HOF+ (Tetrasticks I)

A polystick is a connected finite subgraph of the grid graph, and a tetrastick is a polystick with four edges. There are 25 of them, counting mirror copies.

Tetrasticks 25 01

In other words, these are squiggles you can make with four strokes. They’d make a nice alphabet for people who are addicted to abstraction.

Hof 01

 

Today we are focussing on six of them, fattened and colored above. They are denoted by the letters H, O, F, +, and the mirrors of H and F. For reasons to become clear later we consider O and + also as mirrors of each other in a certain sense. The goal is to tile rectangles with them, like in the 3×7 and 2×12 rectangles below.

Example 01

There are many constraints on what tiles one can use, and how many. For instance, an a x b rectangular grid has a(b+1) vertical and (a+1)b horizontal edges, for a total of 2ab+a+b edges. This number is divisible by 4 only if a-b is divisible by 4, so squares are good for tiling, as are the two rectangles above. They both consist of 52 segments and thus require 13 tetrasticks. Below is a different example.

Example2 01

Note that all our 6 letter except for H and its mirror use two horizontal and two vertical segments. As the 3×7 rectangle has 4 more horizontal edges than vertical ones, we need at least four H-tetrasticks (or its mirrors) to tile this rectangle. We can use more, but then only an even number of them. Likewise, we need at least two H-tetrasticks to tile the 2×12 rectangle.

Example3 01

This brings us to today’s puzzle: Tile the 3×7 rectangle with your choice of 13 tetrasticks from our selection of six, and then use the same set to tile the 2×12 rectangle. The examples on this page are attempts that require to flip an H or an F into its mirror (or an O into +). Can you find a perfect solution that doesn’t require flipping a tetrastick over?

Polysticks (Polyforms II)

One of my favorite polyforms are polysticks on a hexagonal grid. These critters consist of connected collections of grid edges.
I stipulate that whenever two edges of a polystick meet, we add a a joint to the figure. This is in order to avoid indecent intermingling of legs as shown by the two polysticks in the figure below. Blush. The properly decorated green polystick can only watch in dismay.

Example 01

We want to use the polysticks as puzzle pieces, and we want to keep things simple. So here are all four hexagonal polysticks with three legs and just one joint. I like to call them triffids.

FourTriffids 01

Two of them are symmetric by reflection, so I leave it up to you to count them as one or two. We can us three of them to tile a small triangle easily like so:

Mini triangle 01

By tiling I mean that we want to cover all the edges of the given shape, do not allow that two polysticks share a leg or joint (what a thought!), and do not require all vertices to be covered. We could do so, limiting the possibilities dramatically.

Below are two more examples. First a larger triangle, tiled using three kinds of triffids.

Triangletile 01

I have not found a way to tile this triangle (or a larger one) with just one kind of triffid. And here is a hexagon that uese all four triffids to be tiled:

Hexatile 01

Now go and make your own. If you want to use triffids, make sure that the number of edges of your shape is divisible by 3.