epsilon/2.

Now, as you gain confidence in walking on these boards, you will be able to cut epsilon in half, and try again. By repeating this process, you can get as close to the ideal solution to the problem as you like, which is the same as if the boards had zero width and no overlap. Do the math again, and I think you will find that my first answer is right.

For the advanced student: Those dots don't have to be in the center of the first board. Instead, put them on diagonally opposite corners which increases the effective length of the board a little depending on how wide the board is. Similarly, place the second board across the span along a diagonal for a little bit better reach. Now, to calculate the moat width in this case, you do have to know how wide the boards are. If, for example, they are 100 feet wide, the moat would be over 100 feet wide as well.

Warm regards,


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