19

Today, should, presuming this is going out on the 18th June 2020, be my 19th Birthday. Obviously I am not on my laptop and in the office today, this all published automatically. However, I wanted to write a little bit on the number 19!

(no, not 19 factorial you pedants)

Firsty, we know that the number 19 is a prime.

If we take a point for the first iteration of a sequence, then for the second we have a hexagon focused on the first point and the third another layer of hexagon… we get a sequence: $1, 7, 19, 37, …$. Oooh, 19! But what are these numbers called? They are the centred Hexagonal numbers.

What about if we take a similar thing but with octohedrons, maybe not so much about centralising but making octohedtrons out of points. Then we get the octrohedral numbers; $1, 6, 19, 44, …$.

Now time for something that people tell me is useless, but I love! Number Theory. Let us look at the number $3$, we can write this number as $’111’$ under partitions, this is less the number $111$ and more ‘one-one-one’. We can also write $3$ as $12$ or even just $3$. The number of permutations of each of these respectively are $1, 2, 1$ so there are $4$ different partitions of $3$. If we do the same for $19$, not only is it a tedious task, but it gives a much larger number: $490$.

Now, a little puzzle. Taking the first four factorials, add and subtract them to take 19. Below is a spoiler wall, hover over it or tap on it to see what is below.

$19 = 4! – 3! + 2! – 1!$

If you had $19$ powers of 4, you could make any number.

If you take all of the platonic solids and add the vertices, faces and edges they are divisible by 19. The same is true for the sum of all vertices of Archimedic and platonic solids and the faces and edges of the Archimedic and platonic solids.

You can also produce some magic stars with 1 to 19 and a magic hexagon. I shall let you find these. I wonder if you can do a magic octohedron…

Finally, 19 is the fourth centered triangular number!