How many triangles in a triangle of side four?

Question

An equilateral triangle of side 4 is divided by lines parallel to its sides into 16 small equilateral triangles of side 1 (rows of 1, 3, 5 and 7 small triangles).

How many triangles of all sizes can be found in the figure? Count the upward-pointing and downward-pointing triangles separately, and give the totals for a triangle of side 3 as a check.

Accepted Answer

27 triangles: 20 pointing up and 7 pointing down. Upward triangles. An upward triangle of side k is determined by its top corner, which can be any grid point in the top n minus k + 1 rows of points (row j has j points). So there are 1 + 2 + ... + (n minus k + 1) of them. For n = 4: side 1: 1 + 2 + 3 + 4 = 10; side 2: 1 + 2 + 3 = 6; side 3: 1 + 2 = 3; side 4: 1. Total 20. Downward triangles. A downward triangle of side k needs k rows of room beneath its base, so its base can only sit in rows with enough space; the count is 1 + 2 + ... + (n minus 2k + 1). For n = 4: side 1: 1 + 2 + 3 = 6; side …

How many triangles in a triangle of side four?

Your answer

Discussion

Learn the concepts

Related questions