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Covering perforated hexagon with triminoes

Go to solution Solved by gavinksong,


This puzzle is inspired by posted by bonanova.

Again we work on a hexagonal tiling of a plane, and the question is about
possibility of covering some shape with triminoes.
Trimino is a "triangle" formed by three unit hexagons sharing common vertex.

The shape to cover is defined as follows:
Let's pick a unit hexagon and call it H1.
Now we recursively define Hn+1 as a sum of Hn and all unit hexagons adjacent to Hn.
So basically Hn is a "hexagon" with side of length n (unit hexagons).

Let Dn be Hn with one unit hexagon at it's center removed.
So, can you cover D2015 with triminoes?

Edited by rookie1ja
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  • Solution

Wait, I just realized that the question is asking about D(2015), not D(2014). :P

Using my reasoning above (which was not very elegant), D(2015) has an odd number of colors, namely three extra of each color that isn't what the center tile would have been. Thus, a complete tiling is impossible. Yay!

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I just saw this puzzle today.


The coloring trick immediately came to mind, but I couldn't figure out how to count the colors...

Just now, I reasoned that you could split any D(n) into six triangle shaped pieces that look like this after coloring (each triangle will only look the same as two others, but the color pattern will be the same for all six):








After the first three rows, the later rows repeat except with some sets of three different colors appended to the end. In other words, the regex for every third row after the first row would be R(GBR)*, the regex for every third row after the second row would be GB(RGB)*, and the regex for every third row after the third row would be BRG(BRG)*.

Since D(n) has a side length of n, the triangles each have n-1 rows. Since the colors repeat, we can just look at the first n-1 (mod 3) rows and count the colors.

From this, I gleaned that there is only an odd number of colors if n (mod 3) = 2. So for D(2014), there is an equal number of each color.

So after all this, all I can say is that it's not necessarily impossible to tile D(2014) with triminoes...

Edited by gavinksong
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Is there a more elegant reasoning to show why that is?

I was analyzing D

n+1-Dn instead of dividing Dn into 6 triangles, but I don't think that it was more elegant

I see, so you were looking at each ring.

Anyways, that was an excellent puzzle, witzar! Well done!

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