Thursday, April 28, 2016

Results and prizes: 2015-2016 academic year

The results are in for the Emmanuel College Math Department's Puzzle of the Week contest for the 2015-2016 academic year! Congratulations to the individuals listed below for their outstanding performances with the variety of puzzles that I've posted over the last two semesters! Here are the standings, in order:
  1. Darren Parke 
  2. Paul Bleau
  3. Pawthorne (?)
  4. Rachel Gammal
  5. Madison Daigle
  6. Allison Mostowy
  7. Connor Higgins
(Important note: I have no idea who Pawthorne is. Are you an Emmanuel student? Are you a group of people? What is your real name(s)? How can I contact you? I'm hoping you'll confirm your identity and claim a prize!)
    Prizes were to be given out at our ceremony earlier today (which also saw the induction of three Emmanuel students into the Pi Mu Epsilon honor society) but not everyone was present. So, if you are in the top 5 listed above and have not yet claimed your prize, please find me in my office (Library G08-C) to pick up a fun math book from the list below. (These were all donated by Emmanuel Math Dept. faculty, so they are highly recommended, of course!)
    I found it interesting that each of these books also has a substitle. So, to conclude the year with a fun "puzzle", try to match the book titles above with their subtitles below: 
    • Making Light of Weighty Ideas
    • How You're Being Fooled By The Numbers
    • Symmetry and Asymmetry from Mirror Reflections to Superstrings
    • The Story of Nicholas Bourbaki, the Genius Mathematician Who Never Existed

    Friday, April 15, 2016

    Spring 2016 POTW #5: Check (Your Work), Please!

    Solutions due by Sunday, April 24, 2016.

    This is the final POTW of the year! We will be awarding prizes for yearlong standings at an Emmanuel Math Department event at 4:30 pm on Thursday, April 28, 2016 (which is also Senior Distinction Day). At the same time, we will be inducting members of the graduating class into the mathematics honor society Pi Mu Epsilon. Emmanuel folks, please join us in celebrating the outstanding achievements of our seniors and you, the puzzlers!


    The 6 members of the Emmanuel Math Department recently went out to a celebratory dinner. When it came time to pay the bill, they discovered a few curiosities:

    • Everyone except Brendan owed exactly the same amount for their meal.
    • Brendan was an extravagant spender: His meal cost $10 more than twice the average of everyone's cost.
    • Meanwhile, each other person's meal cost $10 more than half the average of everyone's cost.
    What was the total bill for everyone, and how much did Brendan's meal cost?


    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.

    Monday, April 4, 2016

    Spring 2016 POTW #4: Bow Do You Do?

    Submissions due by midnight on Sunday, April 10, 2016.

    At a recent college meeting, there were several faculty members, several administrators, as well as the President, Sister Janet. There were twice as many faculty members as there were administrators.

    In addition, this was a rather formal academic meeting, so there were many bows that occurred: Every administrator made a bow to every other administrator, to each faculty member, and to Sr. Janet; also, every faculty member made a bow to every other faculty member, to each administrator, and to Sr. Janet. Overall, exactly 900 bows took place.

    How many people were in attendance at this formal academic meeting?

    (Use the submission box below to submit your answer. No need to explain: a correct answer will suffice for 10 points. However, feel free to explain your answer if you want a chance at some partial credit, in the event that your answer is incorrect.)


    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.

    Monday, March 28, 2016

    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!)

    Sunday, March 27, 2016

    Spring 2016 POTW #3: Stop, Clock, and Roll

    Submissions due by midnight on Sunday, April 3, 2016.

    On this New Year's Day (January 1, 2016), at precisely 12:00 noon, I set up three clocks in my office. Let's call them Clock A, Clock B, and Clock C.

    The next day, at precisely noon, I checked on them and discovered that Clock A had gained exactly 1 minute, Clock B had lost exactly 1 minute, and Clock C was right on time.

    Assuming that the same rate of gaining/losing time continues on these clocks, When is the next date on which all three clocks will show precisely noon at the same moment? (That is, your answer should be a specific date, like "May 6, 2025" or something.)

    (Use the submission box below to submit your answer. No need to explain: a correct answer will suffice for 10 points. However, feel free to explain your answer if you want a chance at some partial credit, in the event that your answer is incorrect.)