Sometimes when I am teaching I throw out little puzzles. Can do it if I sense some restlessness – some extra intellectual stimulation to absorb that edgy energy (factoring bigger numbers is better than a binky for some 9th grade boys). Can do it if a nice puzzle is just staring at us. Today, the latter…

Naming angles is boring. But we’d just named three angles in a diagram with three rays with a common vertex. I drew the same thing but with four rays, announced this was for anyone who wanted to look at it, and asked how many angles. I also drew the diagram with five rays. (In each case the outer rays were spread about 60 or 70 degrees.)

We did other work, but got drawn back, and started discussing. I was of course expecting “six.” Someone got a higher count, but including the “external” angles (reflex is what our high school texts call angles greater than 180 degrees, but we were not there yet…) Nice twist. Do you want to try his puzzle:

Consider rays AB, AC, AD, and AE, none collinear. How many distinct angles can be formed using these rays as sides of the angles?

September 13, 2007 am30 12:35 am 12:35 am

If you include reflex angles, there are r * (r – 1) angles that can be formed from r incident rays (there are r instances of each of the (r-1) possible angle “sizes”, where the “size” of an angle means how many rays it skips over). Not including reflex angles, there are r*(r-1)/2. This can be seen in several ways; one is to note that you can count all the angles including reflex ones, pair up the angles that add to 360, and throw away all the reflex ones. Another is to note that forming an angle corresponds to choosing two of the r rays, giving a total of $\binom{r}{2} = r(r-1)$ angles.

2. September 13, 2007 am30 3:08 am 3:08 am

Formula at end seems off by a 2. Each time we choose two rays we id two angles (1 regular, 1 reflex), so I’d go for $2\binom{r}{2} = r(r-1)$ (the right side already reflected the doubling).

Nice analysis. These kids don’t have the tools available, so I will shelve it for now (they are not waiting for something from me), and return to it after we attack “the handshake problem” later this fall.