Building Squares with Tangram Puzzle Pieces

Date: 03/21/2006 at 17:14:25
From: Anne
Subject: tangram puzzle

I have been able to use 1 piece, 2 pieces, 3 pieces, 4 pieces, 5
pieces, and 7 pieces of the tangram puzzle to build a square.  I have
tried for a while to build a square using 6 pieces of the tangram
puzzle, but without success.  Is there a solution to this puzzle or is
it impossible?

Mostly I  have tried trial and error when building with the tangrams.
I have also tried to build two equal-sized isosceles triangles using a
total of 6 pieces with the idea that I could put the two triangles
together to form a square.  I still have had no luck finding a
solution using a total of 6 pieces.

Date: 03/21/2006 at 20:05:25
From: Doctor Vogler
Subject: Re: tangram puzzle

Hi Anne,

Thanks for writing to Dr. Math.  I assume you are talking about
tangrams as described in

  Tangram
    http://www.tangrams.ca/ 

You can prove that it cannot be done with 6 of the 7 pieces.  Let's
call the shortest edge on any tangram a length of one unit.  Then
every edge has either length 1, 2, sqrt(2), or 2*sqrt(2).  Also, there
are two pieces of area 1/2 (congruent triangles), three pieces of area
1 (a square, a triangle, and a parallelogram), and two pieces of area
2 (congruent triangles), for a total area of 8 square units.

Thus all seven pieces together make a square with an area of 8, and
therefore a side length of sqrt(8) = 2*sqrt(2).  What other squares
are possible?  The area must be an integer multiple of 1/2, between
1/2 and 8.  The side of the square is the square root of the area, but
the side must be a sum of 1's and sqrt(2)'s.  The only possible sums
between sqrt(1/2) and sqrt(8) are

  1
  sqrt(2)
  2
  1 + sqrt(2)
  2*sqrt(2),

whose squares (the area) are, respectively,

  1
  2
  4
  3 + 2*sqrt(2)
  8.

All but the fourth one are possible squares to construct; I imagine
you can make them all.

But consider, if you are only allowed six of the seven pieces, which
piece do you leave out?  If you leave out anything, then you don't
have enough area left to make a square of area 8, but the next-biggest
square you could make has area 4, and there is no piece of area 4 that
you can leave out.  So you can conclude that it is impossible to make
a square using six of the seven tangram pieces.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

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