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.

Solution to Spring 2016 POTW #3 (Stop, Clock, and Roll)

See here for the original post of the puzzle.

Since Clock A is gaining 1 minute every 24 hours, and Clock B is losing 1 minute every 24 hours, then the next time they will all show noon at the same moment will be when Clocks A and B both gain/lose (respectively) 12 hours (meaning they've gotten "closer together" by 24 hours).

Since they lose/gain 1 minute every day, and there are 12 x 60 = 720 minutes in 12 hours, then it will take 720 days to reach that phenomenon.

Thus, we are looking for the date that is 720 days after January 1, 2016. Be careful because 2016 is a Leap Year! So, the final answer is December 21, 2017.

Solution to Spring 2016 POTW #2 (Weird? You Bet!)

See here for the original post of the puzzle.

Trial and error may work here, but I don't recommend it. Instead:

• Suppose Adam starts with $A and Brendan with $B
• Adam loses first game, pays out to double Brendan's money.
• New results: Adam has $(A-B)$ and Brendan has $(2B).
• Brendan loses second game, pays out to double Adam's money.
• New results: Adam has $(2(A-B))=$(2A-2B) and Brendan has $(2B-(A-B)) = $(3B-A).
• By assumption: 120 = 2A-2B = 3B-A.
• Solve this system of equations to get A=150 and B=90.

Playing it out, just to check:

• Start: Adam has $150, Brendan has $90.
• Adam loses. New results: Adam has $60, Brendan has $180.
• Brendan loses. New results: Adam has $120, Brendan has $120.

(Here's a bonus challenge for you readers: Is there anything special about $120? Could this puzzle be solved for any ending amount? And what if there were 3 players and each one lost a round? Try to generalize!)

Solution to Spring 2016 POTW #1 (Vinyl Buys)

See here for the original post of the puzzle.

Say x and y are the costs of the two records. I bought them for x+y=100 dollars and then sold them for 1.2x+0.8y=110 dollars. (Note the 20% markup on x and 20% discount on y.) Solve this simple system of two equations with two unknowns and you'll find x=75 and y=25.

Solution to Fall 2015 POTW #6 (Ehhh, The Horror!) and #7 (Emmanuel Fay Cult)

See here for the original post for Fall 2015 POTW #6: Ehhh, The Horror!
Solution:

1. "The Silence of the Limbs"
2. "The Bards"
3. "2001: A Spice Odyssey"
4. "The Amen"

See here for the original post for Fall 2015 POTW #7: Emmanuel Fay Cult

Solution:

1. Yulia DEMENTIEVA (MEDIA EVENT)
2. Aren GERDON (GO NERD)
3. Christine LEIGHTON (HOTEL GIN)
4. Lenore MARTIN (TIN ARM)

Solution to Fall 2015 POTW #5: Oh My Gourd!

See here for the original puzzle post.

Solution:

1. Pumpkin
2. Butternut (squash)
3. Acorn (squash)
4. Spaghetti (squash) ("Spa-Getty")

Solution to Fall 2015 POTW #4: trivia with a theme

See here for the original puzzle post.

Solution:

1. Royale
2. The Middle East
3. Church
4. Great Scott

Theme: These are all live music venues in Boston.

Solution to POTW #3: MBTehhhhh

See this link for the original puzzle post.

Solution:

1. Haymarket
2. Government Centre
3. Braintree
4. Coolidge Corner
5. Cleveland Circle

Solution to POTW #2: mOvie vOwels

See this link for the original puzzle post.

Solution:

1. Johnny Depp gives a stellar performance in this true-life tale of mobsters, the FBI, and the search for a deadly, dark-colored fungus.
   Black Moss (from Black Mass)
   
2. This critically-acclaimed Oscar-winner depicts the mayhem and violence caused by corruption, organized crime, and double-agents at a big city’s department of Citizenship & Immigration Services.
   The Deported (from The Departed)
   
3. In this dark tale of entangled crimes and accusations, a softball team’s enigmatic extra infielder is murdered by his childhood pal.
   Mystic Rover (from Mystic River)

Theme: All of these movies are set in the city of Boston, MA. (More specifically, they're crime dramas set in Boston, but I accepted "set in Boston" as sufficiently correct.)

Solution to POTW #1: the broken pocketwatch

See this link for the original puzzle post.

Solution:

We know the watch was correct at 2:00am. Let’s make a table of times, showing both the actual time and the time that the watch shows:

Actual time	Watch time
2:00 am	2:00 AM
3:00 AM	3:36 AM
4:00 AM	5:12 AM
5:00 AM	6:48 AM
6:00 AM	8:24 AM

Voila! When the watch stopped, showing 8:24am, it was actually 6:00am. Since that was one hour ago, it must now be 7:00 am.

“Alternative” method: We can take an algebraic approach instead of making a table of values and hoping to get lucky. Let X denote the number of hours that have passed since 2:00 am, when the watch last showed the correct time. Then, we can set up an equation that represents the time that the watch shows:

The left-hand side says, “The watch shows 8:24am right now”. The right-hand side says, “It was 2, and every hour that passed, the watch gained 1 hour and 36 minutes.”

Solving for X yields X = 4. That is, 4 hours passed since 2:00am, so the actual time was 6:00am when the watch showed 8:24am. Again, since it’s now one hour later, the time must now be 7:00 am.