A hexomino tile is a tile in the shape of a regular hexagon with a number from 1 to n at each corner. The same number can appear on more than one corner.
A set of size n will have one tile with each possible configuration of numbers from 1-n in both clockwise and counterclockwise orientation.
How many hexomino tiles are in a size n set?
When 1 through 3 are the only possibilities for each vertex, there are already 130 possible hexominoes, found by
clc, clearvars
n=3, ct=0;
fileName=['hexom' char(string(n)) '.txt']
fid=fopen(fileName,'w');
src=[];
for a=1:n
entry=repmat(char(string(a)),6);
src=[src entry(1,:)];
end
c=nchoosek(src,6);
c=unique(c,'rows');
for i=1:length(c)
t=unique(perms(c(i,:)),'rows');
for j=1:size(t,1)
tst=t(j,:);
lowest=min(tst);
ix=findstr(tst,lowest);
if ix(1)==1
tst2=[tst tst];
good=true;
for k=2:6
if string(extractBetween(tst2,k,k+5))<string(tst)
good=false;
break
end
end
if good
ct=ct+1;
disp(tst)
fprintf(fid,'%s\n',tst);
end
end
end
end
fclose(fid)
They are:
111111
111112
111113
111122
111212
112112
111123
111132
111213
111312
112113
111133
111313
113113
111222
112122
112212
121212
111223
111232
111322
112123
112132
112213
112312
113122
113212
121213
111233
111323
111332
112133
112313
113123
113132
113213
113312
121313
111333
113133
113313
131313
112222
121222
122122
112223
112232
112322
113222
121223
121232
121322
122123
122132
122213
112233
112323
112332
113223
113232
113322
121233
121323
121332
122133
122313
123123
123132
123213
131322
132132
112333
113233
113323
113332
121333
123133
123313
131323
131332
132133
113333
131333
133133
122222
122223
122232
122322
123222
132222
122233
122323
122332
123223
123232
123322
132223
132232
132322
133222
122333
123233
123323
123332
132233
132323
132332
133223
133232
133322
123333
132333
133233
133323
133332
133333
222222
222223
222233
222323
223223
222333
223233
223323
232323
223333
232333
233233
233333
333333
For various n, we get
n tiles
2 14
3 130
4 700
5 2635
6 7826
7 19684
OEIS finds these to be in A006565, where they are mentioned as being the Number of ways to color vertices of a hexagon using <= n colors, allowing only rotations.
The formula given is
n*(n+1)*(n^2+n+1)*(n^2-2*n+2)/6
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Posted by Charlie
on 2021-10-27 13:19:01 |
computer solution
