Back in this post I asked you the question:

In the song “The Twelve Days of Christmas” how many gifts did my true love give me over the twelve days of Christmas?

As the twelve days of Christmas have now been and gone, I’d thought I’d better give you the solution.

There are 2 ways to arrive at the answer (there’s probably more than two ways, but I’m going to tell you two ways)

**Method 1**

On the first day I get one gift – a Partridge in a Pear Tree

On the second day I get one gift – a Partridge in a Pear Tree, and two gifts – Two Turtle Doves, so a total of 1 + 2 = 3 gifts.

We could summarise this approach like so:

Day |
Gifts Received |
Total |

1 | 1 | 1 |

2 | 1 + 2 | 3 |

3 | 1 + 2 + 3 | 6 |

4 | 1 + 2 + 3 + 4 | 10 |

5 | 1 + 2 + 3 + 4 + 5 | 15 |

6 | 1 + 2 + 3 + 4 + 5 + 6 | 21 |

7 | 1 + 2 + 3 + 4 + 5 + 6 + 7 | 28 |

8 | 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 | 36 |

9 | 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 | 45 |

10 | 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 | 55 |

11 | 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 | 66 |

12 | 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 | 78 |

… and all you need to do now is add up the total for each day to get the total amount of gifts given over the twelve days of Christmas.

The observant amongst you will have noticed that the totals for each day create the series of triangular numbers.

**Method 2**

Work out how many of each type of gift you receive.

e.g.

You get one Partridge in a Pear Tree each day for twelve days, two turtle doves for eleven days, three French Hens for ten days and so. Add up how many of each gift you get and you have your answer:

1 x 12 = 12

2 x 11 = 22

3 x 10 = 30

4 x 9 = 36

5 x 8 = 40

6 x 7 = 42

7 x 6 = 42

8 x 5 = 40

9 x 4 = 36

10 x 3 = 30

11 x 2 = 22

12 x 1 = 24

(The astute amongst you will have spotted that after 6×7 the pattern repeats itself with the numbers reversed, so you could just add up the first six totals and double the answer.)

Whichever way you do it, you get the answer **364 Gifts**

I hope you had a great Christmas and I wish you a happy 2013 (which we already know is an interesting year)

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