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Tina Fanetti wrote:
Unfortunately, most of my students can't solve problems involving more than one step.
Anything that requires assumptions or manipulating equations they have trouble with.
Do you have any concrete suggestions?
A few more, see below.
I read that discussion you sent me..
I assume that refers to the milk-jug puzzle.
==========
Here's one that Arno Penzias used to spring on unsuspecting
researchers when he was in charge of the Research area at
Bell Labs:
"Tell me roughly how much water flows down
the Mississippi river at, say, New Orleans"
There is a strong temptation to say "I've go no idea" but
Arno wouldn't accept that. He would scream "YOU MUST HAVE
!!SOME!! IDEA -- FIGURE IT OUT". And in fact there are at
least two distinct methods for getting a rough, order-of-
magnitude estimate, just by combining bits of information
that most people carry around in their heads.
===========
Almost any back-of-chapter problem can be turned into
a word problem that requires an extra step or two of
thought. For example, rather than asking for the
terminal velocity of an object of radius R and mass M,
you could ask for the terminal velocity of a coconut.
Or a cantaloupe. Or a golf ball. Or a bowling ball.
This accomplishes a couple of things:
1) when the problem is stated in terms of radius R
and mass M, it more-or-less implies that those
variables (and no others) are relevant. But with a
coconut, one needs to _decide_ that its size and mass
are the relevant variables.
2) One then must perform the step of finding out
the size and mass of a typical coconut. This isn't
a hard step, but it's a step.
===========
Here's one my father sprang on me when I was about ten.
I had no idea how to solve it. I worked on it for days
before he coughed up the crucial hint (which is tantamount
to giving away the answer).
You are given a supply of 31 ordinary dominos. You are also
given a checkerboard with the ordinary arrangement of colored
squares, of a size-scale such that each domino just covers
two squares. Can you arrange the dominos so that all squares
are covered except for the two squares at the opposite ends
of the main NW/SE diagonal?
This seems like a pure math/logic puzzle, but there's something
akin to a physics principle hiding in there. Hint, hint.
===========
Sam Loyd invented thousands of logic puzzles, including
the now-ubiquitous 15-puzzle
http://www.cut-the-knot.com/pythagoras/fifteen.shtml
See also
http://thinks.com/puzzles/loyd/loyd.htm
============
There's also "Hermione's Potion Puzzle" (page 285 of Harry
Potter volume 1). This can be solved using less information
than Hermione herself used (i.e. without seeing the sizes of
the various bottles).
============
I am particularly fond of the Twelve Coins puzzle.
I get 300 hits from
http://www.google.com/search?q=twelve-coins
Main assignment: Solve the puzzle according to
the usual rules.
Extra credit: Design a _parallel algorithm_ for
solving the puzzle, i.e. you decide in advance
what combinations of coins will be weighed, and
then all the weighings are carried out. That is,
you don't get to use the results of earlier
weighings to decide which coins participate in
the later weighings.
There's physics hiding in here, too. Entropy
and information.