One of my hobbies (I developed during my PhD) is *designing boardgames*. I designed three boardgames so far, one of which is **Tessel**, a word-building game based on graph theory. I am glad that Tessel is getting good feedback especially from schools and families. One of the most time-consuming part of Tessel’s design is deciding what values to assign to the letters and deciding which pairs of letters to use in the tiles. The pairs of letters are carefully chosen based on computer simulations of frequency of letters in english words and their “relative” importance. The pairs are chosen so as to give fair share to both vocabulary skills and optimization skills.

Today’s post is about a nice theoretical problem arising from this game.

Before you read further, please read the **rules of Tessel**. Henceforth I will assume that you understood the rules and goal of this game.

I guess you observed that the tiles are being placed on the edges of a *planar graph*. Tessel uses a special planar graph that has cycles of length 3,4,5 and 6. In general, this game can be played on any planar graph. I am planning to design another board using Cairo tessellation. Anyways, here is a theoretical problem :

Let S be a set of finite alphabets. You are given two different words (using alphabets from S) of length l1 and l2. Construct a planar graph G and label each edge with two alphabets, such that there are two walks in G that correspond to the given two words. (Read the **rules of tessel** and look at these **examples** to understand this correspondence). Your goal is to construct G with minimum number of vertices (or minimum number of edges).

In general you can ask the above question given k different words. What is the complexity of this problem ? I don’t know. I haven’t given it a deep thought. These days whatever I do for fun (to take my mind off open problems), ends up in another open problem :(

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