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Imperfect Square (Posted on 2004-10-14) Difficulty: 3 of 5
The above piece of chocolate looks like a square with 4 pieces stuck on to it. How can you divide it with 4 straight cuts that they may form four congruent squares when moved? (You may not move the pieces until all cuts have been made, and you may not rotate or flip over any of the pieces when moving them. Also, all the chocolate must be used to make the squares.)

No Solution Yet Submitted by Gamer    
Rating: 3.0000 (1 votes)

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Solution Answer + Solution | Comment 1 of 8

I had no clue how to solve the Zquare problem, but after reading other's solutions I now have a plan of attack for this one.

Ok, first things first, how long are the sides of the squares? Well, let's look at area first. We are given 20 units^2 to begin with, which means that each of the 4 squares will have an area of 5 units^2. So the length of the side will be sqrt(5) units long. How do we get sqrt(5) units length? We need to find a rectangle with sides a and b such that a^2 + b^2 = sqrt(5)^2 = 5. I'm hoping a and b will be integers, so I'll start looking at those. Hey, how about that, a=1 and b=2 satisfies that equation.

Since the starting shape is not symmetrical over a line, I have two paths to examine in order to find a solution. I'm not good at drawing ascii pictures, so hopefully that made sense.

The other key, I think, is to make your cuts like a tic-tac-toe board. You're "center" square will be solid, and the other pieces that are in the 8 regions of your cuts will have to be put together to form the other 3 squares.

I tried one way, where a "1 square tall, 2 square wide" rectangle is cut from the top left to the bottom right (and where a "2 square tall, 1 square wide" rect is cut from the top right to the bottom left to form the right angles), but I couldn't get a solution. And I saw that if I moved my cuts around, I was getting the same situation, just rotated. So I abandoned that path.

Next, I tried cutting a "1 square tall, 2 square wide" rectangle from the bottom left to the top right (so a "2 square tall, 1 square wide" rectangle would be cut from the top left to the bottom right). And here I found a solution using the tic-tac-toe idea.

Now for a good way to describe my cuts... I think the best would be to label all the vertices (there are 33 of them). So I will call them v1 through v33. I will start labeling them left to right in the first row, then left to right in the second row, and so on. So v1 and v2 are in the top row. v3 - v7 are in the second row, v8 - v13 are in the third row, and so on.

If I say a cut goes from one vertex to another, I am defining the LINE of the cut, and not just a segment that is limited between those two vertices. So if you are trying to follow along, draw a line between the two vertices given, but then extend the line through the rest of the chocolate. The reason I did this is because sometimes the end of the cut is on the edge, and not the corner, of the square.

Alrighty, now for the cuts:

v9 to v5
v22 to v18
v9 to v22 (and on to v33)
v5 to v18 (and on to v31)

Now tilt your head to the left a little, so it looks like a properly oriented tic-tac-toe board.

The first square is made solely from the center piece.
The second square is made from the top middle, and bottom middle pieces.
The third is made from the left middle and right middle pieces.
And the fourth is made from the four corner pieces.

Everything slides around without rotating or flipping to create the four squares. Tada

tidying up formatting

Edited on October 14, 2004, 1:18 pm
  Posted by nikki on 2004-10-14 12:45:28

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