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Matches

cinelli
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Matches

#657205

Postby cinelli » April 1st, 2024, 11:02 am

Image
Twelve matches have been arranged to form four squares. Can you re-arrange the same number of matches (all lying flat on the table) so that they enclose five squares? Every square must be entirely “empty” or the illustration itself would show five squares if we were allowed to count the large square round the boundary. No duplicate matches or loose ends or broken matches are allowed.

Cinelli

UncleEbenezer
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Re: Matches

#657225

Postby UncleEbenezer » April 1st, 2024, 12:13 pm

Four matches form an infinitesimal square, with the length of each match protruding. That could be done by moving your four "central matches" half a match width each as follows:

Top one to the right.
Bottom one to the left.
Left one upwards,
Right one downwards.

and pushing each one inwards just far enough to close the gaps we just opened.
.
|
|
|-----
------|
|
|

(whoops, that's a mirror image of the text).

That leaves eight matches to form the four squares similar to those in your picture. While the protruding sides of each square differ by a match-width, we compensate by placing the remaining matches end-to-side:

|---
| |
| |
---|


(That won't look square, but imagine three vertical dashes and three horizontal dashes as a match length each.)

jfgw
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Re: Matches

#657335

Postby jfgw » April 1st, 2024, 10:43 pm

Spoiler

The matches have to be laid flat on the table. Does this preclude overlapping? If so, I am not sure if UncleEbenezer's method can be made to work, bearing in mind that squares have sides of equal length. This is an issue with the question, however — the four "squares" are not exactly square. This can be resolved by arranging the matches in the way that UncleEbenezer suggests but without leaving a tiny square in the middle.

I think that the intended solution could be very similar, however.

I think we have to ignore the match heads otherwise the shapes in the question are not quite squares, even with the slight rearrangement.

Pedantry aside, this would work:

The matches need to be arranged similarly to the photo in the question. The four middle matches should just touch on the corners, leaving a small square hole in the middle, i.e., so that the ends of the matches form the sides of the square. The outer matches need to be placed so that they just touch at the outer corners, and just touch (corner to corner) the middle matches.


Julian F. G. W.

Mike4
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Re: Matches

#657336

Postby Mike4 » April 1st, 2024, 10:51 pm

I found I could make six squares, but not five.

jfgw
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Re: Matches

#657339

Postby jfgw » April 2nd, 2024, 12:49 am

Mike4 wrote:I found I could make six squares, but not five.

There are some solutions if you can cross matchsticks. You can make five squares and some rectangles, or five squares and a trapezium or a rhombus. The OP didn't say you couldn't have additional shapes.


Julian F. G. W.

malkymoo
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Re: Matches

#657533

Postby malkymoo » April 2nd, 2024, 6:37 pm

https://ibb.co/6tTSCLv

Shows photo of my solution.

y0rkiebar
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Re: Matches

#657542

Postby y0rkiebar » April 2nd, 2024, 7:23 pm

malkymoo wrote:https://ibb.co/6tTSCLv

Shows photo of my solution.


Doesn't that fall foul of the "no loose ends" constraint ?

malkymoo
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Re: Matches

#657543

Postby malkymoo » April 2nd, 2024, 7:34 pm

Ah, thought it seemed too easy!

mc2fool
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Re: Matches

#657553

Postby mc2fool » April 2nd, 2024, 8:31 pm

Does each square have to have a match on each of its four sides or can a side be "implied"?

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Re: Matches

#657566

Postby UncleEbenezer » April 2nd, 2024, 9:26 pm

mc2fool wrote:Does each square have to have a match on each of its four sides or can a side be "implied"?

I'm sure you're allowed matching pairs. Erm, sorry, pairing matches. :?

cinelli
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Re: Matches

#657629

Postby cinelli » April 3rd, 2024, 11:48 am

UncleEbenezer's solution is essentially what I had in mind. I even have a picture:
Image
Notice the tiny square in the middle, as the original puzzle made no mention of the size of the squares. But I do like malkymoo's picture even if it doesn't quite meet all the question's rules. I am sure readers will have noticed the date the puzzle was posed :)

Cinelli

jfgw
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Re: Matches

#657648

Postby jfgw » April 3rd, 2024, 1:13 pm

That looks a bit more like my solution.

The arrangement with four squares would only work if you overlapped matches. It is possible to do it without overlapping, however, by adapting UncleEbenezer's solution but without leaving a gap in the middle.
Image
I did ponder UncleEbenezer's solution but I don't think it quite works.
Image
This was my solution,
Image

Julian F. G. W.

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Re: Matches

#657693

Postby UncleEbenezer » April 3rd, 2024, 4:49 pm

You missed the word infinitesimal in my reply. It's asymptotically perfect as you shrink the square. In real life, and in the picture in the OP, there are irregularities in the matches themselves and their layout that will dwarf any design imperfections somewhere on the way to that asymptote.

Your own measurements of 26 and 28, each for the length of a match minus the width of another, demonstrate that your real-world imperfections are indeed quite substantial.


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