OK, two puzzles this week.

First, you have a large box, a small box, and three stones…..

How can you put the stones in the boxes to ensure that each box contains an odd number of stones?  There are at least two solutions – can you find both of them?

Second, this nice puzzle was submitted by my good friend Mr Pythagoras.  Take a look at the diagram below….

Can you figure out the radius of the circle?

As ever, please do NOT post your answers, but do feel free to say if you have solved the puzzles and how long it took.  Solutions on Monday.



    1. For the first puzzle there is in fact a third sollution possible, but that one is very cumbersome. You would also need more objects to manage that.

    2. Or else you’d have to have really good balance (a box on its edge and balls on the edge of a box)? Is that the third one you’re thinking of?

  1. First one, almost while reading it. Although I am not very sure if I am right.
    Second one, just on the first glance at the diagram.

  2. Hmm found 2 clear answer for the first one in about five seconds, then another three debatable answers in another ten seconds.

    The second puzzle took about twenty seconds before the penny dropped, lol..

  3. I have two ways to solve the first one and it took about as long as it took me to read it then rethink it just to make sure….both work for me anyway…lol…now the second one I went into a mathaphobia panic and my brain refuses to even try…..I’ll have to keep trying…..got til Monday anyway…..

    1. kick self in fanny, jump…kick self in fanny, jump….OK stop kicking self around……after looking back at puzzle and trying not to be freaked as it contains math….I now think I have worked it all out and have a fixed answer took about 10 min. including panic time…lol….

    2. I think the second one is all about “mathaphobia” panic. Those who have it or those who see things more complicated than they are, are going to have difficulty on that one. I almost succumbed to the second one too… fortunately, I had not time at the time to think in complicated approaches and that was my luck.

      Still, you beat me in 5 minutes. And I thought I had woke earlier… wait!, are you past your bed time? naughty, naughty… 🙂

    3. Australian English is the same as British English in this respect. And we’re obviously right, because what else could a fanny be other than the part of this http://is.gd/fRbmu that most resembles one of these http://is.gd/fRbpy ? That’s how I’ve always thought of it.

      (The above links do not contain photographs of nudity. Just drawings. So it’s art.)

      I saw the trick behind the second puzzle as soon as I had printed it on to paper and carried it to my bed so that I could lie down while thinking about it. Something about relaxing made me twig.

    4. @Tony…..OMG! No didn’t had to look it up….lol…should have just said a$$ or butt I guess…lol…

      @Joao Pedro….I bet others will too or they will be kicking their butt….and yes it is past my bed time and that is where I’m going now…I’ll have to come back later to read the rest of the post as there are over 80 now….too sleepy to read! Catch everyone later and have a great weekend too!

    5. BTW, I have a comment just after Tony’s that’s “awaiting moderation“. I forgot that on WordPress, comments are moderated by default if they contain two or more links. (On my blog, I’ve set that number a little higher.)

      Don’t know how often Richard checks these things. Oh well. It’s not as if I said anything important.

    6. well I guess I should of said “derriere” but American English the term is used when referring to the part of the body that one sits upon….and I mean NORMAL people….so if anyone of you want to debate this you will be marking yourself as NOT normal……LOL….he he he giggle….slap thigh….the body part that is known as one’s leg or the leg that meets at knee and up to the hip…and so on….lol….I will not use the term “fanny” ….lol

  4. a couple of seconds for one answer to the first puzzle. Another couple of seconds for another answer.

    Went to draw the second puzzle out and spotted the answer as soon as I’d stopped drawing and actually thought. Nice one.

  5. First one instant. Second one… erm… nope. Well I thought I had an answer straight away but as I thought about it more I realised I was wrong and got completey confused.

  6. Hm found 2 solutions for the first one in less than a minute, and have bekame aware of the solution of the second one after playing around with Pythagoras for a minute

  7. First one is trivial.

    Second, I confess I’m having trouble with. In fact I don’t think it’s possible.

    Consider a slightly larger circle. Now visualise the green 3″ length – it’s slightly shorter in relation to the (vertical) radius. Now also visualise the blue 8″ length and have it still meeting the horizontal radius. The blue line will be a little steeper. In all respects the diagram can be drawn again, the various lines and right angles simply appear in slightly altered positions.

    So, what am I missing?

    1. That was my first reaction too, Ian. You are actually wrong, there is no indeterminacy: from the information provided you can calculate all the lengths and angles of all the lines shown.

      As with many of these problems, the solution can be found by constructing some additional line or lines. The trick is to draw the right ones. Then it becomes very easy, embarrassingly easy.

    2. Ian has a point: you can enlarge the circle in such a way that each of the “known knowns” holds true. I think those claiming that they did it in seconds either don’t understand the problem or are making assumptions about the remaining sides of those triangles.

  8. First one done instantaneously. Second one, after an embarrassing amount of time manipulating algebra I succumbed to getting the ruler out which confirmed my instinct, yet I’m putting my inability to find a proof down to early Friday morning hangover.
    At least i’ve got something to work on to look busy at work!

  9. Yay, really quick this week. The first is a bit of lateral thinking and the second was in my year 9 SAT test 15yrs ago. Happy me >:O)

  10. i think i have solved the first one,,, easy though….
    the second one took the time,, never realised it was a clear case of observation!!! nice ones….

    1. That’s makes one solution. What is another, to make at least two? Richard asked to find both…

      (but you are right, I think we are missing one word in the description of the puzzle. As it is written, there are at least 3 solutions)

    2. No… Actually that is a -bad- sollution. It is a bit misworded maybe, but you should put all three stones in the boxes…

    3. @Bonaparte, “put the stones in the box”, taken literally, doesn’t mean “put some of the stones in the box”. It can only mean “put all the stones in the box”. The only way out is to not take it literally.

      If there were 3 puzzles and you solved 2 of them, you could not truly say that you had solved the puzzles. If I gave you 100 chocolate biscuits and you ate 99 of them, you could not truly say that you had eaten the biscuits.

    4. hmmm… “put the stones” is indeed more like “put all the stones” than “put some of the stones”. But the puzzle stressed the objective of having odd number of pebbles in the boxes in the end… when associated to the first part “How can”, I think it introduces some freedom in the interpretation of the fragment “put the stones”.

      To use your example, lets say each biscuit weights 50g. If you ask “You have 100 biscuits; How can you put the biscuits in the body?”, the answer will be “eating all of them”. If the problem is “You have 100 biscuits; How can you put the biscuits in the body to ensure a weight gain equal to 4Kg?”, the answer will now be “eating 80 of them”. Maybe I’m making a cultural blunder but in the second case, I have the impression that the word “ensure” creates an imperative on the “4Kg” objective that overshadows the previous solution of eating all the biscuits, at least on informal conversation. Few persons would be enough suspicious to think they are obliged to use all the biscuits (unless they thought they were facing a difficult or serious puzzle… but is precisely the seriousness of the puzzle that Bonapart is doubting) or to ask about that. “How to put the pebbles to ensure odd numbers in each box”, allow me to think in the solution “one pebble in each box”, then…

      PS.: This freedom of interpretation is essential, even because if not, I think many are going to contest the solution Monday, at least the solution I’m thinking, on the grounds that, a literal interpretation of the problem doesn’t allow it. But that freedom runs farther than that. For example, do you think anything in the puzzle description forbids the puzzle solver from breaking a pebble in two? You know, and I know that this is not the “solution”, but the truth is, doing that, we have an additional arbitrary number of extra solutions…

    5. Yes, in everyday life there are plenty of situations where sentences are not taken (or meant) strictly literally. I would argue, though, that a puzzle is not everyday life, and that if a puzzle seems too easy because you’re not taking it literally enough, then it’s rather obvious what to do about that.

      Interestingly, had the puzzle said “How can you use the stones to ensure that each box contains stones?“, I would agree that you needn’t use all of them. I think that’s because talk of using them emphasises the stones as a group (to use any one of them is to use the group) whereas talk of putting them in boxes emphasises the individual stones, but it might be interesting to hear Neal Whitman‘s take on this.

    1. Go on, BA, you can do it! Not really a lot of math involved, to be honest…
      Use that critical brain of yours. When you just look for the radius, you won’t miss it.
      Happy hunting!
      R 😉

    2. I know which side of the triangle I need to calculate in order to get the radius, but you’d need to do sine and cosine and all that nonsense… Well, I managed to forget how to do that as soon as I graduated high school. 😛

    3. Oh, wait. I cheated and used my fingers on the screen to confirm a suspicion. If you do it that way, it IS easy. 😛

    4. Dragon, I’m at work right now and cheating with an online anagram solver doesn’t seem to work. Will have more time to work on it tonight.

    5. I used an online anagram maker to construct it, though not until after I gave up trying to construct it by hand.

      It’s probably impossible to solve the anagram without already knowing the answer; even if you stumbled upon the correct sentence, its applicability would still not be obvious at first glance. Basically, it’s a reminder of a situation in everyday life where probably most of us have encountered a geometric fact that is the key to the puzzle.

    6. By the way, looking at it again, it finally clicked WHY my brain was giving me that suspicion. My conscious mind is a bit denser than my unconscious mind, it seems.

  11. Solved the first one in the time it took to read the question. The second one took a bit longer as I was racking my brains for trigonometry formulae, before the answer suddenly presented itself. Love it 😉

  12. Spent ages puzzling over the Pythagarus problem before seeing the shortcut answer.

    Why though, Richard, are you measuring in imperial!? Didn’t we go metric 30ish years ago? And aren’t metric measures part of the language of science. And aren’t you a professor of Psychology (which is almost a science) ??

  13. @Bonaparte I’m assuming that Richard intends that all the stones are placed in the boxes – there aren’t any left roaming free.

    I have a sneaky solution that takes advantage of the fact the boxes are different sizes.

    As for the second, it took a few minutes before the penny dropped and I disregarded the red herrings (helpful hint: you don’t need to know any trigonometry whatsoever to solve it)

  14. Less than 1 second for the first. Soooooo obvious. (Loving this feeling of superiority, quite a new one for me!)

    The second one? Love it. Sooooooo elegant. Had to think a bit (30 secs?)
    As I started to apply trigonometric principles and pythagorean postulates, I realised that the answer was staring at me in the face. Quite wonderful.

    A very happy Rob today. 😀

  15. Ooh, I like the second one a lot! First one v quick, second one too much maths training had me thinking of soh cah toa and parallel angles etc. until I spotted it!

  16. I showed the second to one of the math teachers I work with, and he started with the Heavy Thinking – “Okay, r-3 squared, divide that by….” And then the solution struck me out of the blue. A fine math moment!

    Then he got a very nice solution to the first problem. Rock on.

  17. 1st one very quick. Second one still nothing. Any hints? Tried applying trig, but comments suggest that’s not the route.

    1. You’re going to slap yourself when you see it 🙂 It’s one of those hiding in plain sight things.

    2. Got it now! How silly. However, it’s quite surprising. It’s led me to realise an assumption I’d made has to be false.

  18. The misdirection in the second one is a work of art. I was looking at it for a while, thinking about similar triangles and square roots before the fog cleared. Nice one.

    I came up with two solutions to the first pretty quickly (while reading it).

  19. First one was a few seconds.

    The second one had me head scratching for ages, had the calculator out and was working through the figures, came up with the answer. Then looked through the comments and realised I’d wasted a good half hour. Had another look at the second one and saw the obvious.

  20. First one was pretty quick, text and picture guided me there.

    Fortunately for the second, I saw the answer before digging too deep (I might aswell have done just that, hehe).

    I’m not a fan of trying to quantify the time I spent, so I’ll just say half a minute for both 🙂
    Funny how a minute of reading and thinking can be so amusing… I Guess I tricked my brain into thinking I accomplished something great 😛

  21. It took me about 10 seconds to get the trick to the second one but it made me laugh when I did so that’s always good. Got the first one straightaway.

  22. About 5 seconds for the first. Rather embarrassingly, spent several minutes looking at the second one and writing down equations before realising the answer was staring me in the face.

  23. Ah, the benefits of a higher mathematical education. Immediately dived into algebra without thinking. Contemplated doing something clever with limits. Then realised the answer was *really obvious*.

  24. Ha, the second one is lovely. I took a couple of seconds to get both answers to #1, then a good five minutes of thinking and scribbling before giving up on #2, coming back to read the comments and finally working it out!

  25. The first puzzle felt almost instant. The second one still puzzles me, so I measured the radius with the edge of an envelope.

  26. The first one, a couple of seconds. The second one, I cheated and looked up the answer… and immediately said “Aaah, you idiot!” out loud. Nice puzzle, can’t believe I missed that!

  27. Solved the first puzzle really easily- was visualising whilst reading the question, then shocked because that meant I’d solved it!
    Second one- well have an answer and know all the compicalted theories, but am waiting for Monday because I know there must be a simpler explaination. I think I am part way there but am just not quite seeing it.

  28. Got the first one while reading it, the second one I started to write out the algebra before another part of my brain kicked in. Nice piece of misdirection.

  29. 30 seconds for he first one, two minutes for the second one, because first i did consider whether its now the right point of time to try to do my first “equation with two unknowns”. Hm.., and measuring by a ruler is not allowed for a proper mathematical result. Then i looked a bit around on the graphic to find some essential Points and then it made “click” !
    Very genius.

  30. The first one was done in just a second, I’ve seen that one before.

    For The second one, I started to draw the circle on a piece of paper, putting labels on the unknown lengths, setting up a couple of Pythagoras-equations, then I saw the answer staring me in the face and I felt rather stupid 😀

    So about 2 minutes for the second puzzle, possibly fooled by too much education.

  31. Saw the first one in 1 sec. I now feel brainless for not seeing the second one anything like so quickly. It’s obvious now, several hours after I first saw it. What a moron.

  32. About half a second for the first one, haven’t really looked at the second yet. But did I do something wrong? Surely the first can’t be this easy…

  33. I have four authentic solutions to the first puzzle — the two you’re thinking of, one slightly dubious but defensible, and one purely for fun.

  34. The two easy solutions to the first puzzle came… er, easily.

    I was messing w/Pythagoras for the circle puzzle, when suddenly Mr. Obvious dropped by and kicked me in the shins (no fanny references here!).

  35. 2 answers for the first whilst I was reading it, however the second is an excellent piece of work.

    I did get it very quickly, but I might well not have, if that makes sense? I can imagine myself taking hours on it had I not happened upon it by chance.

    Splendid misdirection.

  36. A few seconds for the first answer of the box puzzle, then maybe 10-20 seconds for the second solution.

    Haven’t tried the radius puzzle yet.

    1. Second one took a bit longer. I had a good hunch that the answer was “X”, followed by a Well, duh, of course it’s !”X” realization.

  37. There are two completely obvious and trivial solutions to the box problem that I saw within a second of reading it.

    The circle problem: the answer is yes but I’m too lazy to actually work it out right now.

  38. First one almost instantaneous (2 solutions), second one took me a minute before I realized its just an exercice in misdirection. Should I mention I’m a university math professor… ?

  39. Kept me amused for about a minute. First one is a no-brainer, second one is simply GCSE (or the lower end of ‘O’ level) geometry!

  40. First one – got one solution. Second one – took a few seconds, then kicked myself for not getting it faster.

  41. Both instantly. However, it occurs to me that in absence of stating “all the stones” there are at least 2 additional solutions to question 1.

  42. yes, we know you are all soooo smart. sure, some people get certain answers faster than others, but I’ve never read an answer to a friday puzzle that sounded something like:

    “oh, the second one was rather tricky. took me about 10 minutes or so”.

    no, it’s always:

    “harrrr, the second one was rather tricky. took me about 30 seconds. solved the first one instantly before the page finished loading”…

    is that some self-esteem thing?

  43. Starting counting up answers to the fist one right away and got more than three.

    Mathphobia kicked in as I saw an algebraic answer forming before the light went on with the geometric answer. then, “Doh!” came the answer.

  44. Took me about 5 secs to figure out the second one, and another 5 minutes to figure out why, brilliant feeling when the penny dropped though!! :):)

  45. First one was almost instant, but on the second I spent an age dithering over whether or not RW was likely to set a puzzle that required a scientific calculator — eventually decided not, and realised the answer 🙂

  46. First one 2 answer very quickly, have a horrible feeling there is an other solution though….

    The second took about 10 minutes and then it was a real duh moment 🙂

  47. Put the small box inside the large box, then put all the three stones inside the small box or split each stone into two pieces to get a total of six pieces then put 3 inside each box!!

  48. Just realized not to say the answer even thought it isn’t what came out, but some one must say the answer this isn’t fair 🙂 or has it been said? anyhow as they say”better to be late than never”

  49. um, I guess I’m stupid, but, I can’t find any solutions “on Monday” for any of these puzzles.How do I get in on the Secret Knowledge? Do i need to join a cult, or buy Moen products, or be a Mormon, or a Fan Dancer, or a member of some other obscure cult I’ve never heard of?

    Please advise, without mentioning fACEbOOK.

    -Charles in San Diego

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