Last Name. First Name. Email address:. Related posts:. Luckily we have a formula for the nth tetrahedral number which is. Now imagine if the singer had continued receiving gifts from their true love for 20 days.
The singer will have: gifts! No wonder the true love stopped after 12 days! 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. The numbers of gifts each day are called triangular numbers: they can each be arranged in a triangle. Each day brings a greater triangular number of gifts.
But how can we compute the total number without actually adding them up? We need to sum the first 12 triangular numbers. We are going to arrange them in a pile, but stacking leaping lords, laying geese and dancing ladies is inconvenient, so let us take a Christmas bauble to represent each gift.
We start with 12 triangles of baubles, one for each day. We can stack the single bauble for day one on top of the three for day two. This gives a little pyramid of four baubles. We can place this on top of the six baubles representing day three to give a larger pyramid with 10 baubles. Eventually we come to the largest triangle corresponding to the gifts received on the 12th day. By all means, add up the numbers from 1 to 12 if you have any doubts. We obtain a stack having three triangular sides and a triangular base, or four faces in total.
This is called a tetrahedron. The tetrahedron is one of the five Platonic solids, known since ancient times. I will skip the details, but it is easy enough to show that the number of baubles in a tetrahedron with N layers is:. Numbers of this form are called tetrahedral numbers.
True love indeed. It is denied as a serious problem but could cause reputational damage. The starting point for reverse maths is a base theory that is strong enough to state the theorems of interest, but not strong enough to prove them.
The first few lines go like this:. 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. The song continues, adding 4 calling birds on the 4th day, 5 golden rings on the 5th, and so on up to the 12th day, when 12 drummers add to the cacophony of assorted birds, pipers and lords leaping all over the place.
Notice that on each day there is one partridge so I will have 12 partridges by the 12th day , and each day from the second day onwards there are 2 doves so I will have 22 doves , and from the 3rd there are 3 hens total of 30 hens , and so on.
We observe that we have the same number of partridges as drummers 12 of each ; doves and pipers 22 of each ; hens and lords 30 of each and so on.
Let's now generalize the above result just in case out true love decides to be extraordinarily generous and keeps on giving us gifts - up to days, say.
Multiplying and adding could get quite tedious. 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:. On the first day, 1 present.
These partial sums are called tetrahedral numbers , because they can be drawn as 3-dimensional triangular pyramids tetrahedrons like this:.
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 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. Let's take for example the sum of the first 4 triangular numbers and represent it as a triangle.
Each row in the triangle on the left, below adds to a triangular number and the sum of the whole triangle is the sum of the first 4 triangular numbers. Let's now re-arrange the first triangle in 2 different ways, then add the result, in respective positions. My total is 3 times what I really need. I will divide by 3 later to cater for this. Notice that by doing this, I get a total of 6 in every position in the result triangle. The answer of "6" is 2 more than the 4 triangular numbers that we are adding.
The formula for the sum to n terms of an arithmetic progression with first term a and common difference d is:. Dividing this by 3 since we used 3 equivalent sum triangles to get this far gives us the n -th tetrahedral number:. If my true love gave me the presents in this pattern for 30 days, I would have this number of presents:. See the 66 Comments below. Posted in Mathematics category - 16 Dec [ Permalink ]. This is incredible. But first I will have to send it by my husband to show me all the ins and outs of it.
Thanks for doing it. Thanks for your post.
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