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!