36th IMO 1995 shortlist Problem N5 A graph has 12k points. Each point has 3k+6 edges. For any two points the number of points joined to both is the same. Find k. Solution
: k = 3. Let n be the number of points joined to two given points. [We are given that n is independent of the points chosen.] Take any point A of the graph. We count the number N of pairs (B, C), where the point B is joined to A, the point C is not joined to A, and the points B and C are joined. For any point C not joined to A, there are just n points joined to C and A. There are 3k+6 points joined to A and hence (12k - 1 - (3k+6) ) = 9k-7 points not joined to A. So N = (9k-7)n. Now given a point B joined to A, it is joined to 3k+5 other points, n of which are also joined to A, so it is joined to (3k+5-n) points which are not joined to A. So N = (3k+6)(3k+5-n). Thus we have: 9k 2 + 33k - 12kn + 30 + n = 0. Hence n = 3m for some integer m and m = (3k 2 + 11k + 10)/(12k - 1), so 4m = k + 3 + (9k+43)/(12k-1). If k ≥ 15, then 0 < (9k+43)/(12k-1) < 1, which would make m non-integral. It is now easy to check that (9k+43)/(12k-1) = 52/11, 61/23, 2, 79/47, 88/59, 97/71, 106/83, 23/19, 124/107, 19/17, 142/131, 151/143, 32/31, 169/167 for k = 1, 2, ... , 14. So the only solution in k = 3. We now have to demonstrate explicitly that there is such a graph for k = 3. Consider: 1 2 3 4 5 6 6 1 2 3 4 5 5 6 1 2 3 4 4 5 6 1 2 3 3 4 5 6 1 2 2 3 4 5 6 1 Each digit represents a point of the graph. Join each point to those in the same row or column and to those with the same digit. Each point is joined to 5 points in the same row, 5 in the same column and to 5 others, a total of 15 = 3k+6. It remains to show that given any two points P and Q, there are just 6 points joined to both. If P and Q are in the same row, then they are both joined to: the 4 other points in the row, the point in the same column as P with the same digit as Q, and the point in the same column as Q with the same digit as P - a total of 6. Similarly, if P and Q are in the same column. If P and Q have the same digit, then they are both joined to: the 4 other points with the same digit, the point in the same row as P and the same column as Q, and the point in the same column as P and the same row as Q. Finally, suppose P and Q are in different rows and columns and have different digits. Then the points joined to both are: (1) same row as P and same column as Q, (2) same row as P and same digit as Q, (3) same column as P and same row as Q, (4) same column as P and same digit as Q, (5) same digit as P and same row as Q, (6) same digit as P and same column as Q. 36th IMO shortlist 1995 © John Scholes jscholes@kalva.demon.co.uk 8 Oct 2002