Take a square and cut it as you wish into a finite number of pieces. Rearrange the pieces without rotating or flipping any of them to form an new square that is rotated 45-degrees compared to the original.

Yeah, I’m having a really hard time with this as well. I had a couple ideas, but they got me nowhere. Maybe if I explained my idea, someone else would get an epiphany and take it home. Let’s brainstorm!

First – let’s orient the original square so that its sides are horizontal and vertical. Then the sides of the new square will be +/- 45 degrees from horizontal. Call the edge length of the square L.

Idea 1: Cut off the 4 corners of the square at a 45 degree angles. Cut them off such that you are left with a regular octagon. So now you have an octagon and 4 right triangles that are all identical (except for orientation). The hard part is how to make the 4 triangles end up fitting into the other 4 triangles. Now it becomes finding a way to "rotate" 4 little triangles, which is just as hard I think.

Idea 2: Make a cut from the bottom left corner at a 45 degree angle. Stop when the cut is length L. Now make a –45 degree angle cut that passes through the end of the first cut. Now you a have two large pieces that are mirror images of each other (over the diagonal of the original square) and a triangle. So on a large piece, the two corners on the longest edge could be two corners of the new square. The hard part is how to break up the other pieces… I often get a long skinny trapezoid that mucks things up.

Idea 3: Similar to Idea 2… Cut the full diagonal of the square (bottom left to top right). Call the top left piece A, and the bottom right piece B. Find point M on the diagonal of piece A such that the horizontal distance from M to the left edge of piece A is L/2. Cut perpendicularly to the diagonal at M, so now you have a funny quadrilateral and a right isosceles triangle. Do the same thing to piece B. In this case the two right angles of the quadrilaterals become two opposite corners of the new square.

Idea 4: Very similar to Idea 3. Cut the full diagonal of the square (bottom left to top right). Call the top left piece A, and the bottom right piece B. Find point M on the diagonal of piece A such that the distance from M to one end of the diagonal is L. Cut perpendicularly to the diagonal at M, so now you have a funny quadrilateral and a right isosceles triangle. Do the same thing to piece B (so the pieces from A and the pieces from B are 180 degree rotations of each other). So point M and an end of the diagonal on one of the pieces could be two corners of the new square.

Idea 5: Cut a tic-tac-toe type shape at a 45 degree angle. I don’t know what to do next!

Idea 6: Make a 45 degree cut from the left edge to the top edge such that the cut is length L. Call the endpoints of this cut X and Y. Make cuts at X and Y perpendicular to XY all the way to the other edges. I haven’t gone through this one, I just thought of it.

How are you guys approaching it?

Posted by nikki on 2004-10-19 20:50:28