Z.P.Su’s second problem
The other day Zhendong Patrick Su showed me the following problem of the last Putnam Competition, in which he had participated. (He had solved the problem, I did not.)
Consider the points in or on the right-angled triangle with hypotenuse of length √:
Let each of these points have 1 of 4 colours. Show the existence of a monochrome pair with mutual distance ≥ 2 − √.
Let us call that critical distance r, i.e. r = 2 − √; let us call a monochrome pair at mutual distance ≥ r a “desired pair”.
If A belongs to a desired pair, we are done; similarly for B. If neither A nor B belongs to a desired pair, we proceed as follows.
To begin with we observe that r satisfies r = √ from which we · (1−r),
conclude that, with AP = r and BQ = r, we have PQ = r. Completing the diamonds with AR = r and BS = r, we conclude RQ = r and PS = r. Being the hypotenuse of a right-angled triangle with side = r, RC and SC are both > r.
In the case that neither A nor B belongs to a desired pair, A and B —more than r apart— are of different colour, and P, S, C, R, Q —all ≥ r apart from A and B— are all of the remaining 2 colours. Consequently, in the cyclic arrangement P, S, C, R, Q of these 5 points, 5 being odd, two successive points are of the same colour. Their mutual distance being at least r, they form a desired pair. QED.
* * *
In the above argument we have appealed to the following theorem:
Consider a finite undirected graph in which each vertex is of degree 2. If each vertex is of one of 2 colours and the number of vertices is odd, there exists an edge that connects a vertex to a vertex of the same colour.
The proof is by double application of the pigeon-hole principle. Let the number of vertices —which equals the number of edges— be 2n+1. Since there are 2 colours, there are at least n+1 vertices of the dominant colour; as they are all of degree 2, they give rise to 2n+2 endpoints of the dominant colour. Since there are “only” 2n+1 edges, at least 1 of them has 2 endpoints of the dominant colour.
Austin, 25 December 1994
prof.dr. Edsger W.Dijkstra
Department of Computer Sciences
The University of Texas at Austin
Austin, TX 78712-1188