On the first day of Christmas, my true love sent to me…

A very mathematical song with plenty of interesting questions to analyse.

When I am making mince pies, I often put Christmas songs on and have a sing-along. “The Twelve Days of Christmas” is a brilliant song for me to sing along to, because even if I can’t remember all the words when the song starts, the repetition guarantees I will have learned them all by the end!

It is well known to recreational mathematicians and maths educators that there is a pleasing result when you add together the cumulative total of gifts mentioned in the song – if you’ve never worked it out, have a think about how you might calculate it before reading on…

On the first day, there’s 1 gift; on the second, 2+1; on the third, 3+2+1, and so on. This gives us the sum of the triangular numbers, also known as the tetrahedral numbers, for which there is a formula using binomial coefficients. Alternatively, you could calculate it using a spreadsheet: in the first column, put the numbers 1 to 12 to represent the gifts, and in the second column the numbers counting down from 12 to 1 to represent the number of days each gift is given. Then create a column for the total number of each gift: 12 partridges, 2×11=22 turtle doves, 3×10=30 French hens and so on. Summing this column gives the answer. Are the gifts enough to last all year? Not quite…

I was musing on this very mathematical song while I made my mince pies this year, and I started pondering what other questions might be interesting to explore. The triangular numbers emerge by considering the number of gifts on a particular day, and a natural generalisation might be to wonder how many gifts would be received if Christmas lasted for more than 12 days. There’s an opportunity then to explore inverses – on which day would I receive more than 1000 gifts in a single day? On which day would the cumulative total of gifts go over 1 million? Working out answers to questions like this requires us not just to be able to express relationships in a general form but also to work backwards to solve for an unknown.

As I was washing up after this year’s first batch of mince pie making, running out of breath after racing through the later verses of the song, another question occured to me – how far through the song will I be when I have done half the singing?

To answer that question I did a bit of mathematical modelling. The nth verse of the song begins with the line: “On the nth day of Christmas my true love sent to me”, and then has n lines. For the purposes of my model I will assume each line is equal in length. So the nth verse has n+1 lines. This means the total number of lines in the song is 2 + 3 + 4 + … + 11 + 12 + 13 = 90. So when I have sung 45 lines I’m about halfway there….

Working backwards, when will I get to the end of the 45th line? Well I happen to know 45 is the 9th triangular number, so the sum of all the numbers from 2 to 9 is approximately 45 (it’s actually 44) so I’m going to suggest that a reasonable approximation for the midpoint of the song is after I have sung the first eight verses. This approximation won’t be exact, because some of the lines are longer than others (in particular, I like to really go for it when I sing “Five go-old riiiiiings” so that line lasts a good long time!) However, I checked on my favourite music streaming site, where there is a recording of Bing Crosby singing the song. His version is timed at 3 minutes and 20 seconds, which is 200 seconds. He begins the ninth day of Christmas at 1 minute 43 seconds or 103 seconds. Not bad for a back-of-the-envelope approximation!

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