On Friday I posted this puzzle.

I have just made this video containing 10 bets that you will almost win (kinda).  About 1 min in I present one of my favourite puzzles – the vanishing square.

You can download your own version of the puzzle here.

So, the puzzle is….why does a square seem to vanish?

If you have not tried to solve it, have a go now.  For everyone else the answer is after the break.

The following video explains the puzzle, albeit with a slightly different set of shapes.  Basically, the top shape is not an exact triangle, and so when the pieces are rearranged, the area that makes up the ‘missing’ square are redistributed in a different way.  Did you solve it?

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. Didn’t anybody like this one?? No weird, wacky comments yet???? 🙂 This is the first time I’ve been first. 🙂

    1. This puzzle is a prime example of how you can go astray if you rely only on maths to understand the world.

      There may be two similtaneous equations to define the triangles in their different configurations. Yet the equations disagree. That leads to confusion and puzzlement.

      If we reflect for a moment that consciousness and the outer world are inextricably tied to one another. Then we quickly realise that how we treat the world is a reflection of how we treat ourselves

      By seeing ourselves as puzzle solvers we can solve puzzles. If we see ourselves as equation solvers then the puzzles of the weekend will befuddle us.

      @DAve. Thank you!

    2. @Barry, let me see if I understand …

      This puzzle is a prime example of how you can arrive at a solution if you rely on maths to understand the world.

      There are two equations to define the triangles in their different configurations. And because the equations agree, it eliminates confusion and puzzlement.

      Considering for a moment that consciousness and the outer world are seldom related to one another, we quickly realise that how we treat the world has nothing to do with how we treat ourselves

      By seeing ourselves as puzzle solvers we can solve puzzles. If we see ourselves as equation solvers then the puzzles of the weekend become obvious.

      Is that right?

    3. @Ken Haley

      You keep using words. I do not think they mean what you think they mean.

    4. @Ken Haley

      As you can see from the discussions further below the great mathematical minds are now discussing whether -1 equals 0 or 1 or 2.

      If they could simply free their minds from their equations the doors of their perceptions would be cleansed and they would enter a brave new world.

      It is not too late for you to leave them to their pinhead angel philosophies and join me in the freer intuitive world. We need to think mythologically, because that allows room for the part consciousness plays in generating views of the universe.

    5. @Barry

      You’ll be aware that “Brave New World” was an ironic title that Huxley adopted to describe his dystopia, and a warning that these brave new worlds come with their own detachment from reality.

      Of course, it has its own irony. Huxley himself “dabbled” in his own drug-induced attempts to open the doors of perception. He also then went on to write the virtually unreadable “Island”. Indeed, your own language reminds me a little of the baffling mystical confusion that permeates much of that book.

    6. @Jones. “Island” might be unreadable to you but to millions of others it’s regarded as a provocative philosophical work and is published as a Vintage Classic. Your comment reveals much about your weaknesses with the written English language (although your post actually looks as if it’s been proofread for once, so well done for that at least).

    7. @Noseache

      Yes, Island appeals to a sort of mystical mind that thinks it’s dealing in profundity, but in fact it’s stuffed full self-indulgent mysticism. Great if you’re the Beatles wanting some inspiration (along with a bit of LSD) when in their mystical phase. However, not much use if you are attempting some form of consistent view of the nature of the universe or a workable society.

      If you want to study philosophy, do it properly. There are many very profound thinkers that follow logical trains of thought based on various axioms that don’t have to resort to drug-induced meanderings in order to believe they’ve somehow opened the gates to some new perception of reality.

      Of course it’s a personal opinion, of mine, but it’s a commonly held one. Personally, I’m more of a logical positivist/empiricist than a mystic. But then you probably guessed that.

      I also prefer JG Ballard as an explorer of the alternative nature of societies and sub-cultures. He also manages surreal passages, but without having that crutch of “mind opening” drugs. There is a sort of logical inevitability about the societies he paints if you accept his axioms.

      nb. I was a great fan of Brave New World is, apart from it’s premise, it was full of fun and mischievousness with a liberal helping of satire yet done with a deft touch (well, if you ignore “the savage”, which was a bit of a clunky mechanism to reintroduce the idea of independence and freedom of thought back into a – benignly – totalitarian system). Island reads as turgid and improbably, possibly because it tries to be such a serious book.

    8. @Jones. Brilliant.

      How can you put these two sentences together without a smidgen of self-awareness?

      “Of course it’s a personal opinion, of mine, but it’s a commonly held one. Personally, I’m more of a logical positivist/empiricist than a mystic.”

      High Town sells a good line in large spades for you to continue digging, if you’re interested.

      (Keep working on the it’s/its problems and that pesky spelling, construction and grammar. Raymond Murphy is recommended.)

    9. @Noseache

      Thanks for you’re helpful advice, so obviously meant in the best spirit. The tendency I have to phonetic spelling (hence the unconscious substitution for it’s and its and the like) is something I’ve battled with throughout my life. It’s one thing to be able to spot when written by somebody else, but I always find it tricky when what I’ve written is an expression of my own thoughts. It’s so much easier to proof-read other’s work.

      I am, also, an appalling speller for what it’s worth.

      nb. I wouldn’t have responded to BG if it wasn’t for his snarky remarks about the discussions further down the comments. I’ve yet to work out if his comments are serious, or a wind-up.

    10. Having read Barry Goddard’s statement about consciousness and the outside world being inextricably linked, I think I’m still with Alan Moore when he says:

      “The first thing we can definitely say about reality from a human perspective is that we cannot experience reality directly. We have photons bombarding our retinas. We have vibrations in our inner ear, in our tympanums. The cilia of our nostrils and the buds upon our tongues transmit impressions of the chemicals comprising everything we smell or taste, while the minute electrical impulses racing through our nervous systems tell us whether we are touching silk or sandpaper. Moment by moment, we somehow compose these signals into a grand, shifting tapestry we call reality. It isn’t: It’s our sensory impressions of reality, with a direct experience of the thing itself being impossible. Effectively, to practical intents and purposes, reality is in our minds.”

    11. @noseache “You keep using words. I do not think you mean what they think you mean.”

    12. Just back from a happy few hours with a very old buddy of mine. After a few pints and a shared bottle of wine, we were both agreed that the most important thing in life is the efficacy of your bodily functions. Everything else is a bonus.

    13. @Steve Jones. I doubt that anyone would take any notice of your dreamy mutterings and half-baked conjectures if you didn’t assume such an air of superior oneupmanship, irrespective of the fact that your noodlings are usually scarcely literate or even comprehensible. Your autobiographical stroll-ins are of unpleasant flavour as well. Quite an achievement in a mere few sentences.

    14. @Noseache

      Please do not be too hard on @Simon.

      He admits he knows little of literature, mathematics, philosophy, logic, consciousness and even spelling. And he has a gruff and ready way of expressing himself. He clearly does not listen nor learn easily and he is quick to demean others for holding views that are expanded beyond his horizons.

      Yet he is in dialogue and we may be able to nudge him slowly toward a more positive attitude provided we do not alienate his affections by responding in kind (rather than in kindness).

      I thank you for defending me from the attacks and slights of outrageous posters. Yet I remain in good spirits as I know all that they are doing is exposing their limited minds. Perhaps they will soon settle down and begin to listen to all the god advice I and others have provided over the past few weeks. We should await patiently for signs that they are doing so.

    15. @Barry. Just a point of clarification: I have no problem with Simon. It was Steve Jones’s supercilious, pseudo-literary ramblings, when he couldn’t even be bothered to proof his own posts while being critical of others, that I despised.

  2. If you overlaid the two triangles, you’d see that theta (the angle to the left) is different in each. More precisely, the 8×3 triangle has theta = 20.6 degrees, the 5×2 triangle has theta = 21.8 degrees. Because the difference is only slightly more than one degree, when laid out as in the diagram, they appear to be a straight line – but of course if you checked with a ruler you’d see this wasn’t the case. Following the rearrangement, the resulting shape isn’t a perfect square (the baseline isn’t horizontal), but the small difference in the angles (coupled with the obvious margin for error in cutting out the shapes) leads the eye to believe otherwise.

  3. That is amazing. I mean that is really powerful. With out the video explanation there is no way I would have been able to “see” where the missing square went to.

    That is really REALLY good. Richard, I think this just took its place as my favourite “trick”.

    1. I also would like to see the explanation video, but in germany it is prohibited due to some license issues in the used music.?!?!
      Can i find it somewhere else??
      Thanks for any pointers

  4. The hypotneuses of the triangles do not trace the diagonal of the rectangle. The area traced out is exactly .5 square less than 1/2 the area of the triangle. When the shapes are rearranged, they are, in effect, effect overlayed into an area .5 square LARGER, the difference is 1 square.

    1. Gargh, edit fail, please allow me to retry:

      The hypotneuses of the triangles do not trace the diagonal of the rectangle. The area traced out is exactly .5 square less than 1/2 the area of the rectangle. When the shapes are rearranged, they are, in effect, overlayed into an area .5 square LARGER, the difference is 1 square.

    1. It’s apparent that PO does not necessarily intersect BO internal to the triangle, i. e., the construction was incorrectly drawn (<APO is not right, as drawn, …).

    2. Here’s another.

      Proof that 2=1

      Suppose x = y
      Multiply both sides by x:
      x^2 = xy
      Subtract y^2 from both sides: x^2 – y^2 = xy – y^2
      Factor both sides: (x + y)(x – y) = (x – y)y
      Divide both sides by (x-y): (x + y) = y
      Since x = y, this means: 2y = y
      Divide both sides by y: 2 = 1

    3. @Ken Haley

      As x equals y, it follows that x-y = 0, so dividing by (x-y) is just dividing both sides by 0. If that was an allowed operation it’s pretty well possible to prove anything…

    4. @Mike Benton

      That one is rather more subtle. Step 3 is what’s wrong; sqrt(a * b) = sqrt(a) * sqrt(b) is only true when a & b are positive integers. That’s how we first come across this “rule”, and transposing it across to complex numbers is wrong (in this instance).

      What you have there is pretty well a proof that it does not hold true when a & b are negative…

    5. @Steve: Good observation, with one minor quibble: You said “when a & b are positive integers”. Wouldn’t that be “non-negative real numbers”? No need to be integers and zero should be included.

      Technically, since we’re discussing complex numbers, all numbers have two square roots, and step 3 is simply asserting that the two roots are equal — which is obviously false.

    6. Fair enough, although isn’t zero often conventionally treated as positive. That’s my lame excuse.

      Anyway, it fits in well with the premise. That is we make unconscious assumptions about problems and solutions. I might add, it’s hardly confined to the world of mathematics, although the consequences of such unwarranted things is more immediately provable than such assumptions in the far more complex “real” world. Richard deals in the world of illusions and preconceptions, and it is that aspect of these particular problems which is far more interesting than these (false) mathematical paradoxes. Richard’s problems are often notorious for being ambiguous and rife with room for interpretation. It would be interesting to know if this is by design or by nature.

      On a more philosophical note, the most counter-intuitive finding in the whole of mathematics is surely that found by Godel where real paradoxes must exist in any self-consistent form. That the universe apparently doesn’t even allow us completeness and purity in something (apparently) abstracted from reality is almost shocking (albeit their had been hints earlier which had trouble Russel). That assumption that mathematics must be neat and tidy was virtually and act of faith, and the impact of finding it wasn’t was surely only paralleled by the ancient Greek’s outrage in finding out about “irrational numbers”.

      nb. whilst my mind is roaming free, it’s interesting that mathematics is rife with what sounds like judgemental terms. We have “rational numbers”, “irrational numbers”, “vulgar fractions”, “perfect squares” and much else. It surely speaks volumes of the emotional and aesthetic sensibilities of mathematicians.

    7. @Steve, you have essentially uncovered the lie in that “proof” although Ken expressed it more completely.

      There’s a really good explanation of Kurt Gödel’s Incompleteness Theorem(s) in Douglas Hofstadter’s book, Gödel, Escher, Bach. It is a thoroughly entertaining book about consciousness and cognition, with lots of allegory and word-play. I highly recommend it if you like that sort of thing.

    8. @Mike: The book you mention, Gödel, Escher, and Bach is one of my all-time favorites! I, too, highly recommend it. Even if you don’t care to follow the technical minutia that he sometimes delves into (Gödel’s proof, et al.), it is incredibly entertaining, and quite thought-provoking. I read it a long time ago, when it was first published, and reread in the past couple of years. It is great.

    9. I’ve read the book (about 20 years go?), and it was it was quite fun, but I found the allegories got a bit laboured. The demonstration of recursion (did it involve turtles?) got a bit strained, but then I was a computer programmer at the time and it seemed quite natural. You just had to get the termination conditions right, or you’d recurse for ever – or until the stack memory ran out.

      Anyway, an interesting trio, and something of a one-off as a best seller. Inspired insight to see the common thread among them.

    10. Well Steve, as a former(?) programmer with an awareness of recursion, you might appreciate the following joke.

      Q: Why was the computer programmer found dead in the shower?

      A: He’d been washing his hair and the instructions on the bottle said
      Wet hair

    11. One of the more entertaining examples of recursion in Gödel, Escher, Bach is the notion of self-referential sentences. Examples:
      This sentence no verb.
      Thit sentence is not self-referential because ‘thit’ is not a word.
      No language can express every thought unambiguously, least of all this one.
      This sentence every third, but it still comprehensible.
      This line from Shakespeare has delusions of grandeur.
      The rest of this sentence is written in Thailand, on
      This sentence does in fact not have the property it claims not to have.
      Hofstadter’s Law says it always takes longer than you think it will take, even if you take into account Hofstadter’s Law.

  5. Classic.

    I first saw this about 50 years ago, to date myself.

    SPOILER, SPOILER, SPOILER, SPOILER, read on only if stuck:

    What is the effect of dividing by (x – y)?

  6. “Basically, the top shape is not an exact triangle…”

    um, i can’t be the only person wondering what an “exact” triangle is, or how it differs from a “triangle.”

    1. An exact triangle would be one which wondered about the difference between an exact triangle and a regular one. Or is that a precise triangle, I forget?

      See you at the next meeting of the Pedantic Society (soon to be renamed the Society of Pedants – thank you, Pat Harkin).

    2. An exact triagle is a 3-sided polygon. A polygon is a plane shape bounded by straight line segments. The shape is a quadrangle, it is bounded by four line segments.

  7. “Gödel, Escher, and Bach “, many thanks for reminding me about that. I need to look it up and read it again.

  8. Simon:
    We ran out of milk, and my wife asked me to go to the supermarket. See said “Get a quart carton of semi-skimmed. If they’ve got any free range eggs, get 12.” For some reason she was cross when I brought home three gallons of milk.

    1. f(x)=2x+1 walks into a bar and asks for a drink.
      The barman says “I’m sorry, we don’t cater for functions.

  9. At last a Friday puzzle that satisfies the pedants. It allows us all to present our answers. This is good as it prevents cheating.- it stops people waiting until Monday and then claiming they knew the answer all along. Luckily many of us are too honest to take that approach.

    It took only a few moments of deep introspection to arrive at the answer.


    Others I am sure will have spent many hours to produce different solutions. Their time is their own so we should not judge.

    1. Brilliant, brilliant — I genuinely laughed out loud, Barry. That’s the best laugh I’ll get at work today, I’ll be bound.

      I’ve come to love Barry’s stand-up routine. Part Beckett, part Douglas Adams.

    2. I’m sorry, Barry, but as a pedant I must point out that your answer is wrong. The question asks ” Can you create a 10-digit number, where the first digit is how many zeros in the number, the second digit is how many 1s in the number etc. until the tenth digit which is how many 9s in the number.” The answer must therefore be either “Yes” or “No”, although I suppose “I don’t know” might get you half marks.. Since you were able to complete the requested task, your answer should have been “Yes, I can.”

    3. “It took only a few moments of deep introspection to arrive at the answer.”

      I think Baz means “Googling”

    4. @No registration

      I find your lack of faith disturbing.

      Some simple searching on this very site will show you that I can always solve these problems. Ofttimes I find intuitive solutions that the masses (who do use google) angrily disagree with.

      Yet my answers have never been dislodged as correct.

      There are those who can look things up. And there are those of us who can look directly into reality. That is a learnable skill not available on google. If you wish I can teach more.

    1. Oh Dear
      As I have noted before, being in the Lord Manley Fan Club means we are not the sharpest tools in the box

    2. @TLMFC – are you certain that you’re even a tool?

      I have a Lord Manley Fan, a big one from the 1930s. I wasn’t aware that there was a club for them. When’s the next get together? – I’ll bring mine along if I’m free.

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