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We have a 8x8 carpet, we can do only one cut of any complexity and also we have an uncuttable 6x1 carpet. We need to fill a 7x10 space using those 2. The 8x8 can only be cut into two pieces.

We have tried all the simpler shapes like an L and varying stair shapes, we cannot find the solution. We only got the hint that a normal person wouldn't cut their carpet like that. Diagonal cuts are allowed.

The question comes from my father but no one in our family can find the solution. He said we could use any tools, he himself once solved it when he was in uni decades ago when asked by a friend.

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2 Answers 2

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Here is one way it can be done.

solution image

Path of discovery:

Assuming there is a simple, symmetrical solution on grid squares with the 6x1 piece in the center, the outer ring is forced.
discovery part one

The bold portions of the border in the 8x8 piece appear in two places on the 7x10 piece. After another step in this direction, the solution is complete.
discovery part two

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    $\begingroup$ Can you please explain it more? $\endgroup$
    – Etack Sxchange
    Commented Jul 26 at 11:51
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    $\begingroup$ @EtackSxchange I'm not sure what there is to explain. You cut the 8x8 rug as shown in the left image, then rearrange it together with the 6x1 rug to form the 7x10 rug as shown in the right image. $\endgroup$
    – F1Krazy
    Commented Jul 26 at 12:41
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    $\begingroup$ That was fast. Any insight in how you solved it? Or have you just solved many puzzles like this before? $\endgroup$
    – Oliphaunt
    Commented Jul 26 at 19:55
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    $\begingroup$ @smci See edit. Note that I started with an assumption (hypothesis), and found it to be true. $\endgroup$
    – Daniel Mathias
    Commented Jul 27 at 13:25
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    $\begingroup$ Awesome, thanks. I think once you start a 1x2 staircase, you're forced to continue it. $\endgroup$
    – smci
    Commented Jul 28 at 5:37
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For anyone wanting to play around with this themselves, here is a Desmos graph (without answer spoilers) that allows you to explore different shapes. It makes the mild assumption that:

the 8x8 carpet is cut symmetrically.

Here is the graph with the solution. Using this you can more clearly see the reasoning discussed in the comments of Daniel's answer, and also the path towards how you would discover this.

One of the shapes must include a corner, and WLOG we start from (0, 0). The first adjustable point has to lie on the y-axis as otherwise nothing would be able to fill the void in the left side of the 10x7 rectangle. It cannot have a y coordinate less than 1 as otherwise at least one of the shapes would need to have a height greater than 7 and not fit in the 10x7. The second point must have an x coordinate of exactly 2 as otherwise the combined shapes would not have a total length of 10. It is easy to see how the rest of the construction is forced from this point as others have already pointed out.

Picture of the answer:

8x8 cut and 7x10 assembled carpet

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