[evan_au (OP)]

In this week's podcast, there was a mathematical analysis of the total number of presents in the Christmas song "The 12 days of Christmas"...  the one with the "partridge in a pear tree".

The mathematician on the podcast (Amalia Thomas) assumed that every day, you received all the previous gifts again (so 12 partridges in 12 pear trees, by the end of the song).
https://www.thenakedscientists.com/articles/interviews/mathematical-12-days-christmas

However, I had always assumed that you received each type of present on just one day, and then the song recaps all the presents that had been given on all the previous days, to show what an eclectic collection you had by Christmas day (so just 1 partridge in 1 pear tree, by the end of the song).

With my assumption, the total number of presents still build up quickly (a quadratic function of time), but not as quickly as Amalia assumed (which is a cubic function of time). 

What is your understanding of the meaning of Christmas?

By the way, my favorite version of this song is an Irish rendition by Frank Kelly (here with pictures added):

[evan_au (OP)]

I just did the calculations on a spreadsheet.

If you assume that each kind of present is just given on one day (ie 1 partridge by day 12), the total number of presents ends up with the Triangular Numbers as mentioned on the show, which is a quadratic function.

If you assume that each kind of present is given again on later days (ie 12 partridges by day 12), the total number of presents end up with the Tetrahedral Numbers, which is a cubic function.

12_Days_Table.png (9.13 kB . 241x417 - viewed 1154 times)

12_Days_Graph.png (24.73 kB . 586x356 - viewed 1152 times)

[jeffreyH]

Someone may need to check the maths.

[syhprum]

I have always preferred the cubic version I have seen some delightful video versions with everyone knee deep in pear trees etc.

[alancalverd]

Has nobody actually read the question? Each verse enumerates all the gifts sent each day. So you get 12 partridges (QED), plus 2 x 11 = 22 turtle doves, 3 x 10 = 30 French hens, etc.

The fun question to ask a school class is "which present do you get the most of?". You can do it by enumeration but the subtle approach is by symmetry.

Cynics will of course say the key word is "sent". There is no delivery on Christmas or New Year's Day, and there may be one or two Sundays in between, plus "seasonal difficulties and transport disruptions", so the number actually received by 6 Jan may vary.

[Bored chemist]

Has nobody actually read the question? Each verse enumerates all the gifts sent each day. So you get 12 partridges (QED), plus 2 x 11 = 22 turtle doves, 3 x 10 = 30 French hens, etc.

The fun question to ask a school class is "which present do you get the most of?". You can do it by enumeration but the subtle approach is by symmetry.

Cynics will of course say the key word is "sent". There is no delivery on Christmas or New Year's Day, and there may be one or two Sundays in between, plus "seasonal difficulties and transport disruptions", so the number actually received by 6 Jan may vary.

It isn't stated that the presents must be received in time for this Christmas. I can wait.

More importantly, how does anyone interpret, for example
"On the fourth day of Christmas
My true love gave to me
Four calling birds,
Three French hens,
Two turtle doves
and a partridge in a pear tree"
As meaning anything other than that , on that day, they got that list?
It's not like they sing
"On the fourth day of Christmas
My true love gave to me
Four calling birds.
Having previously received
Three French hens
Two turtle doves
And a partridge in a pear tree"
It's obviously the cubic, and, as a handy mnemonic, the total number is one less than the number of (whole) days in a year.

[alancalverd]

The version I have says my true love sent to me. Seems more probable, even before Amazon, than running backwards and forwards to the shops carrying lords, maids, drummers, etc, when Fortnum & Mason boast that they can deliver anything the heart desires.