Here are two challenges that are completely unrelated except that both involve nodes in some way.
The first challenge is different for every reader, and is one example of a whole family of challenges you can invent for yourself.
Sometimes, in idle moments, my mind is drawn to the geometic layout of buttons on remote control devices. Considering the buttons simply as nodes and abstracting away all extraneous detail, you can invent puzzles for yourself that involve partitioning these nodes according to set rules.
For example: see if you can 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. A grid is an n×m rectangle with a node at every point. 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: what liberties you choose to take is your own business.)
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) as defined below.
Each node is the parent of the two notes immediately below it in the pyramid — i.e. the node immediately below and to the left, and the node immediately below and to the right.
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. Your challenge is to do better.
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.