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Topic: Hexagon's Hexagons (Read 579 times) |
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Noke Lieu
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Hexagon's Hexagons
« on: Mar 24th, 2010, 7:09pm » |
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So I had some regular hexagons.
I kept adding rings of hexagons around the central hexagon, such that there was 1, 7, 19...
I cut the resulting hexagon into n pieces of n small hexagons (n being integer).
But being me, I made sure that every little hexagon touched at least 2 other hexagons in the piece.
What's the minimum number of different shaped pieces that I need to make?
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towr
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Re: Hexagon's Hexagons
« Reply #1 on: Mar 25th, 2010, 2:24am » |
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So you're looking for the first square centered hexagonal number greater than 1? If memory serves my right, that's 13*13.
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rmsgrey
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Re: Hexagon's Hexagons
« Reply #2 on: Mar 25th, 2010, 8:06am » |
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But you then need to find a division of the large hexagon into pieces that uses the minimum number of different shapes.
I'm going to take a guess at 2
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towr
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Re: Hexagon's Hexagons
« Reply #3 on: Mar 25th, 2010, 8:23am » |
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Oh, I was just assuming all pieces needed to be differently shaped, not that that was what we were to minimize.
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rmsgrey
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Re: Hexagon's Hexagons
« Reply #4 on: Mar 26th, 2010, 1:47pm » |
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Having stared at the ceiling for a while (it saves on paper) I'm going to say that, assuming I've not made any silly mistakes (keeping track of over 100 individual hexagons takes more concentration than I'm willing to muster), it's possible to do with 3 different shapes, though I wouldn't be too surprised if it turns out to be possible to do with fewer.
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The large hexagon has 8 small hexagons to a side (6 triangles with 7 hexagons to a side and one central hexagon gives 169 hexagons total - thanks towr for that figure).
The two outer rings of hexagons can easily be split into 6 identical 13-hexagon pieces - a row of 7 overlapping a row of 6.
The centre can be accounted for by a side-3 hexagon with the corners missing (or a side-2 hexagon with one hexagon added outside each side - whichever is easier to visualise).
The intermediate region can then be divided radially into 6 identical chunks to give a third and final shape.
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towr
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Re: Hexagon's Hexagons
« Reply #5 on: Mar 26th, 2010, 3:45pm » |
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There's a lot of ways to use 3 shapes. So we can rule out there were any silly mistakes.
I haven't found any way to do it with less though.
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Noke Lieu
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Re: Hexagon's Hexagons
« Reply #6 on: Mar 27th, 2010, 2:07am » |
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I'll confess that's what I worked out too.
I thought it wise to see if you mob could do better...
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towr
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Re: Hexagon's Hexagons
« Reply #7 on: Mar 27th, 2010, 6:19am » |
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Maybe someone can get a computer to check.
How many ways are there to make a valid 13-hex shape?
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| « Last Edit: Mar 27th, 2010, 6:19am by towr » |
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