9873210 wrote:If anyone would care to comment.

How do you organize all this? I made pretty much all the deductions above but I kept getting to the end and realizing I'd assigned 2-0 twice or some such, and I haven't got my notes organized enough to unwind back to the point of the error.

I'd recommend setting up a main table showing how the knockout competition progresses and a subsidiary table showing which of the 15 scorelines are still freely available, like this:

? ? \

> ? ?

? ? / \

> ? ?

? ? \ / \

> ? ? \

? ? / \

> ? ?

? ? \ / \

> ? ? / \

? ? / \ / \

> ? ? \

? ? \ / \

> ? ? \

? ? / \

> ?

? ? \ /

> ? ? /

? ? / \ /

> ? ? /

? ? \ / \ /

> ? ? \ /

? ? / \ /

> ? ?

? ? \ /

> ? ? /

? ? / \ /

> ? ?

? ? \ /

> ? ?

? ? /

5-0 4-0 3-0 2-0 1-0

5-1 4-1 3-1 2-1

5-2 4-2 3-2

5-3 4-3

5-4

The main table should have the winner of each match on the upper branch and the loser on the lower branch, and the subsidiary table should have all scorelines that have been located (either precisely or to within just a few possibilities) struck out. Repeatedly make a copy of the table so far, make some deductions from what you've got so far and fill those deductions in on the copy (but not the original). If you find you've made an incorrect deduction, you only have to back off to before the incorrect deduction, not all the way to the start.

So if e.g. your first set of deductions is from B's 14 For / 0 Against totals, that B must win the tournament with scorelines of 5-0, 4-0, 3-0 and 2-0 in its four matches, and that the scoreline in its first match must be 5-0 against F because no teams have 0 For / 4 Against, 0 For / 3 Against or 0 For / 2 Against totals, you update that to:

B 5 \

> B *

F 0 / \

> B *

? ? \ / \

> ? 0 \

? ? / \

> B *

? ? \ / \

> ? ? / \

? ? / \ / \

> ? 0 \

? ? \ / \

> ? ? \

? ? / \

> B

? ? \ /

> ? ? /

? ? / \ /

> ? ? /

? ? \ / \ /

> ? ? \ /

? ? / \ /

> ? 0

? ? \ /

> ? ? /

? ? / \ /

> ? ?

? ? \ /

> ? ?

? ? /

5-0 4-0 3-0 2-0 1-0

5-1 4-1 3-1 2-1

5-2 4-2 3-2

5-3 4-3

5-4

in which the asterisks must be 2, 3 and 4 in some order.

And if your next set of deductions is that C's 15 For / 8 Against totals must be the result of three of the scorelines 5-1, 5-2, 5-3 and 5-4 in the first three rounds and one of the scorelines 0-2, 0-3 and 0-4 in the final, the only way to keep that down to 8 Against goals is for those scorelines to be 5-1, 5-2, 5-3 and 0-2, and that the scoreline for C's first round match must be 5-1 against N because no teams have 2 For / 5 Against or 3 For / 5 Against totals, you update that to:

B 5 \

> B *

F 0 / \

> B *

? ? \ / \

> ? 0 \

? ? / \

> B 2

? ? \ / \

> ? ? / \

? ? / \ / \

> ? 0 \

? ? \ / \

> ? ? \

? ? / \

> B

C 5 \ /

> C 5 /

N 1 / \ /

> C 5 /

? ? \ / \ /

> ? # \ /

? ? / \ /

> C 0

? ? \ /

> ? ? /

? ? / \ /

> ? #

? ? \ /

> ? ?

? ? /

5-0 4-0 3-0 2-0 1-0

5-1 4-1 3-1 2-1

5-2 4-2 3-2

5-3 4-3

5-4

in which the asterisks must be 3 and 4 in some order, and the hashes must be 2 and 3 in some order.

Then your next set of deductions might be that J has to play in at least three rounds because of its 12 Against goals (and indeed also its 12 For goals) and cannot play in all four rounds because we already know which teams are the two finalists and they're not J, so it must be a losing semi-finalist. Furthermore, it cannot be the semi-finalist that loses against B, since that one gets at most 5 For goals from each of the first two rounds and none from its semi-final, for at most 10 in total. So it must be the semi-finalist that loses against C, and its scorelines must be two of those we haven't struck out in the first two rounds and either 2-5 or 3-5 in the semi-final. To get those up to 12 Against goals, we must have 7 Against goals from the first two rounds, which can only come from the 5-4 and 4-3 scorelines, and then the total of 12 For goals requires the semi-final scoreline to be 3-5. Furthermore, as no team has totals of 3 For and 4 Against goals, the 4-3 scoreline must be in the second round and the 5-4 scoreline in the first round, which means that J knocks M out in the first round. So we update the tables to:

B 5 \

> B *

F 0 / \

> B *

? ? \ / \

> ? 0 \

? ? / \

> B 2

? ? \ / \

> ? ? / \

? ? / \ / \

> ? 0 \

? ? \ / \

> ? ? \

? ? / \

> B

C 5 \ /

> C 5 /

N 1 / \ /

> C 5 /

? ? \ / \ /

> ? 2 \ /

? ? / \ /

> C 0

J 5 \ /

> J 4 /

M 4 / \ /

> J 3

? ? \ /

> ? 3

? ? /

5-0 4-0 3-0 2-0 1-0

5-1 4-1 3-1 2-1

5-2 4-2 3-2

5-3 4-3

5-4

in which the asterisks must be 3 and 4 in some order.

And so it goes on - I won't complete the solution because the point of this post is to answer your question about how to organise the process of solving the puzzle, not to provide a complete spoiler (though I am leaving it as an unconcealed partial spoiler, because I reckon spoiler concealment produces too much visual clutter for the tables to be at all easy to read). One other point to make is that this way of organising it is much better done electronically than with paper and pencil, because of the far greater ease of making a copy of your work so far and doing further work in the copy only, leaving the possibility of reverting to the current state of your solution if you make a mistake in that further work.

Completing the solution does require about four or five further similar-sized sets of deductions, so one does end up with a pretty long write-up of the complete solution if one does it all this way. But it can be written up quite quickly once one has designed and written the first set of tables, because each stage of copying and editing in the further deductions doesn't involve much typing - most of the time is spent looking for and finding good sets of deductions to make, not on writing them up. And once one has the complete solution and has checked it, one does have the option of shortening the write-up by combining groups of deductions into larger groups, leaving out the intermediate tables between them. (Doing that too much does make it hard for readers to keep track mentally of where one has got to, but one can certainly use larger groups of deductions than those I've used above to illustrate the process - there's basically a balance to be struck between conciseness and easy readabiility here.)

Gengulphus