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/
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