Tom White, May 2025
My family walked up Tor y Foel in the Brecon Beacons when the children were small, and they weren't very impressed by the number of false summits (points that looked like they were the top, but weren't). The fact that I forgot to bring the chocolate didn't help either.
This got me thinking - what shape would a hill with the most number of false summits be? And also - how many false summits were there on Tor y Foel?
Here's a diagram of the top of a hill with one false summit and the final summit. The walker at point A cannot see the final summit while they are walking from A to B since it is exactly in line with the false summit (C). Note that the walker's height is BC (which we assume is 1 for simplicity).
As soon at the walker scrambles up the (vertical) face at B they can see the final summit. Try dragging the walker's foot (initially at point A) to see the effect of climbing the hill.
This diagram shows that point A cannot be any higher up, for if it was then C would not be a false summit, since the final summit would be visible from all points on the line AB. (More generally, the terrain between A and B cannot be above the line AB since it would allow the walker to get a glimpse of the higher summit. It could dip below the line AB though.)
So the line CD determines the gradient of AB. In particular, AB cannot be less steep then CD.
We can use this idea to construct multiple false summits by working backwards (downhill). For example, here are four false summits:
As the number of false summits increases, the shape of the hill gets closer to a parabola. This shows 20 false summits, and they are barely visible.
All of the constructions so far assume that the horizontal distance between false summits is the same. But there is no reason not to vary it.
In fact, if lower false summits had a smaller horizontal separation then they would have a smaller vertical separation, and it would be possible to fit more false summits in to the construction. So the above construction is by no means optimal. Taken to the limit, the optimal solution is probably a series of cliffs, each the height of the walker, in close succession. So more like rock climbing!
It's interesting to count the number of false summits on real hills that you have climbed up. Visualize Tor y Foel's five false summits on this page.