A Brilliant Sonnet
The diff’rent sonnets all have fourteen lines,
Each having syll’bles counting up to ten.
But they all differ in their schemes of rhymes:
The English, Ital’an, and Spenserian.
Let us define a Brilliant sonnet as
One with just two rhymes, only A and B.
Like all the sonnets this one also has
Exactly fourteen lines entirely.
The Brilliant sonnet has one other rule
As to its rhyming scheme of A’s and B’s:
Neither letter, although it may seem cruel,
Ever appear three times consec’tively.
This question I ask, dear reader, so fair:
How many diff'rent rhyming schemes are there?
This is my entry to the Math Poetry Contest .
The answer is 610.
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1 solution
Let S n be the number of poems with n lines whose rhyme scheme fits the pattern given. If we add another line, it will either match the previous line, or it won't. We may always attach a new line that doesn't rhyme with the previous line, as this cannot create three consecutive rhyming lines. Clearly there are S n such extensions. To add a line which does rhyme with the previous line, to avoid creating three consecutive rhymes the line before that must be the opposite rhyme. That means the poem we're adding to must end with two different rhymes, and by the reasoning above there are S n − 1 such poems. To summarize:
S n + 1 = #{ rhyme schemes of length n + 1 }
= #{ rhyme schemes of length n + 1 ending in AB or BA } + #{ rhyme schemes of length n + 1 ending in AA or BB }
= #{ rhyme schemes of length n } + #{ rhyme schemes of length n ending in AB or BA }
= S n + #{ rhyme schemes of length n − 1 }
= S n + S n − 1
With the observation that S 1 = 1 and S 2 = 2 , we see that S n = F n + 1 , the ( n + 1 ) st Fibonacci number. Hence the number of sonnets of the given form is S 1 4 = F 1 5 = 6 1 0 .