One of my all time favourite geometry puzzles this week……

As ever, please do NOT post your answers, but do say if you think you have solved the puzzle and how long it took.Β  Solution on Monday.

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.



    1. Ditto, although I’m sure I’m right. Only logical solution as “going the other way” was eliminated as an option because of the writing.

  1. Sure; about a two minutes of trial and error with a drawing program – unless this is a trick question.

    Will see if there is a strict geometrical construction.

    1. Ahem, upon close verification of my vector drawing and a strict mathematical interpretation of the points I will have to retract my answer.

      This puzzle is indeed more fiendish than it first appears.

  2. Strictly speaking the answer to the question posed is “yes I can”, which I hope isn’t considered a spoiler. This took me 2seconds based on the reasoning that you wouldn;t be asking if it wasn;t possible.

    What? you want to know how? That’ll take a bit longer

  3. I don’t get it. Either this is a trivially easy ‘puzzle’ (at least with the image viewed on an iphone) or I am missing something. Can someone help?

  4. Less than a second to see the answer and then ten minutes of trying to convince myself that, yes, it really was as easy as that and, no, there isn’t some catch or trick that I was overlooking.

  5. I wonder if this has been reproduced quite right on the page. I downloaded it, printed it off and it took seconds to produce a square to a reasonable degree of accuracy.

    I’m not quite sure what the wording means when it refers to a “perfect square”. Geometrically a square is a square; none is more perfect than any other. They all (at least in Euclidean space) have 90 degree corners and four equal length sides. It might also be called a regular quadrilateral. A perfect square is normally used to describe a number where the square root is an integer. I’m assuming there is not trickiness in this wording. Perhaps I’m reading too much into it and it’s just padding out the wording to fill the space available.

  6. OK – after downloading the image as a JPEG and doing some much more careful geometry, this reveals something. However, you aren’t going to see this very easily on a small screen. There is also another problem – which Richard probably hasn’t allowed for. Almost all screens are not quite accurate in their height vs width ratios and introduce a degree of distortion. Unfortunately precision matters with this one.

    I think I know the right answer now, but people might be forgiven for getting this one wrong.

  7. Can’t understand why, each week, so many of you are so desperate to show off that you give away how to solve the puzzle in what you write. That is unless you are 13 year old boys Then I understand perfectly.

    1. how about a 15 years old girl.?
      I think it’s just because the comments would be too boring if everyone just write how long they took to solve the problem, everyone likes to prove they are smart, no matter they are 13 or not.

      The second reason is we want to help. I know how frustrating people who got stuck on the puzzle would feel when they saw other people( who rote comment) said they solve this puzzle in 3 second or things like that. Leaving some clue in the comment might be helpful for those who get stuck, at least they were helpful for me.

      today’s puzzle is kind of tricky.

    2. I agree.

      I find puzzles hard also.

      When I find clues in comments it is great and I can think in other ways.

      It helps the more simple minded people.

  8. This is either fiendishly difficult, or so simple that I fail to see how it’s even a challenge. Solved instantly, while reading.

  9. Looks fairly easy but to be a perfect square each angle has to be 90′ so that means that each side is the same length. I’ve sketched the solution but you can’t be sure from the image printed above that each side is the same length.

    1. Rectangles have 90* corners so that doesn’t imply their sides are all the same length. However it does imply that each side is the same length as its parallel partner.

    1. possibly because it isn’t as ridiculously easy as people think? I rather suspect lots of people think they’ve solved it, when they haven’t. However, we’ll see on Monday.

  10. I haven’t seen this one before, but it took me just one or two seconds to solve. Seeing the comments above that it’s not as easy as it seems, and that folks might be finding false solutions, I took an extra 10 seconds to open the image in a Illustrator to make sure the square I was imagining was in fact “perfect”. Still works. Very easy.

  11. Got it immediately, checked it in PSP, confused by the phrase ‘not as easy…’ and by the mickey mouse reference in the URL.

    I can also draw a square and put a dot on each side but I don’t think that’s the q, tho it doesn’t say the required shape has to surround the text or include the dots around it.

    1. And I came up with a second, also valid solution, after thinking about it just a minute or so.

      Can’t wait to find out which solution was the correct one – the five second one or the 1 minute one.

  12. It’s worth pointing out that nobody who claims to have done this in a second – or even five – can possibly have performed the required geometric measurements. Just a thought…

    1. OK, fine – I solved it, CONCEPTUALLY, instantly. I then downloaded the PNG version and pulled it into a drafting program to illustrate that solution. No measuring was needed; there is a (small) range in which a family of solutions is possible and one can be identified through a (very) little trial and error. So, yeah, fine, a minute for a rigorous solution. Should I upload a solution?

    2. UXO β€” By all means, please share, come Monday. Make sure you use sufficiently thin lines, bisecting the dots as clearly as possible, and verify the square’s geometry.

    3. Kind of pointless as of Monday, innit?

      (And nobody said anything about “bisect”. The challenge reads “one dot ON each of its four sides”, and I freely admit my solution does not have each dot cleanly bisected – though each is definitely on a side.)

    4. I’ve used the normal mathematical practices when we talk about dots and sides of squares (which are lines). Dots are zero-dimensional and lines single dimensional. Any real-world drawing of these does, of course, have to have real thickness or they’d be invisible. However, it’s the convention that the centre of lines & dots that matter.

      I’ve generate my answer using geometry using co-ordinates (as closely as I can measure them) from the original JPEG using Photoshop and overlayed an answer which I’ll put up on my Flickr pages.

      I realise that a strict mathematical sense is not always what these puzzles are about and it’s possible to produce trivially easy results by using thick lines and counting any dot which is at least partially crossed by the line as being on it.

  13. I downloaded to – and stuck it in Photoshop, did some careful measurements and came to and interesting solution. I’ll overlay the ping on the JPEG I downloaded. A lot depends on exactly how accurate that JPEG is.

    1. I think we are on the same track. I did the math and sketched out the treatment. That took about an hour or two. My formula does have terms that depend on the difference of lengths, which sure is susceptible to errors (and leads to interesting math). However, given the magnitudes seen here, I am fairly convinced that I have the existence or non-existence πŸ˜‰ of a solution pinned.

      The jpg snapshot does indeed suffer from some blur. I used the png that graciously posted.

    1. It doesn’t say the dots will TOUCH the side. It says the square has “one dot ON each side”. In geometry, “on” is usually taken to mean the dot is centred on the side (or edge) – although point is usually preferred to dot. The vertex (or vertices) are not mentioned anywhere.

      The verb touch is only used in respect of the sides not touching any of the words.

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