Football formations are generally described by three or four nonzero whole numbers summing to 10 (the goalkeeper isnâ€™t counted), representing, from defence to attack, the number of players in approximate lines across the pitch.

Last season we played a different formation every week, always using four lines, each with at most four players; the difference between one week and the next was that from one line two players moved, one to an adjacent line and the other to the line beyond that (eg, 3-4-1-2 could only be followed by 3-2-2-3). Our number of fixtures was the largest it could have been, given these conditions. The first number in our formations was more often 3 than any other number; 3-1-3-3 gave us ourÂ worst result.

How many games did we play, and what were our first three formations?

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Somewhat short on comments, primitive printout, but it works.

The recursive functon, ‘chain’, above generates a lot of ‘chaff’. The ‘seasons’ list has 1752 entries. As an analogy, let’s say it yielded (1, 2, 3, 4, 5). It would follow that up with:
(1, 2, 3, 4)
(1, 2, 3)
(1, 2)
(1)

By adding a few additional lines the function is more efficient, generating a ‘seasons’ list with only 843 entries. But it’s much less elegant:

Definitive version