# Twelve days of Christmas

As our house is getting rather full of drumming drummers and maids-a-milking, my true love decided on a different gifting strategy this year. I will receive a number of gifts based on the formula

where is the number of gifts, is the day of Christmas (e.g. for the second day of Christmas) and denotes the lowest integer less than or equal to . Our regular gift stockist only has the usual Christmas fayre for sale so my true love has decided that

• when I will receive partridge complete with pear tree
• when I will receive turtle doves
• when I will receive French hens

and so on (following items from previous years).

How many “calling birds” will I receive?

How many “swimming swans” will we have to accommodate?

What will I receive on the th day of Christmas?

Reveal Solution
Calculating a few terms gives a good feel for things
since this gives so

so

since this gives so

since this gives so

The next change will take place when i.e. so so, on the th day and on days and .

To go to , i.e. so . This is day 11.

The sequence would look like this:

There are days on which I would receive calling birds so I would have calling birds in total.

Fortunately, no swimming swans need to be accommodated since for all days.

On the 12th day of Christmas I will receive gold rings.

• partridge complete with pear tree
• turtle doves – only a small dovecote needed for these
• French hens – or “poules françaises” as we like to call them
• calling birds – a tad noisy but never mind
• gold rings

Not a bad haul and much more manageable than previous years!

Problem Page Coordinator: Claire Baldwin – Mathematics in Education and Industry

Acknowledgement: The IMA are indebted to MEI for sourcing and supplying Mathematics Today with these well-known puzzles.

First published in Mathematics Today (December 2017)

Published