Puzzle Break

24 August 2009

I’m on vacation with my family at Disneyworld all week this week, so we’ll do things a little differently. For those of you who aren’t spending the week at theme parks, here is another kind of recreation: two sites full of puzzles of a variety of types.

Everything you’ll find at both sites is rich in mathematical content (but feel free not to notice).

Logic Mazes, a site belonging to Richard Robert Abbott (inventor of Eleusis and other card games). What he calls logic mazes are sometimes called “mazes with rules”. For example, in the Alice mazes, you navigate a grid by following arrows, but the size of your steps changes as you pass through certain squares. So you might find it easy to get to point A with step size 2 but have to work quite hard to get the same point with step size 3. So even small-appearing mazesa can conceal vast maze structure. Many of the puzzles are playable online.

Puzzle Beast is a collection of Puzzles like no other. Perhaps most reamrkable is that the individual puzzles are generated by computer (the mathematics of how this is done is a story in itself!). If you want a suggested starting place, The Fried Okra Perplexity is good, as is the Dry Cleaning. All are original, all puzzle types include problems that will entertain and challenge any solver.

Oh, sure, there are a lot of good math topics that relate directly and indirectly to the puzzles on these sites, but we can talk about that later. Let’s not let my explanations get in the way of your own discovery. Just go play.

FARKLE, Expectation, and Knowing What You Want

22 August 2009

I’ve been asked in- and off-blog to compute the probability of getting three-of-a-kind or better when throwing 3, 4, 5, or 6 dice.  (That is, I only care about scoring combinations other than singleton 1s and 5s.)  This is straightforward, so I just give the answers for anyone interested (after the jump).

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Thought Experiment: the Rational Plane

17 August 2009

By convention, when we refer to the number line, the “numbers” in question are the real numbers \mathbb{R}.  This is very different from the number line that most of us learned in kindergarten, which consisted of the counting numbers together with 0 and their negatives.  There is a drastic qualitative difference between these two versions of the number line — one is continuous and one is discrete.  There are measurable and noticeable gaps between the integers.  It’s the difference between a ribbon of highway and the  mile markers along the way.

What about the intermediate case of the rational numbers, the numbers that can be written as fractions?  Are they more like the discrete number line or the continuous number line?

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The Committee, the Errand, and the Dozen Doughnuts

13 August 2009

A recurring theme in mathematics, or at least a recurring object of my own interest in mathematics, is the way that the same core idea or structure shows up in different contexts, often unexpectedly.

Consider the following three problems.

  1. A certain club has seventeen members, and they wish to select a five-person executive committee.  How many possible committees are there?
  2. You are running an errand which requires you to ride your bike from your office, which is at a major intersection, to the Important Building, which is twelve blocks north and five blocks west.  You ride alongside the roads, and you only turn at major intersections (between blocks); because you don’t want to waste time and energy, you are always moving either east or north (never south or west).  How many different routes are possible?
  3. You are at your favorite doughnut shop, where they sell six types of doughnuts that you like.  (Insert your flavorites here.)  How many different ways are there for you to buy a dozen doughnuts you like?

These are all combinatorial problems; more specifically, they are problems of enumeration, “how many?” questions.  That much is evident; what is much less evident is that all three problems have the same answer!  And it is more than a coincidence.

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Projective Lines and Planes

10 August 2009

Today I’d like to take a taste of projective geometry, both because it gives a pleasant twist on a subject on geometry, and also to give an example of a mathematical construction.

Projective Line (Def. 1) Consider all the ordered pairs (x:y) where x,y are real numbers, not both zero.  (Note that this is different from how we make fractions … with fractions the denominator is never allowed to be zero, but the numerator can be; here either can be zero as long as they are not both zero.)  We consider two pairs (x_1:y_1) and (x_2:y_2) to be the same if there is a number $\lambda$ so that x_1=\lambda x_2 and y_1=\lambda y_2.  In other words, we think of (3:4) and (6:8) as the same object.  Likewise (1:0) and (\sqrt{3}:0) are the same object.  So our objects are ordered pairs modulo rescaling.  Even though they look like ordered pairs and make you think of points in a plane, these really should be thought of as numbers, or points in a one-dimensional thing.

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Three Book Recommendations

6 August 2009

I’ve been thinking a lot lately about books.  About all the books I’ve read, all the books I want to read. And the ever-lengthening list of books that have been recommended to me.  But even though I suspect I will never get through all the books on this lift, I am always grateful for a well-thought-out recommendation.

In that spirit, today I want to recommend three books to anyone who has enjoyed any post on this site.  Each of them is full of information about the history of mathematics, brain-strengthening math content, and most importantly to us deep insights into the Zen of mathematics.  I assure you that none of these books is about apples.

Today’s Recommendations (alph. by author)

  • Prime Obsession, by John Derbyshire
  • Journey Through Genius: the great theorems of mathematics, by William Dunham
  • An Imaginary Tale: The Story of \sqrt{-1}, by Paul Nahin.

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The Two Decks of Cards and the Drunken Hat Check Girl

3 August 2009
  1. Imagine you have two decks of playing cards, each thoroughly shuffled.  You give one deck to your friend and keep the other.  Now each of you goes through your deck, one card at a time, flipping cards face up.  You compare your top cards, then you compare your next cards, and so on all through the deck.  If it ever happens that you both reveal the same card at the same time, you win; if you go through the whole deck without such a match, your friend wins.  Who is more likely to win?  You or your friend?
  2. Thirty-seven men attend a certain social event and check their hats as they enter.  However, the hat check girl has had a bit too much to drink, and when the time comes to leave, she gives back hats at random, with total disregard for which hat belongs to whom.  What are the chances that nobody ends up with his own hat?

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