Kotani’s Ant Problem: Simple but counterintuitive mystery

Where is the farthest point on the surface of a cuboid?

Consider a unit cube. Where is the farthest point starting from a certain vertex and moving over the cube’s surface? This is easy. Intuitively, the opposite vertex is the furthest away. Just to be sure, let’s draw a figure to confirm.

Surface distance point on unit cube from origin (by author)

So what if we consider a 1×1×2 rectangle with two unit cubes stacked on each other? Would the farthest point from the origin (0,0,0) be (1,1,2)?

Are you sure you think of it? You have a chance to reconsider.
Let’s check by drawing a figure in the same way.

Surface distance points from the origin on a 1×1×2 cuboid (by author)

Hmm? Wasn’t something different from what you expected at the last moment? Let’s look at it more slowly, using only the top view to get a clearer picture.

Surface distance points from the origin on a 1×1×2 cuboid, top surface only (by author)

The animation might be ambiguous. Let’s use the development to figure out how far we can reach with 2√2, the distance along the plane from (0,0,0) to (1,1,2).

The area that is reachable within a distance of 2√2 on a 1x1x2 rectangular surface

The development shows that parts of the rectangle’s top surface cannot be reached within a distance of 2√2. This indicates that the farthest point that can be reached from (0,0,0) by traversing over the surface of the rectangle is not (1,1,2).

Let’s look at a 1×1×10 cube as an extreme example to see why this happens.

Surface distance points from the origin on a 1×1×10 cuboid (by author)

In this way, it is easy to understand. The distance from the origin to (0,0,10) is 10, while the distance from the origin (0,0,0) to (1,0,10) is √101=10.0499, and the surface distance from the origin (0,0,0) to (1,1,10) is √104=10.1980, so the difference between these distances is slight. Since these differences (0.0499 and 0.1980) are relatively small compared to the distance from the edge of the top surface to the center of the top surface (0.5), the farthest point from the origin is near the center of the top surface.

Let’s look at this one, too, using only the top view to be sure.

Surface distance points from the origin on a 1×1×10 cuboid, top surface only (by author)

This seemingly counterintuitive problem is called “Kotani’s ant problem,” a mathematical puzzle problem invented by Yoshiyuki Kotani, a computer scientist and puzzle enthusiast.

This article does not show how to find the coordinates of the farthest point. However, if you are interested, try it.

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References

https://ja.wikipedia.org/wiki/%E5%B0%8F%E8%B0%B7%E3%81%AE%E8%9F%BB%E3%81%AE%E5%95%8F%E9%A1%8C