The Twelve Days of Christmas

A good quiz question might be “How many gifts were given during the carol ‘The Twelve Days of Christmas'”. The answer, 364, might be known by a reasonable number of people by now, but let’s look at the problem both from this perspective, and another perspective: how many legs were in those gifts?

Let’s start properly, with the information we know for certain. On the first day of Christmas, the lucky recipient got a partridge in a pear tree. The second day provided two turtle doves, the third three french hens, 4 calling birds came the next day, 5 gold rings, 6 geese, 7 swans, 8 maids milking, 9 ladies dancing, 10 lords leaping, 11 pipers piping, and 12 drummers drumming.

The song is, however, a cumulative song, so each verse builds on the others.

Let’s assign symbols to each gift:

A = \text{Partridge in a pear tree}\\ B = \text{Turtle dove}\\ C = \text{French hen}\\ D = \text{Calling bird}\\ E = \text{Gold ring}\\ F = \text{Goose}\\ G = \text{Swan}\\ H = \text{Milking maid}\\ I = \text{Dancing lady}\\ J = \text{Leaping Lord}\\ K = \text{Piper}\\ L = \text{Drummer}

This means, for the first day we get $latex A$, then $latex 2B+A$ on the second, etc:

A\\ A+2B\\ A+2B+3C\\ A+2B+3C+4D\\ A+2B+3C+4D+5E\\ A+2B+3C+4D+5E+6F\\ A+2B+3C+4D+5E+6F+7G\\ A+2B+3C+4D+5E+6F+7G+8H\\ A+2B+3C+4D+5E+6F+7G+8H+9I\\ A+2B+3C+4D+5E+6F+7G+8H+9I+10J\\ A+2B+3C+4D+5E+6F+7G+8H+9I+10J+11K\\ A+2B+3C+4D+5E+6F+7G+8H+9I+10J+11K+12L\\

To count the gifts, we simply set:


With the result:

1=1\\ 1 + 2=3\\ 1+2+3=6\\ 1+2+3+4=10\\ 1+2+3+4+5=15\\ 1+2+3+4+5+6=21\\ 1+2+3+4+5+6+7=28\\ 1+2+3+4+5+6+7+8=36\\ 1+2+3+4+5+6+7+8+9=45\\ 1+2+3+4+5+6+7+8+9+10=55\\ 1+2+3+4+5+6+7+8+9+10+11=66\\ 1+2+3+4+5+6+7+8+9+10+11+12=78\\ \\ 1+3+6+10+15+21+28+36+45+55+66+78 = 364

Therefore we see that there were a total of 364 gifts given.

However, I asked how many legs were on those presents.

Let’s instead set the values of the gifts not to 1, but to the number of legs on each specific gift:

A=2 \\B=2 \\C=2 \\D=2 \\E=0 \\F=2 \\G=2 \\H=9 \\I=2 \\J=2 \\K=2 \\L=2

(assuming 9 legs for a maid, cow, and milking stool)

(2) = 2\\ (2)+2(2) = 6\\ (2)+2(2)+3(2) = 12\\ (2)+2(2)+3(2)+4(2)=20\\ (2)+2(2)+3(2)+4(2)+5(0)=20\\ (2)+2(2)+3(2)+4(2)+5(0)+6(2)=32\\ (2)+2(2)+3(2)+4(2)+5(0)+6(2)+7(2)=46\\ (2)+2(2)+3(2)+4(2)+5(0)+6(2)+7(2)+8(9)=118\\ (2)+2(2)+3(2)+4(2)+5(0)+6(2)+7(2)+8(9)+9(2)=136\\ (2)+2(2)+3(2)+4(2)+5(0)+6(2)+7(2)+8(9)+9(2)+10(2)=156\\ (2)+2(2)+3(2)+4(2)+5(0)+6(2)+7(2)+8(9)+9(2)+10(2)+11(2)=178\\ (2)+2(2)+3(2)+4(2)+5(0)+6(2)+7(2)+8(9)+9(2)+10(2)+11(2)+12(2)=202\\

This gives the total number of legs to be:


Nine hundred and twenty-eight legs.

“trackfan” from states:

Over the twelve days of Christmas, according to the song, you would receive a total of 364 gifts; 184 of these are birds; 40 are inanimate objects (golden rings); and, dividing equally among the sexes those occupations where sex is not specified (pipers and drummers), 93 women, and 47 men.

These 40 inanimate objects isn’t taking into account the trees in which the partridges are perched, the milking stools for the maids, the pipers’ pipes, and the drummers’ drums.

But how much land would you need to dig a pond for all 42 swans to swim in?