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

Two small Christmas cakes.

I don’t make a Christmas cake every year, but most years since the mid 1990s one of my Christmas traditions has been to dig out the recipe handed down to me from Grandma Kiddle and weigh out a small mountain of dried fruit, soak it overnight in brandy, and then mix butter, sugar, eggs, flour and bake for several hours. Then over the following days and weeks I gradually introduce spoonfuls of brandy into the cake, before covering it with marzipan, royal icing, and some tatty little plastic decorations that I’ve had for years. My final Christmas Cake tradition is to take the cake with me when visiting various family who don’t make their own, and cut a chunk off for them to enjoy. What’s left goes home with me after Christmas where I eat a couple of slices and then get sick of Christmas cake, push the tin to the back of the kitchen worktop and forget about it, and then rediscover it later in the year to find it’s still perfectly edible because Christmas cake Never Goes Off!

This Christmas, like the year that led up to it, has been very different from previous ones. I still wanted to make cake, but I knew it was unlikely I could visit all the family who usually get a chunk of cake. Added to this, certain family members prefer their cake without icing, so I started to come up with a cunning plan. Instead of making one large cake and cutting chunks off, I would make smaller cakes, which could then be iced or not to people’s preferences! This is where the maths comes in…

The recipe I follow makes enough mix for an 8 inch diameter round cake tin. I was limited in what tins I had available; I dug through the kitchen cupboards and found I had a 6 inch round tin, various square tins and a loaf tin. I could have worked out volumes and then scaled the recipe but instead I decided to try to find a pair of tins that got me close to the same base area – the tins were all approximately the same height so I could do all my calculations with area rather than volume.

Approximating π as 3, my 8 inch cake tin has a base area of 4×4×3=48 square inches. The 6 inch cake tin has a base area of 3×3×3=27 square inches. This means I had 21 square inches to play with for the second cake, without needing to do any recipe scaling. Luckily one of my loaf tins is around 8 inches long and around 2½ inches wide, giving a base area of approximately 20 square inches (it tapers out but I figured that my approximations were only ballpark figures and the worst that could happen was a cake that was a bit flatter than I planned – it would still taste alright!)

The only other recalculation that two small cakes required was in cooking time. The 8 inch cake takes 3½ hours to cook, so I figured my two small cakes would need a bit less than that. I dealt with this problem by checking the cakes after 2½ hours and testing with a skewer – they needed a little longer and were done in about 3 hours in the end. The round cake was subsequently marzipanned and iced, to be enjoyed by me over the coming weeks, and the loaf cake was divided into 2 squareish chunks and sent off without icing to be enjoyed by family. It all worked out rather well, and I suspect my new Christmas tradition will be to make two smaller cakes every year from now on!

## 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!