12 Days of Christmas Maths Problem

Lucy

December 22, 2015

Merry Christmas everyone!

If you’re like me, then you love a bit of caroling at Christmas. One of my all-time favourites is “The Twelve Days of Christmas," a song where one extremely lucky lady receives a lot of gifts from her true love! Not that I’m jealous, but I would like to find out how many presents she actually received, so let’s have a look at the song in a bit more detail (feel free to sing along!)

On the first day of Christmas my true love sent to me:
A Partridge in a Pear Tree

On the second day of Christmas my true love sent to me:
2 Turtle Doves
and a Partridge in a Pear Tree

On the third day of Christmas my true love sent to me:
3 French Hens
2 Turtle Doves
and a Partridge in a Pear Tree

On the fourth day of Christmas my true love sent to me:
4 Calling Birds
3 French Hens
2 Turtle Doves
and a Partridge in a Pear Tree

On the fifth day of Christmas my true love sent to me:
5 Golden Rings
4 Calling Birds
3 French Hens
2 Turtle Doves
and a Partridge in a Pear Tree

On the sixth day of Christmas my true love sent to me:
6 Geese a Laying
5 Golden Rings
4 Calling Birds
3 French Hens
2 Turtle Doves
and a Partridge in a Pear Tree

On the seventh day of Christmas my true love sent to me:
7 Swans a Swimming
6 Geese a Laying
5 Golden Rings
4 Calling Birds
3 French Hens
2 Turtle Doves
and a Partridge in a Pear Tree

On the eighth day of Christmas my true love sent to me:
8 Maids a Milking
7 Swans a Swimming
6 Geese a Laying
5 Golden Rings
4 Calling Birds
3 French Hens
2 Turtle Doves
and a Partridge in a Pear Tree

On the ninth day of Christmas my true love sent to me:
9 Ladies Dancing
8 Maids a Milking
7 Swans a Swimming
6 Geese a Laying
5 Golden Rings
4 Calling Birds
3 French Hens
2 Turtle Doves
and a Partridge in a Pear Tree

On the tenth day of Christmas my true love sent to me:
10 Lords a Leaping
9 Ladies Dancing
8 Maids a Milking
7 Swans a Swimming
6 Geese a Laying
5 Golden Rings
4 Calling Birds
3 French Hens
2 Turtle Doves
and a Partridge in a Pear Tree

On the eleventh day of Christmas my true love sent to me:
11 Pipers Piping
10 Lords a Leaping
9 Ladies Dancing
8 Maids a Milking
7 Swans a Swimming
6 Geese a Laying
5 Golden Rings
4 Calling Birds
3 French Hens
2 Turtle Doves
and a Partridge in a Pear Tree

On the twelfth day of Christmas my true love sent to me:
12 Drummers Drumming
11 Pipers Piping
10 Lords a Leaping
9 Ladies Dancing
8 Maids a Milking
7 Swans a Swimming
6 Geese a Laying
5 Golden Rings
4 Calling Birds
3 French Hens
2 Turtle Doves
and a Partridge in a Pear Tree

It’s such a great song! If I wish to find out exactly how many presents she receives in total, then it might be a good idea to add up the total number of presents she receives each day.

12 Days of Christmas graph

Does anyone recognise the numbers?

That’s right! It’s the triangle numbers!

So, to find out how many presents she has in total all we have to work out is the sum of the first twelve triangle numbers. (You’ll find it is 364.)

Here is a small challenge:
Students studying for their maths GCSEs, find the nth term for the sequence above.

A-level further mathematicians, find the nth term and use the formulae from summations of series to work out the grand total. Show us your answers!

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