You people in the high numbers are counting trapezoids and other shapes. I don’t know if my number is right, but I’m pretty sure numbers like 18 must be wrong.
Or, I could be wrong. Lol. I might be missing some.
At first I was counting the trapezoids, and then I thought I had caught my mistake—caught the trick, but then I obviously had missed a bunch of the triangles in the end.
18. If we label the columns c1, c2 and c3, we can combine columns as: c1, c2, c3, (c1 and c2), (c2 and c3) and (c1, c2 and c3). That is 6 ways of combining columns. The triangle can include 1, 2 or 3 rows. 3×6=18.
The number of ways of selecting the columns is 3 + 2 +1. Here is why. If the leftmost column is c1 then the rightmost column is c1, c2 or c3. If the leftmost column is c2 then the rightmost column is c2 or c3. If the leftmost column is c3 then the rightmost column is also c3. Adding up the possibilities, we get 3+2+1. If there were n columns, the number of ways of combining them would be n + (n-1) + n-2) + ... +1 = n(n+1)/2, using the formula for arithmetic series.