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How can you divide the board into exactly $2$ equally-shaped and sized pieces such that it fits the hole?

Bonus: Can you do the same problem in $3$ pieces, such that one of the pieces of the board has the sides of a square? If no, then what is the minimum number of pieces you have to make to do so?

Checkmark goes to the one who solves both parts :) .

Edit :- Forgot to mention that in the case of $3$ pieces, the square piece will be one of them, and the other two pieces will be equally-shaped and sized.

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  • $\begingroup$ This is given as an example here. $\endgroup$
    – WhatsUp
    Commented Oct 16, 2020 at 14:44
  • $\begingroup$ I see, any idea for the Bonus part? $\endgroup$
    – Anonymous
    Commented Oct 16, 2020 at 14:45
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    $\begingroup$ I'm not familiar with the whole site, but I haven't seen this puzzle here before. My link was to another site. So don't worry (: The bonus part is not spoiled by the above link. $\endgroup$
    – WhatsUp
    Commented Oct 16, 2020 at 14:47
  • $\begingroup$ On the three pieces problem, can the different one be a rectangle? $\endgroup$
    – Pspl
    Commented Oct 16, 2020 at 14:58
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    $\begingroup$ I have a solution which requires that the long sides of the rectangles are in 7:5 ratio. The picture is "somewhere in the vicinity" of that, but not acceptably close by any means. Given that the shown areas are off by about 10%, the 7:5 ratio might also be within error margins. I wonder if that could be the case? $\endgroup$
    – Bass
    Commented Oct 16, 2020 at 16:30

2 Answers 2

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Assuming the picture is a bit off (seems like a safe assumption given that the areas don't quite match), I'm going to assume that the rectangles are scaled in

7:5 proportion (to preserve area, the other dimension must of course scaled in the inverse 5:7 proportion.), and that the thin rectangle is in 1:7 aspect ratio.

Some nice integer side lengths that achieve these proportions are

$$5 \times 35 = 7 \times 25 $$

These particular shapes allow for this interesting dissection:

enter image description here

All the horizontal bits are supposed to be of the same length (5 units). For clarity I marked the lengths of the vertical sides (in units of one fifth of the height of the thinner rectangle) so it's easier to verify that the widths match in the rearranged shape too.

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  • $\begingroup$ Interesting Answer, by the way the bonus part is what I thought of myself. Probably that's the correct answer (much better than I expected) . $\endgroup$
    – Anonymous
    Commented Oct 16, 2020 at 17:06
  • $\begingroup$ Wondering if you realize your ratio is not that special?..(not that e.g. 0.6 is closer to the shown ratio ) $\endgroup$
    – Retudin
    Commented Oct 16, 2020 at 18:03
  • $\begingroup$ @Retudin If you have an alternate solution in mind, it's as a rule better to actually draw it than to drop comments in a condescending tone of voice. $\endgroup$
    – Bass
    Commented Oct 16, 2020 at 18:28
  • $\begingroup$ It was not intended to be condescending, sorry. The variant is so much like the one above it seemed better to ask. Posting a debatably worse variant of a 'stolen' idea seemed more condescending. $\endgroup$
    – Retudin
    Commented Oct 16, 2020 at 18:46
  • $\begingroup$ @Retudin ah, sorry for being so thin-skinned then :-) This is the idea you meant? (Only now figured it out.) $\endgroup$
    – Bass
    Commented Oct 16, 2020 at 18:57
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Is the following the solution for the 2-piece problem?

enter image description here

Regarding the 3-piece puzzle, I'm not sure if the following solution will meet your criteria:

enter image description here

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  • $\begingroup$ @Anonymous, check this new answer. $\endgroup$
    – Pspl
    Commented Oct 16, 2020 at 14:41
  • $\begingroup$ That works, congrats you solved the $1$st part. Any idea for the Bonus one? $\endgroup$
    – Anonymous
    Commented Oct 16, 2020 at 14:41
  • $\begingroup$ I'm thinking about it! :D $\endgroup$
    – Pspl
    Commented Oct 16, 2020 at 14:42
  • $\begingroup$ Since the sizes matter for the bonus part, I carefully measured them and find out that the board is roughly $98 \times 282$ while the hole $71 \times 424$. This forces both answers to be "not possible"... I'm not sure what are the intended sizes of the board. Perhaps they should be given in the statement of the puzzle. $\endgroup$
    – WhatsUp
    Commented Oct 16, 2020 at 14:58
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    $\begingroup$ @WhatsUp in some sense, telling the exact size of the board would technically ruin 50% part of the puzzle, so I didn't. Probably you have to guess it. $\endgroup$
    – Anonymous
    Commented Oct 16, 2020 at 15:03

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