If you want to be a great success, talk to people about sex [my grandfather].
I think 'sexehexes' would be a better
name for them.
Yes, that would be a perfect name! [Ed Pegg Jr]
Thanks, Ed, it's a good palindrome.
The longest Italian palindrome is "accavallavacca." Which is the longest English palindrome? (word which you can read from right to left too).
Among the regular polygons, only the triangles, the squares and the hexagons tile the plane. I don't want to think what the sex-pentominoes would be. Now I want to speak about the sexehexes.
How many are they? Below you see (one side): 1 neutral, 1 male, 1 female and 11 bi-sex:
32 penta-sex (one side, they are 2^5):
At last 14 6-sex (note: the number 14 is a real obsession):
If you left out the "*" pieces, you have the two-side series. The total is 130 one-side and 92 two-side sexehexes (See More...).
With the two-side series we could tile a pseudo-rectangle of 14+15+14+15+14+15+14+15+14=130 hexagons (the number 14 again!). The figure below shows the best today (18 mar 2000) known solution (by Richard Dickson). It's good, but I think you could do it better.
130=12^2-14 (The nr. 14 again!) The picture below has 130 hexagons too. It's nice, but I think it's impossible (with straight edge).
The next is a 127 pieces sexehex hexagon. It's made with 127 of the 130 sexehexes, if three pieces are left out.
QUESTION: Can a solution be, without male or female external edges?
ANSWER 1: I think it's impossible. I don't have a proof yet.
ANSWER 2: Perhaps it exists, but it's practically impossible to find it.
ANSWER 3: Look at it:
Ed Pegg Jr has sent the first email. He says: The 92 two-side sexehexes might be more interesting. 91 pieces will make a hexagon. How straight can the edge be made, if the plain hexagon is left out?
I painted the Ed map according to the number of the sexes. Notice the not random arrangement.
How straight can the edge be made, if the plain hexagon is left out?
The cases are two: the full-solution exists OR the solution below is the best (and it tiles the plane!).
The last picture is a nice not-solution. It is made with 91 of the 130 one-side sexehexes. My Darwin Machine employed 11 seconds to solve it. If you give this sub-set to a friend, I don't think he'll solve it.
You could purchase the SexeHexeS here.
First edition: 20 Feb 2000 - Last modify: 12 May 2000
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