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Puzzle with a Difference

Date: 03/19/2003 at 22:55:37
From: Janice
Subject: It's a Puzzler

Place each number from 1 through 10 in a box. Each box must contain a
number that is the difference of two boxes above it, if there are two
above it.

The underscore represents the boxes that are set up as follows:

_______     _________     __________     __________

     _________      ___________    _________

            __________       ___________


        10      3       8      9

             7      5       1

                 2      4


Thank you for your help.

Date: 03/20/2003 at 22:25:42
From: Doctor Peterson
Subject: Re: It's a Puzzler

Hi, Janice.

This is a sort of puzzle for which there is not a lot of help I can 
give other than to encourage persistence. Especially at your level, 
it is probably meant just to give you a lot of practice with 
subtraction; the bad part is that it can be very frustrating. You 
need a way to be orderly in your testing.

I tried playing with it, and found three solutions without filling 
more than a page with attempts. Maybe some of my tricks can help you, 
though I wouldn't expect someone without a lot of math experience to 
think of all of them.

One thing I did was to think about odd and even numbers. When you 
subtract two numbers, whether the difference is odd or even depends 
only on whether the numbers you started with are odd or even. So I 
could see what sort of pattern of odd and even numbers we will get by 
just using 0 and 1 as my sample odd and even numbers. There are 16 
possible patterns of odd and even to start with; here are three of 

    0 0 0 0   0 0 0 1   0 0 1 0
     0 0 0     0 0 1     0 1 1
      0 0       0 1       1 0
       0         1         1
    10 even   6 even    5 even

Now, the first two of these won't work for the numbers 1-10, because 
we have five even and five odd numbers, and these don't. But the third 
does have the right number of odd and even numbers. So if we start 
with even-even-odd-even, we have a chance of using all ten numbers. 
The only other patterns that will work are even-even-odd-odd, even-
odd-even-odd, and the reverses of these three (which we can ignore 
because if we discover a solution that fits one pattern, we can 
reverse it to make the other). That helps limit what patterns I try. 
(My three solutions happen to be one in each pattern; I wouldn't 
have expected that.)

Another important fact is that 10 has to be in the top row. Do you 
see why? There is no pair of numbers that could go above it. And 9 
can't be below the second row, and even there it can appear in only 
one way, as the difference of 10 and 1. So it helps to start at the 
top row with the large numbers. In two of my solutions 9 was in the 
top row, along with 8.

One more idea is to save writing by filling in just a part of the 
triangle and then trying all the possibilities for what's left without 
writing. For example, suppose I try 10, 2, 9 in the top row, leaving a 
space either on the left or the right of them for the fourth. I get 

    10 2 9
      8 7

Now the numbers I have left are 3, 4, 5, 6. I can try each of these 
in each end of the top row to see what will happen, and quit as soon 
as I get a repeat number. For example, if I try 3 on the left I get

    3 10 2 9
     7  8 7
      .  1

and already I can quit. If I try 3 on the right, I get

    10 2 9 3
      8 7 6
       1 1

and I can quit now. If you're systematic enough, you might be able to 
go through all the possible groups of three in the top row; but most 
likely you'll have an answer or two before you get too far.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 

Date: 03/21/2003 at 00:20:21
From: Janice
Subject: Thank you (It's a Puzzler)

Thank you!  Your advice about persistance paid off for my daughter and 
the Pascal Theory as well.  She was so excited when she got the answer 

8     10     3     9

   2      7      6

       5      1


and of course, that reversed.  We still have not gotten the 3rd way 
you mentioned, but we're still playing with it. 

Thank you again.

Date: 03/21/2003 at 08:38:53
From: Doctor Peterson
Subject: Re: Thank you (It's a Puzzler)

Hi, Janice.

Good job! That's actually the first one I found. Its mirror image was 
not one of my three; I didn't count that because I consider it the 
same solution.

Another math doctor gave me a list of ALL solutions (probably found 
by computer); he has four solutions, again ignoring mirror images, so 
I only missed one.

Interestingly (and I'll have to think about the math behind this), 
the solutions come in pairs, with the same top four numbers in 
different orders. Two of mine formed a pair, and his fourth solution 
pairs up with my third solution. So you should be able to find a 
second easily, and two more with persistence.

It's an interesting puzzle after all, isn't it?

- Doctor Peterson, The Math Forum 
Associated Topics:
Elementary Puzzles
Elementary Subtraction
Middle School Puzzles

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