Moat-crossing puzzle

There's an old riddle about a square moat that asks how to cross it with two planks that are slightly shorter than the moat's width. Here's an illustration:

If you'd like to consider the riddle yourself, stop reading now.

The full length and width of the moat don't matter, only the distance across the water. In the version I read, it was given as 5 meters. The puzzle is how to cross the moat with two planks 4¾ meters long, without fastening them together into a single long plank. Here's the solution:

Verifying the solution devolves into repeated applications of the Pythagorean theorem on isosceles right triangles. Zooming in on the corner, we need to know the distance from the outer corner of the moat to the inner corner:

Let's call the width of the moat a and the length of the plank b. The distance c from the outer corner of the moat to the inner corner is

We also need to know how far out we can put the first plank, each end distance d from the outer corner:

Finally, we need to know how far our first plank gets us out over the moat, e:

We can easily check that b/2 is a sensible result given that the above is an isosceles right triangle, so the distance from the outer corner of the moat to the center of the plank should be half the length of the plank.

To find how much the first plank shortens the distance across the moat, find c − e:

In the case of a moat 5 meters across and a plank 4¾ long, the gap remaining to be crossed is

meters, which is just shorter than our second 4¾-meter plank.

How short can the planks be? They have to be longer than c − e, or, put mathematically,

The planks have to be at least 0.94 times the width of the moat. Given that this puzzle is only interesting when the planks are shorter than the width of the moat, we have a narrow range of possibility:

Shown graphically, with our 5-meter moat and 4¾-meter planks marked as a dot,

A narrow slice of possibility like this is part of what makes an interesting puzzle.