triangles removed | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
number of fractals | 6 | 21 | 47 | 69 | 69 | 47 | 21 |
The triangles are numbered as follows:
Here is a Python program for counting the number of fractals, where two fractals that are mirror images across a vertical reflection are counted only once. The integers correspond to the triangles that are removed as indicated in the diagram above.
from itertools import combinations a = [1,2,3,4,5,6,7,8,9] m = 3 # number of rectangles to remove b = list(combinations(a,m)) # all 9 choose m combinations in lexicographical order b2 = [list(x) for x in b] c = [] dic = {3:1, 6:5, 7:4, 1:3, 5:6, 4:7} # dictionary for symmetric locations for k in range(len(b2)): d = [dic.get(n,n) for n in b2[k]] # replace triangles in kth item with symmetric locations d.sort() # put in lexographic order after swap if d not in b2[0:k]: # check if already in list up to kth position c.append(b2[k]) # if not, append to the list c2 = [tuple(x) for x in c] print(c2) print(len(c2))
The output is shown in the following table:
m | Remove the following triangles | Count |
1 | [(1,), (2,), (4,), (5,), (8,), (9,)] | 6 |
2 | [(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (1, 9), (2, 4), (2, 5), (2, 8), (2, 9), (4, 5), (4, 6), (4, 7), (4, 8), (4, 9), (5, 6), (5, 8), (5, 9), (8, 9)] | 21 |
3 | [(1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6), (1, 2, 7), (1, 2, 8), (1, 2, 9), (1, 3, 4), (1, 3, 5), (1, 3, 8), (1, 3, 9), (1, 4, 5), (1, 4, 6), (1, 4, 7), (1, 4, 8), (1, 4, 9), (1, 5, 6), (1, 5, 7), (1, 5, 8), (1, 5, 9), (1, 6, 7), (1, 6, 8), (1, 6, 9), (1, 7, 8), (1, 7, 9), (1, 8, 9), (2, 4, 5), (2, 4, 6), (2, 4, 7), (2, 4, 8), (2, 4, 9), (2, 5, 6), (2, 5, 8), (2, 5, 9), (2, 8, 9), (4, 5, 6), (4, 5, 7), (4, 5, 8), (4, 5, 9), (4, 6, 8), (4, 6, 9), (4, 7, 8), (4, 7, 9), (4, 8, 9), (5, 6, 8), (5, 6, 9), (5, 8, 9)] | 47 |
4 | [(1, 2, 3, 4), (1, 2, 3, 5), (1, 2, 3, 8), (1, 2, 3, 9), (1, 2, 4, 5), (1, 2, 4, 6), (1, 2, 4, 7), (1, 2, 4, 8), (1, 2, 4, 9), (1, 2, 5, 6), (1, 2, 5, 7), (1, 2, 5, 8), (1, 2, 5, 9), (1, 2, 6, 7), (1, 2, 6, 8), (1, 2, 6, 9), (1, 2, 7, 8), (1, 2, 7, 9), (1, 2, 8, 9), (1, 3, 4, 5), (1, 3, 4, 6), (1, 3, 4, 7), (1, 3, 4, 8), (1, 3, 4, 9), (1, 3, 5, 6), (1, 3, 5, 8), (1, 3, 5, 9), (1, 3, 8, 9), (1, 4, 5, 6), (1, 4, 5, 7), (1, 4, 5, 8), (1, 4, 5, 9), (1, 4, 6, 7), (1, 4, 6, 8), (1, 4, 6, 9), (1, 4, 7, 8), (1, 4, 7, 9), (1, 4, 8, 9), (1, 5, 6, 7), (1, 5, 6, 8), (1, 5, 6, 9), (1, 5, 7, 8), (1, 5, 7, 9), (1, 5, 8, 9), (1, 6, 7, 8), (1, 6, 7, 9), (1, 6, 8, 9), (1, 7, 8, 9), (2, 4, 5, 6), (2, 4, 5, 7), (2, 4, 5, 8), (2, 4, 5, 9), (2, 4, 6, 8), (2, 4, 6, 9), (2, 4, 7, 8), (2, 4, 7, 9), (2, 4, 8, 9), (2, 5, 6, 8), (2, 5, 6, 9), (2, 5, 8, 9), (4, 5, 6, 7), (4, 5, 6, 8), (4, 5, 6, 9), (4, 5, 7, 8), (4, 5, 7, 9), (4, 5, 8, 9), (4, 6, 8, 9), (4, 7, 8, 9), (5, 6, 8, 9)] | 69 |
5 | [(1, 2, 3, 4, 5), (1, 2, 3, 4, 6), (1, 2, 3, 4, 7), (1, 2, 3, 4, 8), (1, 2, 3, 4, 9), (1, 2, 3, 5, 6), (1, 2, 3, 5, 8), (1, 2, 3, 5, 9), (1, 2, 3, 8, 9), (1, 2, 4, 5, 6), (1, 2, 4, 5, 7), (1, 2, 4, 5, 8), (1, 2, 4, 5, 9), (1, 2, 4, 6, 7), (1, 2, 4, 6, 8), (1, 2, 4, 6, 9), (1, 2, 4, 7, 8), (1, 2, 4, 7, 9), (1, 2, 4, 8, 9), (1, 2, 5, 6, 7), (1, 2, 5, 6, 8), (1, 2, 5, 6, 9), (1, 2, 5, 7, 8), (1, 2, 5, 7, 9), (1, 2, 5, 8, 9), (1, 2, 6, 7, 8), (1, 2, 6, 7, 9), (1, 2, 6, 8, 9), (1, 2, 7, 8, 9), (1, 3, 4, 5, 6), (1, 3, 4, 5, 7), (1, 3, 4, 5, 8), (1, 3, 4, 5, 9), (1, 3, 4, 6, 8), (1, 3, 4, 6, 9), (1, 3, 4, 7, 8), (1, 3, 4, 7, 9), (1, 3, 4, 8, 9), (1, 3, 5, 6, 8), (1, 3, 5, 6, 9), (1, 3, 5, 8, 9), (1, 4, 5, 6, 7), (1, 4, 5, 6, 8), (1, 4, 5, 6, 9), (1, 4, 5, 7, 8), (1, 4, 5, 7, 9), (1, 4, 5, 8, 9), (1, 4, 6, 7, 8), (1, 4, 6, 7, 9), (1, 4, 6, 8, 9), (1, 4, 7, 8, 9), (1, 5, 6, 7, 8), (1, 5, 6, 7, 9), (1, 5, 6, 8, 9), (1, 5, 7, 8, 9), (1, 6, 7, 8, 9), (2, 4, 5, 6, 7), (2, 4, 5, 6, 8), (2, 4, 5, 6, 9), (2, 4, 5, 7, 8), (2, 4, 5, 7, 9), (2, 4, 5, 8, 9), (2, 4, 6, 8, 9), (2, 4, 7, 8, 9), (2, 5, 6, 8, 9), (4, 5, 6, 7, 8), (4, 5, 6, 7, 9), (4, 5, 6, 8, 9), (4, 5, 7, 8, 9)] | 69 |
6 | [(1, 2, 3, 4, 5, 6), (1, 2, 3, 4, 5, 7), (1, 2, 3, 4, 5, 8), (1, 2, 3, 4, 5, 9), (1, 2, 3, 4, 6, 8), (1, 2, 3, 4, 6, 9), (1, 2, 3, 4, 7, 8), (1, 2, 3, 4, 7, 9), (1, 2, 3, 4, 8, 9), (1, 2, 3, 5, 6, 8), (1, 2, 3, 5, 6, 9), (1, 2, 3, 5, 8, 9), (1, 2, 4, 5, 6, 7), (1, 2, 4, 5, 6, 8), (1, 2, 4, 5, 6, 9), (1, 2, 4, 5, 7, 8), (1, 2, 4, 5, 7, 9), (1, 2, 4, 5, 8, 9), (1, 2, 4, 6, 7, 8), (1, 2, 4, 6, 7, 9), (1, 2, 4, 6, 8, 9), (1, 2, 4, 7, 8, 9), (1, 2, 5, 6, 7, 8), (1, 2, 5, 6, 7, 9), (1, 2, 5, 6, 8, 9), (1, 2, 5, 7, 8, 9), (1, 2, 6, 7, 8, 9), (1, 3, 4, 5, 6, 7), (1, 3, 4, 5, 6, 8), (1, 3, 4, 5, 6, 9), (1, 3, 4, 5, 7, 8), (1, 3, 4, 5, 7, 9), (1, 3, 4, 5, 8, 9), (1, 3, 4, 6, 8, 9), (1, 3, 4, 7, 8, 9), (1, 3, 5, 6, 8, 9), (1, 4, 5, 6, 7, 8), (1, 4, 5, 6, 7, 9), (1, 4, 5, 6, 8, 9), (1, 4, 5, 7, 8, 9), (1, 4, 6, 7, 8, 9), (1, 5, 6, 7, 8, 9), (2, 4, 5, 6, 7, 8), (2, 4, 5, 6, 7, 9), (2, 4, 5, 6, 8, 9), (2, 4, 5, 7, 8, 9), (4, 5, 6, 7, 8, 9)] | 47 |
7 | [(1, 2, 3, 4, 5, 6, 7), (1, 2, 3, 4, 5, 6, 8), (1, 2, 3, 4, 5, 6, 9), (1, 2, 3, 4, 5, 7, 8), (1, 2, 3, 4, 5, 7, 9), (1, 2, 3, 4, 5, 8, 9), (1, 2, 3, 4, 6, 8, 9), (1, 2, 3, 4, 7, 8, 9), (1, 2, 3, 5, 6, 8, 9), (1, 2, 4, 5, 6, 7, 8), (1, 2, 4, 5, 6, 7, 9), (1, 2, 4, 5, 6, 8, 9), (1, 2, 4, 5, 7, 8, 9), (1, 2, 4, 6, 7, 8, 9), (1, 2, 5, 6, 7, 8, 9), (1, 3, 4, 5, 6, 7, 8), (1, 3, 4, 5, 6, 7, 9), (1, 3, 4, 5, 6, 8, 9), (1, 3, 4, 5, 7, 8, 9), (1, 4, 5, 6, 7, 8, 9), (2, 4, 5, 6, 7, 8, 9)] | 21 |
Click here to see all the fractals when 1, 2, or 3 triangles are removed, and the 8 symmetric fractals when 4 triangles are removed.