Toothpick Puzzle

Date: 09/27/2004 at 22:26:17
From: Michael
Subject: 3x3 group of squares made up of toothpicks

I got this one the other day, and I'm not sure it's possible.  If you
have a square that is made up of nine little squares, each one
toothpick per side, so the big square is three toothpicks per side,
can you remove 5 toothpicks and leave 3 squares (of any size necessary)?

Taking out 5 toothpicks leaves 19 toothpicks to make 3 squares.  I
can't figure out a functional way to add 12, 8, 6, 4, 3, 2, and 1 to
get 19.  I can, but it doesn't seem possible, because some
combinations eliminate other possibilities. 

I don't think the big square (of 12 toothpicks) can be one of the
three, because that leaves 7 toothpicks to make two more squares, and
making one out of four removes any possibility of making one out of
three.  If I start with one of the squares being 8 toothpicks, that
leaves me 11 to make two squares, so that seems to necessitate another
square of 8, but If I do that the only way I can see how, I am left
with 3 squares already, and still 3 toothpicks left.

Date: 09/28/2004 at 01:53:49
From: Doctor Ian
Subject: Re: 3x3 group of squares made up of toothpicks

Hi Michael,

It's possible that some of the squares can overlap, as in the 
following diagram:
   
   --- --- ---
  |   |   |   |
   ---        
  |       |   |
   --- ---   
  |           |
   --- --- ---

which uses 18 toothpicks (--- or |) to make 3 squares. 

Does this get you closer to your 19-toothpick solution?

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

Date: 09/28/2004 at 09:16:37
From: Michael
Subject: 3x3 group of squares made up of toothpicks

Ah, there it is.  Thanks.