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Who Finished In What Place?

Date: 12/15/2002 at 20:12:33
From: Nicole Hoover
Subject: Using problem-solving strategies

Sarah, Carla, and Kim were the finalists at a gymnastics meet.
Sarah was not second, and Carla was not third.
Sarah's score was 0.6 point higher than that of the oldest finalist.
In what order did the girls place in the gymnastics finals?

I know Sarah was first or third place in the finals, with Carla being 
first or second.

The line that is confusing to me is "Sarah's score was 0.6 point
higher that that of the oldest finalist" since no age is stated for
any of the finalists. I can't figure Kim's place in the gymnastics
finals. I am sure that Sarah was in first place, with Carla in second
place, and Kim in third place. Is that right?


Date: 12/16/2002 at 12:29:18
From: Doctor Ian
Subject: Re: Using problem-solving strategies

Hi Nicole,

To check your answer, I started by listing the places, and the girls
who _could_ be in those places. Sarah wasn't second, so she could
have been first or third:

  1: Sarah
  2:
  3: Sarah

Carla wasn't third, so she could have been first or second:

  1: Sarah, Carla
  2:        Carla
  3: Sarah

Kim apparently could be anywhere:

  1: Sarah, Carla, Kim
  2:        Carla, Kim
  3: Sarah,        Kim

So what about this final clue?  It's telling you that Sarah finished
ahead of _someone_... which means she couldn't have been third:

  1: Sarah, Carla, Kim
  2:        Carla, Kim
  3:               Kim

And that eliminates all the possible orders but one.  

The key to working a problem like this is to be systematic. Otherwise 
you can drive yourself crazy. 

So what what does that mean, to be 'systematic'? 

Basically, you write down all the things that _could_ happen, and then
you start looking for ways to rule out what _couldn't_ happen.  And
each time you rule something out, you go back and check to see if the
change you just made rules something _else_ out. 

In this case, we just figured out that Kim must be third. That tells 
us she can't be first or second:

  1: Sarah, Carla
  2:        Carla
  3:               Kim

And this change makes it clear that Carla must be second, which tells
us that she can't be first:

  1: Sarah
  2:        Carla
  3:               Kim

So I end up with the same answer you got.  

It's possible to make up problems that are _really_ complicated, like
this one:

   Who Owns the Fish?  (Einstein's Problem)
   http://mathforum.org/library/drmath/view/60971.html 

But if you'll look at the solution to Einstein's problem, you'll see
it's just the same thing that we did here: Write down what _can_ be
the case; eliminate what _can't_ be the case; and see if each change
makes it possible to eliminate something else. 

As Sherlock Holmes observed, "When you have excluded the impossible,
whatever remains, however improbable, must be the truth."  That's the
whole idea behind this kind of puzzle. 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
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