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I have been trying to make a general formula/function that helps in calculating the number of pentagons that may be formed using 2 or more hexagons. Like it is shown in the picture below: enter image description here

In Fig. 1 there are 2 hexagons and 6 pentagons. Therefore, the ratio becomes $1:3$. However, this should never be the ratio as no pentagon can be formed with a single hexagon. Let $y$ be the number of pentagons formed and $x$ be the number of hexagons, we can say: $y = 3x$ when $x =2$.

In Fig. 2 there are 3 hexagons and 12 pentagons. Therefore, the ratio becomes $1:4$ Now, in this case the equation becomes: $y = 4x$ when $x = 3$.

If we bring in another hexagon, we shall have 6 more pentagons. So, in that case, the ratio will be $2:9$, thereby, changing the equation too. What I mean to say is the ratio is changing every time we introduce a hexagon.

I know that $x$ (number of hexagons) and $y$ (number of hexagons) $\epsilon$ $N$; and $x \geq 2$, $y \geq 6$.

Since the ratios are changing every time we introduce a hexagon, I don't seem to define a general formula of a function that gives us the number of pentagons if we have the number of hexagons. Maybe, I am missing out something.

So the question: How to define a function that gives us the number of pentagons formed in between two or more hexagons?

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up vote 1 down vote accepted

It looks like you are drawing 6 pentagons for each hexagon, except the outermost. So if $x$ and $y$ are the number of hexagons and pentagons, respectively, then $y=6(x-1)$.

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That's it! Thank you. I got very confused looking at the ratios. Thanks. :D – xyres Jul 16 '13 at 11:09

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