Tricky Triangles

One of the items I find in the comics area of my newspaper, the Richmond Times-Dispatch, is an item for children called Kidspot, by Dick Rogers. I have been watching these Kidspot cartoons ever since one such item suggested nonsensical notions like that of an eyeball room (I never want to be in one of those) or an eardrum stick, which would be met with disapproval by health professionals. See my companion blog, Beyond Opinion, for details.

Today I found another questionable cartoon. It consists of this arrangement of dots:

.

The text runs:

TRICKY TRIANGLES. How many triangles can you make by connecting three dots at a time? Connected triangles can count as larger triangles.

I am not sure what Dick Rogers means by connected triangles being larger triangles. This is what he gives as the answer:

Ans: There are 24 possible triangles.

This is wrong. There are 50 possible triangles. To show this, note that there are eight dots, and that any three dots either define a straight line or define a triangle. The total number of collections of three of these dots is:

C_8,3 = 8!/3! = 1*2*3*4*5*6*7*8/(1*2*3)*(1*2*3*4*5)

=(8*7*6)/(1*2*3) (since the 1*2*3*4*5's cancel)

= 56.

There are 6 ways of constructing straight lines. There are two vertical lines, and four straight lines that make one X on top of another one. Subtracting this from 56 gives 50 possible triangles.

Here are some possible triangles:





Mr. Rogers should have checked his calculations. Maybe he did not consider the magenta or green triangles.