Thursday, January 13, 2011

Are you a Ringer? A Reverse Ringer?

A ringer is an impostor, especially one whose pretense is intended to gain an advantage in a competition. This definition is from Wikipedia and agrees with what I thought the term meant. (Wiktionary has a similar definition. Spellcheck insists that Wiktionary should be either Dictionary or Visionary.)

The following problem appeared in The Bent Winter 2010 issue. (The Bent is a publication of Tau Beta Pi, and Engineering Honor Society.)
Al's job is testing bowling balls. He has two identical bowling balls and is to test their impact resistance by dropping them out of windows on various floors of a 100-story building. He is to determine from which exact floor a dropped ball will shatter on impact with the pavement below. Al knows nothing about the strength of the balls. They may shatter when dropped from the first floor or not until dropped from the 100th floor. What is the minimum number of ball drops needed to guarantee that Al can uniquely determine the floor fro which the balls will shatter. Balls that do not shatter may be dropped again. Both balls may be destroyed during the test. Include a brief outline of how testing is done.
This problem raises questions and metaquestions.
  1. What is the answer?
  2. A while back I had an undergrad work on the general problem of f floors and e eggs (we used eggs not bowling balls). Hence I know the answer with matching upper and lower bounds for all f and e. Should I submit my answer? If I did would I be a ringer? All you get for submitting a correct solution is your name in the next issue so that would be okay(?). Even so, its seems like cheating. (See here for my students paper. We didn't publish it since a similar paper had already appeared: The Egg Drop Number by Michael Boardman, Mathematics Magazine, Vol 77, No. 5, Ded 2004, 368-372. You can find it here.)
  3. When is one a ringer? In a later issue there was a problem I had not seen before but was able to solve easily with graph theory. For that problem am I a ringer?
  4. The intent of the problems (I think) is to test your cleverness not your repository of knowledge. With this in mind, clearly if I send in a solution to the Bowling Ball problem, I am being a ringer. For the graph theory problem it is less clear.
  5. Once in a restaurant I was doing the kids math puzzles on the paper placemats with my great nieces and nephews. There was one problem that I could solve by brute force but tried instead to find a clever solution (there probably wasn't one). Hence I could not solve it. Or at least that is what my great nieces and nephews think. This might be called being a reverse-ringer. Is there a better term?
  6. Here is a problem from Activity book from the NSA on Codes, Ciphers, Puzzles) that is geared towards kids. I was able to do every puzzle in it very quickly except this one. If you know the answer please tell me before some kid asks me:
    Logic Puzzle Number 1: If a railroad train is moving northward, there is a part of each car on the train that is moving southward at each instant, no matter how fast the train is going. what is the contrary piece moving southward? Hint- sketch a train moving on a track and examine the parts you sketch.
What has been your experience with solving (or not) problems that are, in some sense, below your ability and knowledge level?


  1. The lowest part of the wheels are lower than the top of the track.

  2. A similar phenomenon occurs in any one-on-one competition (fencing and chess are the two I'm familiar with). A strong, but not expert, player will often have difficulty beating a mediocre opponent since they don't react "correctly".

    Also, I believe the answer to the riddle is "the bottom of the wheel".

  3. I'm guessing the answer to the puzzle is (the bottom part of) the wheels on the train. This is an incorrect answer, of course, but it might seem correct if you didn't work things out all the way... the puzzle is probably just buggy.

  4. The bottom edge of the wheel?

  5. Hint: The train problem assumes that the train is not applying its emergency brakes.

  6. The bottom of the wheels.

  7. Inre. the train problem: I find this troubling. The bottom half of each wheel isn't moving backward; in fact the closest it comes is that for any fixed point A on the rim of the wheel, at the instant point A touches the track, its forward velocity is zero. For the rest of the time that the wheel is moving, point A is in fact moving forward with a nonzero velocity. This is an easy thing to check: take a disc and mark a point on it, then roll it along a surface and track the motion of the point. It never actually goes backward.

    Now, perhaps, given some kind of friction holding the train back from being propelled forward at exactly the rate of speed of the wheels, you might be able to argue that for a nanosecond the bottom point is actually going backward, but I don't imagine that's what they intended.

    Since I know plenty of smart people that work at the NSA, I'd like to think that they had something else in mind. I was thinking about the rods that propel the wheel, but even if the rods are moving faster than the wheel and manage to go backward, they only go backward half of the time and thus don't satisfy the hypotheses of the problem.

    Now I want to get to the bottom of this...

  8. How about "stooge" or "patsy" for reverse ringer?

  9. for any fixed point A on the rim of the wheel, at the instant point A touches the track, its forward velocity is zero.

    As Arvind observed, the lowest parts of the wheel are lower than the place where the wheels touch the track. Wheels on a train are shaped like a slot into which the tracks fit -- that's how the trains stay on the track.

  10. @joxn:

    Just as a side remark, the guards on the side of the wheels are *not* the primary reason trains stay on track. Those are there just for additional safety.

    The real reason is that the rim of a train wheel is sloped: the wheel's inner diameter is slightly more than the outer diameter. Also, the distance between two wheels opposite each other is kept fixed. So, if one of the wheels shifts to one side, the sloping on the other wheel shifts the train back to the center. For the same reason, the train can turn around a bend, where one wheel must cover more distance than its opposite, even though the distance between the wheels is fixed.

    See the discussion here:

  11. What bothers me a bit is that this was called a "Logic Puzzle." To solve it, you really need to know something about train wheels, i.e., that they have ridges that extend beyond (and below) the point of contact with the track. What does this have to do with logic?

  12. Steve, there is often domain knowledge required in a puzzle problem. That knowledge here is minimal.

  13. 20? it's the best solution I can think of..

  14. There are gears in the gearbox that drive the wheels that are spinning faster wheels are. Depending on where they are relative physically to the final gear, either the top or bottom is moving towards the rear of the train faster than the train is moving forward. That is the wayward component of the train.

  15. Arvind Narayanan's answer lead-off answer is "correct" in the sense that it by far the simplest and most geometrically natural answer given, and thus is almost certainly the answer intended by the question-posers.

    It is necessary to appreciate that the question-askers had in mind, not an idealized mathematical train, but a physical train. The wheels of physical trains all look pretty much alike ... they are as tightly constrained in their design as the wings of jet aircraft ... and the flange to which Arvind's answer refers is a vital part of their design.

    Deep mathematical issues associated to non-holonomic constraint forces arise, for which a satisfactory geometric understanding is still lacking; see for example Anderson, Elkins, and Brickle, Rail Vehicle Dynamics for the 21st Century.

    Today, even simpler dynamical systems like rattlebacks and Chaplygin Sleighs evade our full dynamic, thermodynamic, informatic, and geometrical understanding ... if you are thinking "how hard can it be?" ... well ... just try to quantize their dynamics ... or predict their thermodynamical properties in ensemble! :)

    Thus, students who believe that quantum mechanics is more subtle than classical mechanics are deplorably misguided, not about the subtleties of quantum dynamics, but about the subtleties of classical dynamics. As Saunders Mac Lane says in his Mathematics, Form and Function:
    It has taken me over fifty years to understand the derivation of Hamilton's equations ... the point of this cautionary tale is the difficulty in getting to the bottom of it all.

  16. ruiaf: You can do better than 20.
    Much better (for the bowling ball problem). Its a nice problem- think about it some more.

    others: I agree that the Train problem is not a logic puzzle. I also do not think it is a nice problem, especially for a children.
    (Your comments on it help confirm this point of view since nobody said

  17. Bill, it was probably a "nicer" puzzle when it was thought up and children were more familiar with trains having seen them more often and more obviously.

    For the bowling ball, the answer would seem to be 19, which isn't that much better than 20, in the worst case.

  18. AH- I checked and the puzzle was reprinted from a book from 1979.
    Did kids know more about trains then?

    You can do much better than 19.

  19. Perhaps times have changed ... I seem to remember this same railroad wheel puzzle being posed in Boy's Life in the early 1960s, but cannot locate the precise episode.

    There is a terrific wheel-related Donald Duck comic episode, illustrated by Carl Barks, titled "The Runaway Train", in which Donald's nephews Huey, Dewey, and Louie use their Junior Woodchucks training to predict, solely from considerations of wheel and track geometry, the precise point at which a runaway freight train is destined to collide with a passenger train (and all the details of the calculation are given in the comic book).

    Their calculations prove correct, and by switching one of the trains to a side-track, a terrible disaster is averted. The ducks are heroes! :)

    This wonderful illustrated story made a deeply favorable impression upon my young mathematical imagination ... and I see it has been reprinted in at least ten languages. :)

  20. The number of lines in a sonnet?

  21. 1) John S and others- YES, seems likely that in an earlier time kids were more interested in mechanical things (like trains) then now. Stories of kids taking apart their parents Vaccum cleaners for example.
    Kids no longer seem to do this. What are they interested in now? Are we better of worse because of it?
    (I do not know.)

    2) YES the answer to the bowling ball puzzle is the same as the number of lines in a Sonnet.

  22. GASARCH, you ask a very difficult question—in essence "What (if anything) is different about kids today, and about what they are learning?"—for which diverse good answers exist.

    For me, there is one crucial difference that is only peripherally related to physical-versus-abstract problem-solving, but rather, has mainly to do with rank-driven versus narrative-driven educational environments.

    In rank-driven education, the objective is to achieve a high-rank score in testing, then be admitted to a high-rank school, then earn a high-rank degree, then be hired by a high-rank corporation.

    Narrative-driven educational environments focus upon a very different class of objectives. As Garrison Keillor says in the introduction to his anthology Good Poems:

    What makes a poem memorable is its narrative line. A story is easier to remember than a puzzle.

    Carl Barks' story The Runaway Train is (it seems to me) an outstanding example of narrative-driven mathematical education.

    Of course, narrative-driven mathematical expositions are not solely for children ... researchers like Saunders Mac Lane and Bill Thurston have used narrative techniques at the highest levels of mathematics.

    Famously, Richard Feynman was yet another narrative-driven researcher, and so perhaps this is a good opportunity to remind people of today's TEDxCalTech Event, Feynman's Vision: the Next 50 Years.

    If we are lucky, at least some of the TEDxCalTech speakers will echo the memorable words of Steve Martin's character Navin R. Johnson:

    Waiter, take away these old narratives ... bring us some *fresh* narratives! The freshest you've got. This year!

    Because heck ... don't pretty much *all* of Feynman's most famous lectures lay out fresh narratives?

    Moreover, we should all admire the courage of the TEDxCalTech speakers for even taking the stage at this event ... because they're going to have to share that stage with none other than Kongar-Ol Ondar:

    Kongar-Ol Ondar (Tuvan: Коңар-өл Ондар) is a master Tuvan throat singer and a member of the Great Khural of Tuva.

    As Feynman might have said ... it's gonna be terrific! :)

  23. The lowest part of the wheels are lower than the top of the track.

  24. I have the book the train problem came from which contains the intended answer.
    It is

    ``The contrary piece on each car is the flange of the wheel. The flange lies below the rail and moves south as the wheel turns.''

  25. I've heard two more formulations of the bowling balls problem. The one is with magician crystal balls and the other with Galileo and the Pisa tower. I prefer the Galileo one.

    Anyway, in its essense, this is a simple search problem. By using binary search, you use only logN balls on the worst case , which for 100 is 7 balls.

  26. Chazisop- Binary search does not work since if the Bowling ball
    (or egg or glass crystal...)
    shatters then you cannot resuse it.

    Say your first move is to drop the ball off of the 50th floor (which I
    assume you would do with binary search). The ball SHATTERS. All you know now is that the desired floor is
    one of 1,2,3,...,49. You only have one ball left so you HAVE TO drop it off floor 1, then2, then 3.,,,
    until it shatters. Worse case: 50 drops. (I may be off by one here.)

    Think some more- its a nice problem that I am sure you can do.

  27. Anony just above Chazisop- do you understand that solution? I don't
    Whats a flange? Could a kid really get this now? How about back in 1979?
    (These last two questions ARE rhetorical- I think the answer is NO.)

    Thanks for telling us the answer!

  28. GASARCH, perhaps you and Lance should ask your kids "What it is it that keeps a railroad car from leaving the tracks?"

    If they answer "Oh, the train probably has computers to prevent *THAT*!" ... then oh well ... maybe it means the US is destined to import a lot of trains from China. :)

  29. john, i really question your comments. I am not aware of anyone else who comments so much. You probably know, quality prevails, it's not quantity in such circumstances.

    regarding ur remark that we will need to import trains from china. well, the chinese have done remarkable well in importing all kinds of different technology related to high speed trains. eventually, they reverse engineered everything and now start producing items domestically.

    They have shown once again the lack of commitmment to copyright and their total disprect to intellectual property.

    This is something that will backfire on us, if we continue to tolerate their political trade agenda and their stealthy and dirty way of lurking technology companies into mainland china so that tthese companies have to partner up with domestic chinese ones which in turn will have 100% access to trade secrets.

    what we see is the analog of brain drain for leading u.s. companies. in the short run, setting up these companies in mainland china might seem like a profitable venture (particularly for share holders), but surely nobody seems to be remotely interested about the consequences ...

  30. GASARCH asks: What has been your experience with solving (or not) problems that are, in some sense, below your ability and knowledge level?

    As an exercise, and as a counterpoint to anonymous' post (immediately above)—a post whose rhetorical excesses to me made no sense—by writing a explicitly constructive response to GASARCH's question.

    It very commonly happens that I encounter natural mathematical questions that seemingly are "below my ability and knowledge level" ... and yet to my dismay, I cannot answer them.

    Whenever this happens, "the danger signal is up"—to borrow a phrase from von Neumann's essay The Mathematician (1947)—and it is natural to ask myself, "Hmmmm ... so maybe this question is in fact above my ability and knowledge level?"

    Moreover, when this question arises naturally in the context of daily activities, then I have ask myself whether perhaps the problem is that the style of my mathematical understanding is not "classical", but rather is excessively "baroque and even very high baroque" ... again to use von Neumann's descriptive language.

    The question "What keeps a railroad train on its tracks?" is a good example of a mathematical question, arising naturally in daily life, that we tend to think is below our ability to answer ... when actually it is above our ability to answer.

    I maintain a BibTeX database of STEM roadmaps, and it is these troublesome "below our ability" mathematical questions that often have been the primary obstruction to creating viable STEM enterprises.

    So for me, the answer to GASARCH's question is something like: "The great virtue of simple mathematical questions, arising naturally in daily life, that we can't answer, is that these questions induce in us a feeling of humility that is healthy, by reminding us that a broadly integrated mathematical understanding is a classical virtue."

    Our planet has seven billion people on it—which is a lot—and yet there is *NO* risk of any shortage of these humility-inducing questions ... and it is very fortunate that these questions, and the global-scale enterprises that are naturally associated to them, are plenty enough to keep a planet-full of people busy.

  31. Regarding the bowling ball puzzle.
    You have already said that the answer is 14 (lines in a Sonnet), without mentioning how it works.
    Correct me if I am wrong. (I am not even sure you do mean the 14-line sonnet.)
    First throw one ball from floor 14.If it breaks start from the first floor and throw the ball from every floor going up until it breaks.You have 13 throws left which are enough. If the ball does not break from floor 14, then go to floor 14+13=27 and try. If it breaks here you have 12 throws left to test floors 15 until 26.Otherwise go to floor 14+13+12=39 and continue similarly.If you reach floor 99 and the ball does not break, you know the answer is the 100th floor having only thrown the ball 11 times.

  32. John Sidles, dont u have to do anything better ? it is nice to see you spend so m

  33. 14 for the bowling balls problem. Can we prove that one cannot do better than 14 ?

  34. YES, one can show that 14 is optimal.
    Exact upper and lower bounds are known for f floors, b bowling balls.
    The papers I pointed to in the post tell you how to do this, but its more fun to do it yourself.

  35. Prof. Bible Freak,

    How do you get 14? Is that (say) the mathematical expectation of the number of attempts needed to surely know the correct floor, assuming a discrete uniform distribution across floors?

    I get 19, like some others here.

    I have the honor to be, with the highest consideration, Your Excellency's most humble and most obedient servant.

    - The low-IQ atheist man-pig.

    PS: Chazisop, you have, perhaps, a lower IQ than even me, and I'm just a low-IQ man-pig.

  36. Last Anonymous- the Anonymous a few back had it right- first drop off floor 14,
    then floor 14+13, etc...

    You can also see either of the two papers that this post pointed to.

  37. It's only anecdotal, but I have 2 nephews that are both (independently) fascinated by trains, and could tell you quite a bit about them.

  38. Does light from the caboose count?