How many triangles can you make with numbers on a telephone keypad?
There are ten numbers on a telephone keypad, arranged as a 3x3 grid with an additional number under the bottom centre.
To count the number of distinct triangles, I think I can follow the logic explained here to get 109. 10C3 = 120, minus 3 horizontal lines, 2 diagonal lines, and 6 vertical lines (2 on the sides and 4C3 in the centre).
But I'm more curious about how many types of triangles there are. Right now I'm just counting by hand and I see at least 13.
Four right triangles, with side lengths (i.e. how many buttons are covered by the non-hypotenuse sides) 2x2, 2x3, 3x3, and 4x2,
Three non-right isosceles triangles, all with a base length of three and with heights 2, 3, and 4.
Two non-right isosoles triangles that are tilted diagonally, e.g. [067] and [037].
One scalene triangle with a base of 4 and a height of 2.
The others are sort of skewed and not simple to describe but e.g. [068], [035], and [038].
Is this all of them? And more interestingly, is there a systematic way to count them?
