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Consider the following diagram of your prized 9' x 12' Persian rug:

                                 Persian rug diagram

During some recent festivities a guest spilled wine on the rug, ruining the 8 square feet depicted in red.

To remedy the problem, you intend on cutting the rug into multiple pieces, moving and possibly rotating the pieces (2D rotations only; no turning pieces over), and gluing them back together again. You can cut any way you please—vertically, horizontally, diagonally, or along any 2D curve—but you must respect three rules:

  1. the reassembled rug may not comprise more than two pieces
  2. none of the red area may appear on the reassembled rug
  3. the reassembled rug must be square or rectangular in shape and be a simple polygon (i.e. no holes).

Since this is indeed a very expensive Persian rug, your objective is to leave as much area as possible in the reassembled rug.

What square footage can you save, and how do you accomplish it?

Puzzlers are politely encouraged to place answers in spoiler blocks to avoid spoiling the fun for other readers. :)

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  • $\begingroup$ When you mention curve would 90 degree turns in straight lines be considered a curve? $\endgroup$ – skv Nov 5 '14 at 5:07
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    $\begingroup$ Thank god for wine that spills in perfect squares, how did we get along without it? :-) $\endgroup$ – Joe Nov 5 '14 at 8:33
  • $\begingroup$ When you say no more than three pieces, do you include the squares being thrown away? It seems like the red squares shouldn't count as pieces. $\endgroup$ – TheRubberDuck Nov 5 '14 at 13:54
  • $\begingroup$ @EnvisionAndDevelop: Yes, any piece that includes some red, including a piece that is the entirety of the red rectangle, counts as a piece. $\endgroup$ – COTO Nov 5 '14 at 16:01
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    $\begingroup$ Naturally, send the rug home with Aunt Darlene and make her buy you a new rug. C'mon Darlene, you do this every year. $\endgroup$ – corsiKa Nov 5 '14 at 21:02
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Quite an interesting cut I had to make to get it to fit.

image of rug cut

Shift the bottom piece 2 tiles up and one tile right to produce a 10 x 10 square. You can save all 100 square feet.

The end result should look like this

the yellow is the second piece
enter image description here

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  • $\begingroup$ Interesting indeed, can you add "What square footage can you save, and how do you accomplish it?" to the answer $\endgroup$ – skv Nov 5 '14 at 5:17
  • $\begingroup$ Thanks for adding an explanation, a picture would say it much better, I can kind of understand but its so hard to visualise $\endgroup$ – skv Nov 5 '14 at 5:26
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    $\begingroup$ @skv This might be easier to imagine: "Move the bottom piece one tile to the right, then push it up until it meshes with the top piece." (The rug will now be 1 tile wider and 2 tiles shorter.) $\endgroup$ – MetaGuru Nov 5 '14 at 15:18
  • $\begingroup$ Is this related to that weird trick with the disappearing gnome? There are 8 gnomes, but if you cut the paper and move the pieces there suddenly is an extra gnome. (couldn't find source) $\endgroup$ – goodguy5 Mar 23 '16 at 16:11
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I think for the sake of simplicity this should be part of the answer

You lost 8 squares, you have 108 to start with, so the maximum we can save is 100 I propose 99 of them can be reused
enter image description here
cut the (2) above and paste it on top of the spoiled squares, and throw (1) away.
The spoiled portions wont "appear"

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Otaia's answer is optimal for square footage. This provides the same results as skv's without any unsightly bumps...

enter image description here Cut a strip off the end (1) Cut a 9 square section (2) out of the center and discard. (include one non-soiled square in line with the 8 square wine soiled section.) rotate (1) and insert in the hole left by (2) results in a 9'x11' rug.

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solution only requires:

1.the reassembled rug may not comprise more than two pieces
2.none of the red area may appear on the reassembled rug
3.the reassembled rug must be square or rectangular in shape

so, max saved if all but red section retained. to do that...

one cut deletes the center red section. DONE
problem does not prohibit hole. one piece left, max area satisfied, still a rectangle

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  • $\begingroup$ Haha, can't argue with that. Time to edit the question. $\endgroup$ – McMagister Dec 30 '14 at 4:21
  • $\begingroup$ I stand corrected! :) $\endgroup$ – Hugh Dec 30 '14 at 4:41

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