Can You Really Solve This Tricky Rep-Tile Puzzle?

1961, American mathematician Solomon W. Golomb coined the term: Rep-Tile as a pun on “reptiles”. Golomb had been researching replicating geometrical figures and had landed upon an interesting phenomenon. He went on to publish three papers on the topic of replicating polygons. This in turn turned out to be pioneering work in this field.

In this essay, I will initially be covering the fascinating world of rep-tiles by giving a basic introduction to the topic. Following this, what awaits you is a very tricky, yet engaging puzzle. Let us begin.

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What is a Rep-Tile?

The term rep-tile refers to any replicating geometrical figure that could be used to tile an entire plane without using any other shape. As an example, consider the typical square tiles. Using the same shape (a square that enjoys reflective and rotational symmetry), we would be able to tile an entire plane without needing any other shape.

As it turns out, among regular geometrical polygons (all sides equal / all angles equal), only the square, the equilateral triangle, and the regular hexagon can be used to tile an entire plane without using any other respective shapes. Among irregular polygons, there are more possibilities.

What is a Rep-k Polygon?

Golomb defines a rep-tile of order ‘k’ as a replicating polygon that is comprised of ‘k’ miniatures of itself that are congruent to one another and are similar to the original polygon. In shortened notion, we call such a polygon a rep-k polygon.

As an example, consider the rep-4 polygon illustrated below. It features one big trapezoid which sub-divides into exactly 4 miniature trapezoids that are congruent to one another and similar to the bigger trapezoid (if you are kind enough to see beyond the sub-par illustration).

As far as rep-2 polygons are concerned, we know of only two of them: the isosceles right triangle with a hypotenuse-to-side ratio of √2, and the parallelogram with a length-to-breadth ratio of √2. One notable feature of the parallelogram is that its rep-2 properties are independent of its angle.

This means that at an angle of 90°, the rep-2 parallelogram transforms into the rep-2 rectangle. This rectangle has historically played an influential role in the world of art. To learn about more practical uses of this rectangle, check out my essay on how we really use mathematics to define paper.

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Can You Really Solve this Tricky Rep-Tile Puzzle?

Now that you have had an introduction to the concept of rep-tiles and rep-k polygons, it is time to jump into the puzzle. The problem statement is really simple:

Construct a rep-3 triangle.

Just to clarify, it is not a given that any arbitrary rep-k polygon exists. It just happens to be the case that a rep-3 triangle exists.

To elaborate on the task, you will need to construct a triangle that divides itself into three miniature versions that are similar to the original and congruent to one another.

Hint:

First, consider the various kinds of possible triangles (equilateral, isosceles, etc.). Then, start eliminating the types that would make it impossible to construct a rep-3 triangle. This should help you narrow the problem down to just one type of triangle that permits a rep-3 construct.

Spoiler Alert:

Beyond this section, I will be directly discussing solutions to this puzzle. If you wish to solve this puzzle on your own, I suggest that you tune off of this essay now. Once you have tried to solve the puzzle on your own, you may choose to come back to this essay and continue reading from this point.

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My First Attempts at Solving the Tricky Rep-Tile Puzzle

When I started solving this puzzle, I wrote down all of the relevant knowledge that I had about triangles beside the conditions necessary to solve the puzzle. Then, I started trying out one idea after another (trial and error).

As one of my first promising solutions, I came up with the following structure. It is an equilateral triangle of side length: 1 unit. I used the angle bisectors and the incentre to construct three triangles inside the bigger triangle. I thought I’d take advantage of the circular symmetry around the incentre.

This solution is close, yet so far away from the goal. You see, the three smaller triangles are congruent to one another. However, they fail to be similar to the original triangle; they are not equilateral triangles.

From this approach, I learned that as long as I aim to take advantage of circular symmetry, an equilateral triangle would make it impossible to construct a rep-3 triangle. After this, it took a lot more effort, trial, error, and luck(!) to arrive at the solution to the puzzle.

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The Solution to the Tricky Rep-Tile Puzzle

I would have loved to share a structured approach to solving this puzzle. But unfortunately, my approach was anything but structured. It involved wild thrashing of ideas and trial and error.

I assumed that perhaps I was simply not smart enough to solve the puzzle in a structured manner. So, I went looking for literature on the topic. Unfortunately, I am yet to find any literature that points to a structured approach to solving this puzzle.

I did land on literature that covers structured approaches to solving higher order rep-k triangles. I will cover this topic in a separate essay in the future. But for now, here is the solution to our puzzle:

First, I constructed a 90–60–30 triangle with side lengths of 1, 2 (hypotenuse) and √3 units respectively. After constructing this triangle, I split the side that is √3 units long at one-third of its length, and constructed a triangle from the top vertex.

Then, from this point on the bottom edge, I constructed a perpendicular line to the opposite side to complete the remaining two triangles. This results in three miniature triangles that are congruent to each other and similar to the original triangle as well.

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References and Credit: Solomon W. Golomb and Martin Gardner.

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Further reading that might interest you: The Shipwreck Puzzle — A Fun Linear Math Challenge and How To Really Benefit From Curves Of Constant Width?

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