Solution to Spring 2016 POTW #4 (Bow Do You Do?)

See here for the original post of the puzzle.

Let's say x is the number of administrators, so the number of faculty members is 2x.
Then, we can careful count all the bows that took place:

• Every administrator (x) bows to each other administrator (x-1): total x(x-1)
• Every administrator (x) bows to each faculty member (2x): total x(2x)
• Every administrator (x) bows to Sr. Janet (1): total x
• Every faculty member (2x) bows to each other faculty member (2x-1): total 2x(2x-1)
• Every faculty member (2x) bows to each administrator (x): total 2x(x)
• Every faculty member (2x) bows to Sr. Janet (1): total 2x

Total number of bows: x²-x+2x²+x+4x²-2x+2x²+2x=9x²
Therefore, 900=9x², so x=10.

This means there were 10 administrators, 20 faculty, and Sr. Janet in attendance, for a total of 31 people.