Andy's Math/CS page

Friday, October 05, 2007

Another Heads-Up for Puzzlers

Two puzzle collections (both drawing on diverse sources) caught my eye recently. The first is 'The Art of Mathematics: Coffee Time in Memphis', by the famous mathematician Bela Bollobas. It's got a number of gems, for instance: is it possible to 'load' two dice, i.e. skew their faces' probabilities, possibly in distinct ways, such that when both are rolled the resulting distribution is uniform on {2, 3, ... 11, 12}?
The book is definitely worth looking at, but, perhaps as a function of its ambition to teach some important (and cool) math along the way, it is rather difficult on the whole, and ridiculously so in some cases (one of his 'puzzles' is to disprove the Borsuk conjecture, a problem that had been open for something like 60 years. Granted, there is a hints section, but come on...). The Amazon page lets you browse a number of the more approachable puzzles, so check it out.

The second is 'Mathematical Mind-Benders', Peter Winkler's much-anticipated sequel to the instant classic 'Mathematical Puzzles: A Connoisseur's Collection'. I haven't spent enough time on the new one to see how it stacks up to its predecessor, but it definitely contains some beguiling problems. Many are 'old friends' of the puzzle genre (bugs, prisoners, hats, etc.), but remade and fiendishly harder--think of the fight with Super Shredder at the end of 'Turtles II'...

However, while Winkler's books are also quite hard, they generally involve much less 'higher math' than Bollobas', which is really written for readers interested in mathematics proper. Another emphasis of Winkler's new book is on surprise answers, as in the following two puzzles from the first, warm-up chapter:

-A pencil has a regular pentagon as its cross-section, and its maker's logo on one face. If the pencil is rolled on a table, what is the probability that it comes to rest with the logo facing up?

-You have 15 bags. How many marbles do you need to produce an arrangement where each bag contains a different number of marbles?

Winkler also unveils a neat word puzzle called 'the HIPE game', something he co-invented as a teenager. One player comes up with a string of letters and challenges the other players to find a word containing it (as a block of adjacent letters): for instance, can you place HIPE in a word? Of course, the problem poser should actually have a word in mind...

Some other challenges he gives: BG, CM, FC, FW... and the book has many more (and harder). I'm normally pretty decent at word games, but this one is killing me so far.

Thanks, Bollobas and Winkler!

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14 Comments:

  • -A pencil has a regular pentagon as its cross-section, and its maker's logo on one face. If the pencil is rolled on a table, what is the probability that it comes to rest with the logo facing up?

    If you mean directly it's obviously 0; if you mean "visible from above" it's 2/5. What's the tricky part?

    -You have 15 bags. How many marbles do you need to produce an arrangement where each bag contains a different number of marbles?

    What's the catch?

    By Anonymous Anonymous, at 4:59 PM  

  • Anon, I think many people would say it's "obviously" 1/5... I almost did.

    As for "what's the catch", well, what's your answer?

    By Blogger Andy D, at 6:57 PM  

  • The pencil problem doesn't bother me, I can see how it would have a counterintuitive answer. But what is the answer to the bag question?! Is it not 105 (= 0+ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14)?

    By Blogger George, at 3:08 AM  

  • presumably some players will forget
    that zero is a number (& get 120).

    "hipe" is great; thanks for bringing
    it to my attention.
    i've linked to this post in VME.

    By Blogger vlorbik, at 3:08 PM  

  • Thanks, Vlorbik!

    What you say is true, but the treachery of the marbles puzzle cuts deeper... Don't let me spoil it for you (& George).

    By Anonymous Andy, at 4:20 PM  

  • Please spoil it for me! I just have no clue why what I said isn't the answer. Email me? Typical swat algorithm for email addresses. My last name is Dahl.

    By Blogger George, at 8:06 PM  

  • oh, now I see. it's 14 marbles.

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  • ah-ha. cute.
    14 it is, then.
    v.

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  • Magnific!

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  • This is perfect to play with numbers because it makes us think very well what we are gonna do, and increases our logic.

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