Take the 15 smallest dominoes in a set (double blank through double four.)
In how many ways can they be arranged in a row such that the numbers on consecutive pieces match.
Count the two directions separately.
It might be wrong, but here's my first attempt. I'll explain more
after a day at the beach, unless it turns out that I am wrong:
Consider complete 10-chains without doubles. (See my previous post)
Cut these complete chains in the two places the zeroes join. This
forms either a 3-row and a 7 row, or a 4 row and a 6 row, or two 5 rows.
While I could be wrong, I calculate 12 different chains yield a pair of
five rows, 48 different chains give a 4 row and a 6 row, 36 different
chains give a 3 row and a 7 row. Altogether, 12 + 48 + 36 = 96
different unique 10-domino chains (without doubles).
Doubles can be inserted in 2x2x2x2x2 = 32 different ways, giving 3072 unique 15-chains, including doubles.
And each of these 15 chains can be cut in 15 ways, with each cut
yielding a unique row. Final (and possibly correct) answer
= 3072 * 15 = 46,080 15-domino rows where consecutive pieces
match.
This = 2^10 * 15 * 3, which suggests a better method that I'll explore later.
Pencil and paper solution
