Puzzle of the Week #3

by Nolan Musslewhite '20

​Solvers of last week’s puzzle (in order):    
1. David Hla
2 . Brandon Torng
3. Maggie Wang
4. Robert Shekoyan

5. Jonathan Merril
6. Matthew Chalk
7. Liam Chalk
8. Mark MacGuidwin

9. Mr. Rick DuPuy
10. Dr. Jarad Schofer
11. Lars Nordquist


Answer to and commentary on Puzzle #2:

    Classification: Medium. I will use Robert Shekoyan’s excellent explanation; You know that the top factorial is going to contain the highest term in it (e.g., the 17 doesn't get canceled when you cancel with the factorial in the denominator) since ♤ > ☖ (because the hypotenuse is larger than either side). Thus, you want the terms between ♤ and ☖ to yield a product that is two times ☗ (e.g, if ♤ was 7 and empty pentagon was 4, I'm looking at 6,5). 8-15-17 is the only Pythagorean Triple for which this is true. It's easy to recognize it as 8-15-17 because we see that 16 is the only term between spade and empty pentagon, which is twice 8. Basically, the gist of it is to look at the integers between the length of the longer leg and the length of the hypotenuse and see if the product of those terms is twice the length of the shorter leg.

Puzzle #3
Nolan Musslewhite (nmusslewhite@stalbansschool.org)

This week’s puzzle: Mathematics (note: the puzzlemaker apologizes for the overemphasis on mathematics. Future puzzles will concern a more diverse array of topics). Solve for x.
Hint: Think of a way to generalize an infinitely repeating chain. For example, if y = -0.5x – 0.25x – 0.125x – …, then it could be said that y = -0.5x + 0.5y