## Proof By Lice!

From camels to lice. (A proof promised here.)

Theorem (Hopf and Pannwitz, 1934): Let $X$ be a set of $n$ points in the plane  in general position (no three points on a line) and consider $n+1$ line segments whose endpoints are in $X$.  Then there are two disjoint line segments.

Micha Perles’s proof by Lice:

Useful properties of lice: A louse lives on a head and wishes to lay an egg on a hair.

Think about the points in the plane as little heads, and think about each line segments between two points as a hair.

The proof goes as follows:

Step one: You take $n$ lice from your own head and put them on the $n$ points of \$X\$.

Step two: each louse examines the hairs coming from the head and lay eggs (on the hair near the head)

Step three (not strictly needed): You take back the $n$ lice and put them back on your head.

To make it work we need a special type of lice: spoiled-left-wing-louse.

A spoiled-left-wing louse lays an egg on a hair if and only if the area near the head,  180 degrees to the right of this hair is free from other hairs.

Lemma: Every louse lays at most one egg.

Proof of lemma: As you see from the picture, if the louse lays an egg on one hair, this hair disturbs every other hair.

Proof of theorem continued: since there are $n+1$ line segments and only at most $n$ eggs there is a hair X between heads A and B with no eggs.

We look at this hair and ask:

Why don’t we have an egg near head A: because there is a hair Y in the angle 180° to the right.

Why don’t we have an egg near head B: because there is a hair Z in the angle 180° to the right.

Y and Z must be disjoint. Q. E. D.

Remarks: We actually get a ZIG formed by Y, X, and Z

If we use right-wing-spoiled lice we will get a ZAG.

We can allow the points not to be in general position as long as one hair from a head does not contain another hair from the same head.

The topological version of this problem is the infamous Conway’s thrackle conjecture. See also Stephan Wehner page about it.

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### 7 Responses to Proof By Lice!

1. One can ask the question in the dual plane (points become lines and lines become points) and so extend it to pseudo-lines. Thm 1.10 here: https://www.cs.bgu.ac.il/~shakhar/my_papers/pseudolines.ps
extends those idea to pseudolines…

• Gil Kalai says:

Many thanks Shakhar. PS. many people cannot access ps files sp perhaps you can translate it to pdf?

2. shacharlovett says:

are louse = lice?

• Gil Kalai says:

Shachar, my inquiries strongly indicate that the plural of louse is lice. I dont know why the ‘s’ is transformed into ‘c’.

• jj says:

mouse and mice