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        riddles >> easy >> Hexagon's Hexagons
        
(Message started by: Noke Lieu on Mar 24^th, 2010, 7:09pm)

Title: Hexagon's Hexagons
Post by Noke Lieu on Mar 24^th, 2010, 7:09pm

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?

Title: Re: Hexagon's Hexagons
Post by towr on Mar 25^th, 2010, 2:24am

So [hide]you're looking for the first square centered hexagonal number greater than 1? If memory serves my right, that's 13*13.[/hide]

Title: Re: Hexagon's Hexagons
Post by rmsgrey on Mar 25^th, 2010, 8:06am

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 [hide]2[/hide]

Title: Re: Hexagon's Hexagons
Post by towr on Mar 25^th, 2010, 8:23am

Oh, I was just assuming all pieces needed to be differently shaped, not that that was what we were to minimize.

Title: Re: Hexagon's Hexagons
Post by rmsgrey on Mar 26^th, 2010, 1:47pm

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 [hide]3[/hide] different shapes, though I wouldn't be too surprised if it turns out to be possible to do with fewer.

[hideb]
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.
[/hideb]

Title: Re: Hexagon's Hexagons
Post by towr on Mar 26^th, 2010, 3:45pm

There's a lot of ways to use [hide]3[/hide] shapes. So we can rule out there were any silly mistakes.
I haven't found any way to do it with less though.

Title: Re: Hexagon's Hexagons
Post by Noke Lieu on Mar 27^th, 2010, 2:07am

I'll confess that's what I worked out too.
I thought it wise to see if you mob could do better...

Title: Re: Hexagon's Hexagons
Post by towr on Mar 27^th, 2010, 6:19am

Maybe someone can get a computer to check.
How many ways are there to make a valid 13-hex shape?