Three ends

Adding a third end

Two ends is arithmetic. Three ends is arithmetic with a much worse answer.

Part four

8 notes in this part

Two hundred and thirty-three of the three hundred and twenty-five three-end arrangements leave no hour at all. The collapse is steeper than it looks.

Analysis

Nearly three quarters of the three-end arrangements leave nothing. They share one property, and it predicts every one of them.

Reference

Ninety-two three-end arrangements leave at least one shared hour. They are tightly clustered and easy to characterise.

Reference

The three-end table is a triangle, not a square, and the diagonal means something specific. How to find any arrangement in one step.

Reference

Thirty-five of the three hundred and twenty-five three-end arrangements leave four hours or more. The set is narrower than the count suggests.

Analysis

When one of three ends sits between the other two it contributes nothing to the arithmetic and everything to the work.

Analysis

A fourth end adds no new behaviour. Once only the outermost two matter, every further end is inside the span or becomes the new extreme.

Reference

Twenty-two three-end arrangements leave exactly one shared hour. One hour is a worse position than it sounds.

Analysis