## Graph Theory, Numberphile and Mathematica

Edit: I made a mistake with some of the language here. A comment from a true graph theorist:

“Hamiltonian” usually means there’s a hamilton/hamiltonian cycle. Graphs with a hamilton path are “traceable”. Hamiltonian implies traceable, but not conversely.

Thus, I have edited below accordingly.

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There was a nice video up on Numberphile about a problem which could easily be explained to a school student, and yet we don’t yet know the answer to it. See the videos here:

and

The game is the following: Given a sequence of consecutive integers, draw a graph where the nodes are the integers and there is an edge between each integer if their sum is a square number.

If we take the numbers from 1 to 12, then the following would be the associated graph (note that there are three disconnected pieces of it).

Note that this is a disconnected, undirected graph. Looking at some of the edges.…

## How much does a Dougie weigh on Jupiter?

I spent an evening with old friends just after Christmas. One of their sons is fascinated by Space, and always has great questions for me about it. This year he asked how much he would weigh on Jupiter. I admitted that I didn’t know the answer, but that a) It wasn’t a single number, as the surface of Jupiter (if one exists) is not very well understood and different heights in the atmosphere would give different weights and b) I would find out. It took a little Googling to find some information about the density profile of Jupiter, which includes information about Saturn too. The paper is the following, with abstract:

Taken from http://www.nature.com/nature/journal/v520/n7546/full/nature14278.html

In this paper is the following pair of graphs:

Taken from http://www.nature.com/nature/journal/v520/n7546/images/nature14278-sf4.jpg

Which is all we need to calculate how much Dougie weighs on Jupiter. In the lower plot we see two different models of the density of Jupiter at different heights.…