Here are two challenges that are completely unrelated except that both involve some sort of nodes. The first challenge is different for every reader.
In idle moments, I sometimes create challenges for myself involving the geometic layout of buttons on remote control devices. For many of these, I consider the buttons on the remote control as nodes in a grid (idealising their positions slightly but no more than necessary), and then partition that grid into a number of smaller shapes following set rules.
You can invent your own challenges of this type (and I’d be interested to read about them in the comments), but here is an example that I have solved for all three remote controls in my possession.
Partition the remote control into multiple rectangular grids, such that there’s a grid containing one button, a grid containing two buttons, a grid containing three buttons, and so on, until there aren’t enough buttons left for another grid. It’s most elegant if you make each grid as “square” as possible (i.e. better 2×2 than 1×4, better 2×3 than 1×6, etc). Are yours solveable?
Below are solutions for my remote controls: respectively my CD player control, TV/DVD control, and VHS control. (With the last, I’ve taken the liberty of treating the centre of the ‘+’-shaped control as a button in its own right.)
Now for the second challenge, which is a word challenge I came up with the other day.
Create a triangle of letters that is shaped like this:
a a a a a a a a a a a a a a a a a a a a a a a a a a a a
And meets the following conditions:
- Each row of the triangle is a word.
- The letter at each node does not precede, alphabetically, the letter(s) at its parent node(s). Parent nodes are defined below.
Each node is the parent of the node immediately below and to the left, and also of the node immediately below and to the right. This means that each node has two child nodes, that each node on the edge of the triangle has one parent (except the apex node which has none), and that each node in the middle of the triangle has two parents.
The challenge is to see how many rows high you can make the triangle. Six rows is easy; can you manage seven?
Here’s an example of a valid six-row triangle:
a a d a d d b e e f n e i g h p o i s o n
And here’s a seven-row triangle that I don’t find quite satisfactory because I sort of made up the word “strutty” on the spot (as in: “he walked in a strutty manner”, adding the adjective-forming -y suffix to the verb “strut”). The less esoteric the words used, the more impressive the solution.
i i d i n k m o n k m o o n s s p o r t s s t r u t t y
I make no claims as to what is or is not possible, but give it your best shot.