Ten Card Arrangement Puzzle

Date: 10/04/2005 at 15:06:05
From: Cece
Subject: Ten Card Arrangement

My daughter has been struggling for days on a math challenge given 
to her.  The teacher suggested asking her folks for ideas on finding 
a solution based on logic.  My husband, her grandparents, and I are
all baffled. Could you give us an idea of how to go about solving this
so we may offer some guidance?

The challenge is to arrange 10 cards numbered from 1 through 10 so 
that the first card turned over is 1 and the second card is put under
the pile.  The third card turned over is a 2 and the fourth card is
put under the pile.  The fifth card turned over is a 3 and the sixth
card is put under the pile.  This continues until all cards turned
over are numbered consecutively, 1 through 10.

After that she needs to do the same with 1 through 20.  We just can't 
seem to make it work.  She alternates the numbers in a way that seems
to make sense in the beginning but when we get to the end it's just a
jumble of numbers out of sequence.

Date: 10/04/2005 at 17:04:31
From: Doctor Achilles
Subject: Re: Ten Card Arrangement

Hi Cece,

Thanks for writing to Dr. Math.

Here's how I'd solve it.

Assign each card a letter, and then figure out what order the letters
come out in.  Then you can figure out the order to put the numbers in.

So you have a deck that is arranged like this:

  A, B, C, D, E, F, G, H, I, J

You take off the first card and put it face up:

  A

And then you move the top card to the end:

  C, D, E, F, G, H, I, J, B

Then you repeat:

  C

  E, F, G, H, I, J, B, D

And again:

  E

  G, H, I, J, B, D, F

And again:

  G

  I, J, B, D, F, H

And again:

  I

  B, D, F, H, J

And again:

  B

  F, H, J, D

And again:

  F

  J, D, H

And again:

  J

  H, D

And again:

  H

  D

And then finally:

  D

So that means you want to order the cards such that:

  A = 1
  C = 2
  E = 3
  G = 4
  I = 5
  B = 6
  F = 7
  J = 8
  H = 9
  D = 10

Or:

  1, 6, 2, 10, 3, 7, 4, 9, 5, 8

Hope this helps.  If you have other questions or you'd like to talk
about this some more, please write back.

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

Date: 10/05/2005 at 00:29:08
From: Cece
Subject: Thank you (Ten Card Arrangement)

You are a genius!  I can not thank you enough.  Your logic works on
any set of numbers.  My daughter is in awe!