Arranging Numbers

Date: 06/26/2002 at 18:22:09
From: Rhonda Johnson
Subject: Math using tile numbers

Arrange the numbers 1, 2, 3, 4, 5, 6, 7, and 8 in this format:

             ? ?
           ? ? ? ?   
             ? ?

Consecutive numbers can't be adjacent vertically, horizontally, or
diagonally.  Use only the numbers given.  For example, the arrangement 

        1 5
      4 7 3 6
        2 8

cannot be a solution because the 5 and 6 are adjacent 
diagonally, as are the numbers 2 and 3.

I have tried for hours to solve this. Please help!

Date: 06/26/2002 at 20:06:22
From: Doctor Greenie
Subject: Re: Math using tile numbers

Hi, Rhonda --

This is a great puzzle in logic....!

I would like to just suggest something for you to think about to get 
started on this puzzle - to see if you can follow that suggestion 
through to the solution.  This puzzle is so much more satisfying when 
you solve at least part of it on your own, rather than having 
somebody show you the solution.

Let's start by labeling the positions as follows:

      A  B
   C  D  E  F
      G  H

The keys to the solution are positions D and E.  Position D is
adjacent either vertically, horizontally, or diagonally, to all other 
positions except position F; position E is adjacent either 
vertically, horizontally, or diagonally, to all other positions 
except position C.

Now think carefully about how many numbers are adjacent to each of 
the numbers you have to work with.  For example, there are two 
numbers adjacent to 3 (i.e., 2 and 4); there are also two numbers
adjacent to 7 (i.e., 6 and 8).  But there is only one number adjacent
to 1 (i.e., 2); and there is only one number adjacent to 8 (i.e., 7).

Putting the ideas of the preceding two paragraphs together tells you 
which numbers must go in positions D and E; once you have those 
numbers in place, the numbers in positions C and F are uniquely 
determined by the requirements of the problem; and then the rest of 
the solution is fairly easy.

Good luck with this!  Write back if you have any further questions on 
this... but please make an effort to finish the problem yourself 
before you write back.

(Or, it would be great to hear back from you if my help enabled you 
to find the solution...!)

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