At the choir I sing with, we sang a Christmas-themed warm-up song to the tune of Jelly on a Plate. It’s a simple song with just one repeated sentence: ‘I like Brussels sprouts’. The catch is that you don’t break the sentence, even when it doesn’t fit with the phrasing of the song. The syllable counts for each verse are 5, 5, 4, 4, 5, so the first verse goes

I like Brussels sprouts
I like Brussels sprouts
I like Brussels
sprouts I like Bruss
els sprouts I like Bruss

Then the next verse carries on from the first:

els sprouts I like Bruss
els sprouts I like Bruss
els sprouts I like
Brussels sprouts I
like Brussels sprouts I

And so on, until the end of the sentence (‘sprouts’) coincides with the end of a verse.

As well as it being surprisingly difficult to sing the right words, the structure of the song aroused my curiosity. There are 5 syllables in the sentence ‘I like Brussels sprouts’. It takes 5 verses to finish the song, and each syllable gets a turn at being the first syllable in a verse. Is this coincidence? Or is there some maths underlying it?

Each verse has 23 syllables, so the sentence can be repeated 4 times within a verse, with 3 syllables left over. The 5 verse endings are:

I like Bruss

els sprouts I

like Brussels

sprouts I like

Brussels sprouts

The song will finish when the total number of extra syllables is a multiple of the number of words in the sentence. This means that the total number of extra syllables must be the lowest common multiple of [number of syllables in sentence] and [number of syllables left over]. In this case, 3 and 5 have no common factors, so the lowest common multiple is their product, 15. The song will repeat $15 \div 3=5$ times before coming to a satisfying close at the end of a sentence.

What would happen if there was a different number of syllables in the sentence? For example, ‘I like Yorkshire puddings’. This has 6 syllables, and the remainder when dividing 23 by 6 is 5, so there should be 5 syllables left at the end of the first verse:

I like Yorkshire pudd
ings I like Yorkshire
puddings I like
Yorkshire puddings
I like Yorkshire pudd

Again, 6 and 5 have no common factors, so the number of repeats will match the number of words in the sentence. The 6 verse endings are

I like Yorkshire pudd

ings I like Yorkshire

puddings I like York

shire puddings I like

Yorkshire puddings I

like Yorkshire puddings

It happens that the remainder after dividing 23 by any number has no common factors with that number, so for this song the number of verses will always match the number of syllables in the sentence.

What if the verse has a different number of syllables, say 24? Despite racking my brain, I can’t think of a suitable song with 24 syllables per verse, so imagine a ‘Yay!’ at the end of the verse, to make it 5, 5, 4, 4, 6:

I like Brussels sprouts
I like Brussels sprouts
I like Brussels
sprouts I like Bruss
els sprouts I like Brussels

For Brussels sprouts, the remainder when dividing 24 by 5 is 4, which has no common factors with 5. The song will still have 5 verses, ending with

I like Brussels

sprouts I like Bruss

ells sprouts I like

Brussels sprouts I

like Brussels sprouts

For Yorkshire puddings, however, the remainder when dividing 24 by 6 is 0 (because 6 is a factor of 24) so this sentence will now fit exactly into one verse.

I like Yorkshire pudd
ings I like Yorkshire
puddings I like
Yorkshire puddings
I like Yorkshire puddings

How about a sentence with 9 syllables? Say ‘I like a nice big Christmas dinner’. The remainder when dividing 24 by 9 is 6, and 9 and 6 have a common factor of 3; their lowest common multiple is 18. There will be a total of 18 extra syllables at the ends of the verses and the song will repeat $18\div6=3$ times before closing, with the verses ending like this:

I like a nice big Christ

mas dinner I like a

nice big Christmas dinner

So here’s a challenge for you over the remainder of the festive period. Find pairs of songs and sentences that require different numbers of verses, from 2 to 10, to reach a satisfying finish. I look forward to hearing what you come up with!