Tuesday, August 23, 2016

Outside the Box: From Sudoku to recreational mathematics

By Moristotle

Hardly anything calms me more than sitting down alone in a quiet place to solve a fresh Sudoku puzzle. At least, it calms me when I don’t screw up the puzzle. Let me recast the opening statement: Hardly anything calms me more than sitting down alone in a quiet place and flawlessly solving a Sudoku puzzle.
    The calming effect puts me in mind of that scene in Claire Danes’s 2010 biopic film, Temple Grandin, in which she portrays the calming effect on [the autistic] Ms. Grandin of confining herself in a “hug box” when she is feeling anxious. Sitting down to a fresh Sudoku puzzle is for me perhaps like confining myself in Temple Grandin’s hug box.
    I thought of that scene recently upon reading the NY Times article “The Importance of Recreational Math” (Manil Suri, October 12, 2015​). Although the author acknowledges that “‘recreational math’...can broadly include such immensely popular puzzles as Sudoku,” he seems inclined to exclude Sudoku because it “always has the same format and gets easier with practice.” He favors “disparate puzzles that...require different, inventive techniques to crack [emphasis mine]. The solution in such puzzles usually pops up in its entirety, through a flash of insight, rather than emerging steadily via step-by-step deduction as in Sudoku.”
    And he provides an example, which I in turn provide to you: “How can you identify a single counterfeit penny, slightly lighter than the rest, from a group of nine, in only two weighings?”
    ...
    Not got it yet?
    ...
    Ah, that’s more like it! Nice! In its own way perhaps as calming as flawlessly solving a Sudoku?


The NY Times article extols Martin Gardner’s “Mathematical Games” column, “which ran in Scientific American for more than 25 years, introduc[ing] millions worldwide to the joys of recreational mathematics.”
Mr. Gardner’s great genius lay in using such basic puzzles to lure readers into extensions requiring pattern recognition and generalization, where they were doing real math. For instance, once you solve the nine coin puzzle above, you should be able to figure it out for 27 coins, or 81, or any power of three, in fact. This is how math works, how recreational questions can quickly lead to research problems and striking, unexpected discoveries.
“Unexpected discoveries” – that seems to be the rub. And the idea behind this new recurring column, “Outside the Box,” is to highlight areas in life where we may be overly inclined to restrict ourselves to one box or another precisely for the sake of rendering the world “expected” to ourselves, to protect us from being (unpleasantly) surprised or disturbed by ideas that we aren’t comfortable with. Such boxes abound, in the forms of
  • habitual thinking patterns, modes of communication, transportation, economy, entertainment, diet....
  • automatically following the ways or traditions of one’s own family, tribe, peers, neighbors (such as the Joneses), political affiliations, religious affiliations, city (like Pamplona, Spain’s barbaric “running of the bulls”)....
  • adopting the opinions of experts who are widely regarded as authorities, like forensic scientists, medical doctors, financial advisers....
  • “impersonal” psychological factors that have some influence on almost everyone, like confirmation bias, wishful thinking, the tendency to skew perceptions to minimize cognitive dissonance....
    Those boxes aren’t necessarily comforting, or comfortable to be in - or the results of acting from within them may not be comfortable. (Or it might not be comfortable being prevented from acting because one is in them.) Wishful thinking, for example, can land you in various fantasy lands. And the following assortment of boxes can be quite uncomfortable, for both oneself and others:
  • personal psychological factors, like inflated or deflated self-image, addictions, procrastination, “writer’s block,” neuroses, even psychoses....
We invite you to consider that our first offering of “Outside the Box” might be to help you discover that you can put a little excitement in your life simply by trying a new kind of puzzle.
    In today’s googly world, you don’t even need to buy anything. Just google on “math puzzles.”
    (And you can go outside that box and think of other phrases to google on. There are lots of places outside boxes!)

[Note: With graditude, I acknowledge Jim Rix’s suggestion of the name for this column, which grew out of our discussions of the possibility of a column that would be science-oriented. And thanks to Jim, and also to Vic Midyett, for their contributions to the bulleted list of sorts of boxes that we humans are prone to.]


Copyright © 2016 by Moristotle

14 comments:

  1. Belatedly, I have added a note at the bottom of today’s post:

    With graditude, I acknowledge Jim Rix’s suggestion of the name for this column, which grew out of our discussions of the possibility of a column that would be science-oriented. And thanks to Jim, and also Vic Midyett, for their contributions to the bulleted list of sorts of boxes that we humans are prone to.

    ReplyDelete
  2. I truly enjoyed reading this. It went well with my coffee.

    ReplyDelete
    Replies
    1. But a good cup of coffee tends to go well with most anything! Do you still drink Costa Rican - even there in Mississippi?

      Delete
  3. Nice puzzle. So how many weighings would it take if all you is that the counterfeit coin is of a different weight (lighter of heavier) than a real coin?

    ReplyDelete
    Replies
    1. Ooh, nice one, Jim! And just the sort of devilish question I would expect from someone as free of boxed-in thought as you are, and ever have been, since I first met you playing paper-scissors-rock with Don Richert on a school bus in, I guess, 1955.
          I won’t divulge the answer to your question, if you won’t. <smile>

      Delete
    2. Evil, Jim. I've got an answer, but not proof.

      I've always been a puzzler (and most of my family. Genetic?) Some of Gardner's games have become long-running fads among us. Rubik's Cube, fractals, The Game of Life....

      Solving things like this is a great ego trip. These days, though, I usually don't want to work that hard. Gimme my Suduko.

      Delete
    3. Nice comment, Chuck. Maybe the effort involved explains an unexpected discovery I made at the gym this afternoon. Unlike the previous time I did 45 minutes on the ellyptical - when I watched CNN headlines & quotes, and the 45 minutes dragged - today the time passed quickly: I was trying to solve Jim Rix's variation of the 9 coins problem. Outside the box of boring 45-minute sessions on the ellyptical!

      Delete
    4. Your elliptical experience is odd - I find that I can't read, or think about anything complicated, while exercising heavily. Wish I could find something to do more interesting than watching video. Perhaps then I'd go to the gym occasionally.

      Delete
  4. Another good article! I am in awe of persons who make up the puzzles, mathematical and word. My PTSD has robbed me of my mathematical skills. I envy those who are able to master these puzzles. Again, great article!

    ReplyDelete
    Replies
    1. Thank you, Sharon! Jim, Vic, & I do feel we have a winner of an idea for a column here. We are hoping for some good submissions from staff members, and from followers of Moristotle & Co., like you. Of course, you are a "character" of Moristotle & Co.!

      Delete
  5. Chuck, did you come up with no more than 4 weighings necessary? With a possibility of just 2?

    ReplyDelete
    Replies
    1. I was trying to avoid spoilers. I did come up with a maximum of 4, but thought I might have missed a cleverer approach.
      I did come up with a general method that works for any number of coins N. 2(N/2)+1 for an even number, 2(N/2)+2 for an odd number... but that is 5 moves for 9 coins.

      Delete
    2. Chuck, as I initially reported to Jim (privately), I thought I had solved it in THREE weighings (and THAT was really what I was doing on the ellyptical yesterday - trying to replicate it), but so far I have not succeeded in replicating it. I even thought I had discovered an "outside the box" technique: using imported, authentic coins to "weigh against" some optimal subset of the nine coins in the problem. An unequal balancing - you KNOW on which side of the scale the imported coins are - would tell you immediately whether the counterfeit coin is light or heavy....

      Delete
  6. I still want to know if Ed is drinking Costa Recan coffee! Good stuff, Mo!

    ReplyDelete