Santa's quickest route around the world (and other festive brainteasers)

Ever wondered how Santa plots his route between houses on Christmas Eve? Or how many gifts you'd receive if your true love followed 'The Twelve Days of Christmas' to the letter? Oxford mathematician Professor Marcus du Sautoy helped a sizeable audience tackle these festive problems when he gave the annual Oxford Mathematics Christmas Public Lecture in the Andrew Wiles Building.

After admitting he 'really enjoys this time of year because there's so much maths hiding everywhere', Professor du Sautoy kicked things off with a fairly gentle puzzle about Hanukkah candles, before moving on to French hens, turtle doves and, of course, partridges in pear trees.

He told the audience: 'Round my neighbourhood, carol singers just will not come to my door any more. My kids explained that it's because I always point out something mathematical in the carols they're singing. But when they did come, they would sing: "On the first day of Christmas, my true love gave to me…"

'One of the challenges is working out how many presents you get in this song. So I always challenge the carol singers – perhaps that's why they don't come any more – to see if they can work it out.

'There's a nice way to quickly calculate how many presents there are. On each day of Christmas you have a "triangular number" of presents: one, three, six, 10 etc. The interesting thing is that over the whole 12 days it becomes a three-dimensional problem. If you stack the presents on top of each other, you're building up these triangles until you get a kind of pyramid.

'We call these the "tetrahedral numbers", and we can fit six of these pyramids together into a box shape with 12 presents along one side, 13 on another, and 14 on the remaining side.'

Multiplying 12 by 13 by 14, and dividing by the six pyramids, gives the total number of presents: 364 (or 'one for every day apart from Christmas', according to Professor du Sautoy). It's a lot of fowl to accommodate under one roof – not, perhaps, a recommended gift-giving strategy.

Making good use of volunteers from the audience, Professor du Sautoy then demonstrated how to pair up the right people at an office Christmas party – and how to avoid people you don't like.

He also pointed out some of the quirks of clever online algorithms, such as the internet retail company that recommended he buy copies of three of his own books, and the online bookseller offering a colleague's maths tome for the sum of $23m – not exactly a Christmas bargain.

Professor du Sautoy concluded with the lecture's headline puzzle: the travelling Santa problem (known at less festive times of year as the travelling salesman problem).

Santa has millions of houses to visit across the globe in a relatively short space of time. So how does he work out his most efficient route?

The twist, it turns out, is that such a puzzle is not currently solvable. In fact, there's a prize of $1m on offer for anyone who can get to the bottom of it, along with five other so-called 'Millennium Prize Problems' (one of the original seven has been solved). This one is known as P versus NP, and it refers to calculations that are simply too complex for present computers to complete in a reasonable amount of time – after all, the number of potential routes for Santa is a figure higher than we could comprehend.

Until someone solves this problem, Santa will just have to make do on his own. But he seems to be doing OK...

The Oxford Mathematics Public Lectures aim to bring the beauty and rigour of the subject to a wider audience. Aimed at sixth-formers and above, the series has featured some of the leading mathematicians of the day, including Roger Penrose, Andrew Wiles and Cedric Villani, each of them keen to convey the pleasures (and occasional pains) of their subject. Mathematics underpins many of the models from which scientists, medics, economists and social scientists build an understanding of the world. These lectures demonstrate those connections and provide inspiration for the mathematicians of today and tomorrow, and the wider general public.