Ireland

If only one player has an odd score, the total number of points won must be odd. This leaves two possibilities, $N=6$ and $N=7$.

Consider first the case $N=6$. The player with an odd score must have won 1, 3 or 5 games. If 5, the player would have come first. If 1, the player would have come 5th or 6th. Thus the odd score was 3. The three players ranked higher must have won 4 matches leaving 0 for the remaining two players which is impossible.

Hence $N=7$ is the only possibility. The score of the player with the odd score must be 1, 3 or 5. 1 is impossible as the players ranked 5, 6, 7 must all have scored 0 in this case. Similarly, 5 is impossible for a player ranked fourth as the players ranked 1, 2, 3 must all have scored 6 in this case. It remains to show that 3 is a possibility. The players ranked 5, 6, 7 must have scored either 2, 2, 2 points or 2, 2, 0. In the first case, the first 3 players all score 4 points. In the second case, the first three score 6, 4, 4. Thus two possibilities occur: scores of 6, 4, 4, 3, 2, 2, 0 or 4, 4, 4, 3, 2, 2, 2. Both of these outcomes can be realised as shown below; hence the answer to the problem is 3.

	1	2	3	4	5	6	7	tot
1	×	1	1	1	1	1	1	6
2	0	×	1	1	1	0	1	4
3	0	0	×	1	1	1	1	4
4	0	0	0	×	1	1	1	3
5	0	0	0	0	×	1	1	2
6	0	1	0	0	0	×	1	2
7	0	0	0	0	0	0	×	0

	1	2	3	4	5	6	7	tot
1	×	1	0	1	1	0	1	4
2	0	×	1	1	1	1	0	4
3	1	0	×	1	1	0	1	4
4	0	0	0	×	1	1	1	3
5	0	0	0	0	×	1	1	2
6	1	0	1	0	0	×	0	2
7	0	1	0	0	0	1	×	2