The 12 Days of Christmas – How Many Gifts?
When I accept all the gifts in the song “The 12 Days of Christmas” how many gifts do I receive? And, can I work out the total using some neat Mathematics? First, a reminder about the song:
“On the first day of Christmas
my true love sent to me:
A Partridge in a Pear Tree
On the second day of Christmas
my true love sent to me:
Two Turtle Doves
and a Partridge in a Pear Tree
On the third day of Christmas
my true love sent to me:
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree
On the fourth day of Christmas
my true love sent to me:
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree
On the fifth day of Christmas
my true love sent to me:
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree
On the sixth day of Christmas
my true love sent to me:
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree
On the seventh day of Christmas
my true love sent to me:
Seven Swans a Swimming
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree
On the eighth day of Christmas
my true love sent to me:
Eight Maids a Milking
Seven Swans a Swimming
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree
On the ninth day of Christmas
my true love sent to me:
Nine Ladies Dancing
Eight Maids a Milking
Seven Swans a Swimming
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree
On the tenth day of Christmas
my true love sent to me:
Ten Lords a Leaping
Nine Ladies Dancing
Eight Maids a Milking
Seven Swans a Swimming
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree
On the eleventh day of Christmas
my true love sent to me:
Eleven Pipers Piping
Ten Lords a Leaping
Nine Ladies Dancing
Eight Maids a Milking
Seven Swans a Swimming
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree
On the twelfth day of Christmas
my true love sent to me:
12 Drummers Drumming
Eleven Pipers Piping
Ten Lords a Leaping
Nine Ladies Dancing
Eight Maids a Milking
Seven Swans a Swimming
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree”
Here is the solution …
Partridges: 1 × 12 = 12
Doves: 2 × 11 = 22
Hens 3 × 10 = 30
Calling birds: 4 × 9 = 36
Golden rings: 5 × 8 = 40
Geese: 6 × 7 = 42
Swans: 7 × 6 = 42
Maids: 8 × 5 = 40
Ladies: 9 × 4 = 36
Lords: 10 × 3 = 30
Pipers: 11 × 2 = 22
Drummers: 12 × 1 = 12
Total = 364
And, here is another way …
The number of presents each day is 1 on the 1st, then 3 on the 2nd, then 6 on the 3rd, then 10 on the 4th. We call this set of numbers the triangular numbers, because they can be drawn in a dot pattern that forms triangles:
Another way of writing this is:
On the first day, 1 present.
On the 2nd day, 1 + 3 = 4 presents
On the 3rd day, 1 + 3 + 6 = 10 presents
On the 4th day, 1 + 3 + 6 + 10 = 20 presents.
These partial sums are called tetrahedral numbers, because they can be drawn as 3-dimensional triangular pyramids (tetrahedrons) like this:
So how many dots (representing presents) will there be in the 12th tetrahedron?
Of course, we could just start adding with our calculator, but what if my true love is very generous, and starts giving me presents for 30 days after Christmas? Or for 100 days? How would I calculate it then?
Our aim is to produce a formula that will allow us to find any tetrahedral number. Here’s one of the possible ways of doing this.
In general, for the sum 1 + 2 + 3 + … + n:
which is the same as
Multiplying by the (n+ 2) that we get from what I called ‘the result triangle’ earlier:
Dividing this by 3 (since we used 3 equivalent sum triangles to get this far) gives us the n-th tetrahedral number:
On the 12th day, the number of presents will be
Phew! See the complete working here and, above all, enjoy a wonderful Christmas!