9 ms·
Disappearing Bicyclist – Sam Loyd (1906)
- Luc 5y agoSee also the Missing Square puzzle: https://psychology.wikia.org/wiki/Missing_square_puzzle https://psychology.wikia.org/wiki/Missing_square_puzzle Don't read the text if you want to figure it out yourself.
- samcheng 5y agoThere's a good Numberphile video explaining this illusion: https://www.youtube.com/watch?v=cE44nr4d3iY https://www.youtube.com/watch?v=cE44nr4d3iY (Hint: look at the heads)
- rhn_mk1 5y agoThis one includes flags as an easy hint.
- kazinator 5y agoThe flags don't really "do" anything. In the one configuration, every flag is held by a boy. In the other configuration one boy has no flag, and there is a severed hand holding a flag.
- rhn_mk1 5y agoThat's actually wrong for the other configuration. In one, every boy has a flag. In the other, there's one boy with no flag, and 12 boys with flags. The appearance of a boy with no flags is an insight.
- kazinator 5y agoNone of the flags straddle the boundary between the two discs, so it is obvious that no trick is present affecting their apparent count: there are always 13. The diagram's inner disc contains one complete torso with arms and hands, which holds two flags in either configuration, so "every boy has a flag" is only true if you actually mean "has at least one flag", not if you mean "has one flag". Since the number of flags is not affected by configuration, one boy not having a flag is unsurprising. There are always 13 flags, so if there are apparently 13 boys, and one is still holding two flags, then someone must necessarily not have a flag. (In the A configuration, there are actually two boys with no flags. One clutches his head with both hands, holding no flag, and the other, next to the A, is flanked by a floating, detached hand which holds a flag. Therefore, that boy has no hand, and no flag. This severed does not seem essential, either; I believe the picture could be somehow repaired so that the detached hand is reattached. Then in the A configuration would be exactly one boy with no flag, as you say.)
- rhn_mk1 5y ago> it is obvious that no trick is present affecting their apparent count I see you found the insight from the hint: flags are governed by different rules than boys.
- bloak 5y agoUnfortunately this doesn't work with banknotes ... unless you can find lots of people who are willing to accept a banknote that appears to have been torn into two pieces and then stuck together again with sticky tape, with the line of the tear being a weird curve that just happens to cross both serial numbers in roughly the same place.
- dane-pgp 5y agoIt does work with chocolate, though... http://mathandmultimedia.com/2014/07/22/explanation-infinite-chocolate-bars/ http://mathandmultimedia.com/2014/07/22/explanation-infinite...
- Someone 5y ago‘Nobody’ looks at banknotes. Also, historically, banknotes were torn and repaired more often. Because of that, you can just cut a 1/10th width strip out of 9 banknotes and glue them together to make a 9/10 width tenth banknote. Examples: https://books.google.com/books?id=e7QzAQAAMAAJ&pg=PA318&lpg=PA318&dq=fraud+cutting+of+piece+of+banknote&source=bl&ots=uyC6827Y59&sig=ACfU3U0J0707LQmgRRlNp2TwWFzdtcpw1A&hl=en&sa=X&ved=2ahUKEwjy2bneiZjzAhXQ_aQKHWXKDtEQ6AF6BAhFEAI https://books.google.com/books?id=e7QzAQAAMAAJ&pg=PA318&lpg=... (1804) https://books.google.com/books?id=osjhDwAAQBAJ&pg=PA114&lpg=PA114&dq=fraud+cutting+of+piece+of+banknote&source=bl&ots=zHs2ppbcPh&sig=ACfU3U1XHk1eWurrbH4fI4Uei28Q8MexpA&hl=en&sa=X&ved=2ahUKEwjy2bneiZjzAhXQ_aQKHWXKDtEQ6AF6BAhDEAI#v=onepage&q=fraud%20cutting%20of%20piece%20of%20banknote&f=false https://books.google.com/books?id=osjhDwAAQBAJ&pg=PA114&lpg=... (1850s) A more tricky recent variant replaces the cut-out part with a fake part: https://bc.ctvnews.ca/can-you-spot-the-fake-splice-and-tape-money-scam-in-b-c-1.3415262 https://bc.ctvnews.ca/can-you-spot-the-fake-splice-and-tape-... (why do these criminals take the effort and risk of creating a fake fiver? I would discard the remains of the fiver, and spend all effort on improving the technique for transplanting the hologram to the fake 100) Of course, this works better with small denominations, if only because people expect larger denominations to look newer. The risk of getting caught also is fairly large, I think, but a good criminal can feign innocence, claiming to have gotten the note elsewhere.
- pc86 5y ago
- Chinjut 5y agoPut it in the B configuration. There are 12 boys, which can be thought of as 24 halves bundled in pairs: a half on the outside of the circle, and a half on the inside of the circle. (Conveniently, also, each bundled pair of halves includes one half with a flag and one half without a flag). Some of these halves are more substantial looking than others, mind you. Rotate it to the A configuration: There are still 24 halves, bundled in pairs. But now we count it as 13 boys instead of 12. Why? Because in the bottom left, two of the halves that got paired together are both flag-and-head halves, so even though that was just two halves in our original count, it feels like the two of them bundled together should count as two boys instead of just one now. (Correspondingly, in the top right, two of the halves that get bundled together in the A configuration are both non-flag halves. But the result still seems sufficient to call one full boy instead of zero boys.)
- thehappypm 5y agoWhich boy is gone? :)
- lowbloodsugar 5y agoMy guess: The problem is one of discrete values. You are asking about "a boy", but there are no "a boy"s. There are only fractions of boys. The total sum of "boy fractions" is the same, but the number of pairs of "boy fractions" that meet the distinction of "a boy" changes. "A boy" does not vanish because there were never "a boy"s in the first place. One might have better luck with "boy heads", and you can see that at roughly 5 oclock there is a "boy head" that turns into a "boy arm". So part of the trick is that some of the "boy fractions" change in your interpretation. Without the trick, you would reasonably say, "Wait, there is still a boy head there! So there are still 13 boys!"
- pis0m0jad0 5y agoI also couldn't help but notice this after reading your comment, but in the image it seems pretty clearly that the A configuration is pretty incoherent. The 5 o'clock boy you mentioned, for example, clearly has a sleeve for the right side of his face, the boy at 2 o'clock is also missing a large chunk of his head, and of course the two boys at 8 o'clock overlap, but, now that I look again, in an incoherent way. To me it seems that the answer to where the boy goes is simply that "the A configuration is invalid, but slightly so that we might not notice"
- nixpulvis 5y agoI was pretty confused until I noticed the bottom right head turning into an arm.
- ypcx 5y agoThis is the most elegant explanation.
- jmkd 5y agoCan only count 13 boys whether from point A or B..what am I missing?
- erikerikson 5y agoMoving the arrow on the center disk of the two physical disks would rotate the insides of the circle and realign the person parts. Took me a second too.
- jmkd 5y agoThanks. Doesn't help me understand it (nor do any of the above comments) but does help me see it.
- kthejoker2 5y agoThe little line at the top is actually an interactive slider control so you can rotate the inner disk from point A to point B. Definitely not an intuitive interface.
- dang 5y agoAnybody want to figure out the year?
- kthejoker2 5y agoThe puzzle was originally conceived in 1906. Love me some Sam Loyd, there is no greater chess puzzler in my mind.
- egypturnash 5y agoIt's the less-racist remix of his 1896 "Get Off The Earth" puzzle, which contains twelve or thirteen "chinamen" in the exact same poses around a globe. http://www.marianotomatis.it/blog.php?post=blog/20110715§ion=english http://www.marianotomatis.it/blog.php?post=blog/20110715&sec... Whoever currently owns his name is still selling it, except now it refers to "warriors". https://www.samloyd.com/product/get-off-the-earth-card/ https://www.samloyd.com/product/get-off-the-earth-card/
- acomjean 5y agoVideo demo of the "Get Off the Earth" which is a little jarring. It looks to be the same as a bike one. The video has lots of other versions/variations on these puzzles: https://youtu.be/KdwJQbxLFHI?t=27 https://youtu.be/KdwJQbxLFHI?t=27
- dang 5y agoAdded above. Thanks!
- fsckboy 5y agoin a sense, there are only two boys, an inside boy and an outside boy, and where those two boys are together (at around 8 o'clock) you actually don't see 100% of either of them, yet you count it as two boys in the A position, look at around 8 oclock where the two boys next to each other. look at the "inside" boy, and then move clockwise and you just see just his foot on the inside, continue and you see a leg, going all the way around you can see almost all of the "inside boy" emerge. in a like manner, look at the outside boy of the two boys together, and proceeding counter-clockwise you see the foot, the leg, etc as almost all of the outside boy emerges. it reminds me of a trick where you take a stack of dollar bills and slice a thin strip from each one, with the slice taken on each successive bill moving across, and tape each bill together again. At the end, tape all the little strips together and you have an extra bill!
- unyttigfjelltol 5y agoYeah, I don't understand why moving the two boys side-by-side is considered a trick. It's much harder to understand how to be misled than to discover the solution.
- pagade 5y agoCounting clockwise from A, 5th bicyclist head disappears into hand.
- nicwolff 5y agoSimple, the "B" boys each average 1/13 bigger.
- tommoor 5y agoFor anyone else that missed it – there is a green slider at the top to rotate the inner disk.
- Igelau 5y ago> Where does he go? 1906. Typhoid probably got him.
- DoreenMichele 5y agoPart of the trick is that the boys are split into two parts and progress from more inside to more outside along the wheel. Since the unit "a boy" is based on real world experience where boys can be different sizes and especially since how much space a boy takes up in an image can vary depending on angle, we mentally accept that this mashup of different amounts of body parts adds up to "one boy" rather than thinking "No, that's just 85 percent of a boy and over there we have 105 percent of a boy!" As someone else noted, we accept in one position that two partial boys each count as a whole boy and then in another position we accept that they each count as just part of a whole boy when paired differently. (Edited in hopes of improving clarity.)
- jonahx 5y agoTo add to this, the boy at 3 o'clock, whose head is split in two equal pieces, is the key to the sleight of hand. Does his head stay or does it move? If it moves, the boy at 2 o'clock should count as 2 heads in position B. If it stays, then he should count as 2 heads in position B, since the head below him moves up. But because it's ambiguous, we count it as both moving and not moving.
- foobarian 5y agoYeah the size of the boy half goes from say 5% to 95% around the circle, on both sides. In the 12 boys configuration the halves are paired correctly to add up to 100% sized boys. But in the other, there are 12 boys smaller than 100%, and then at 8 o'clock two 95% pieces appear together so we round them up to two. Neat illusion.
- Someone 5y agoLooked at it differently, we go between two states: 1+12, 2+11, 3+10, ..., 10+3, 11+2, 12+1 (12 pairs, each adding up to 13) and 0+12, 1+11, 2+10, ..., 10+2, 11+1, 12+0 (13 pairs, each adding up to 12)
- kazinator 5y agoYou can isolate the discrepancy simply by considering just the bottom left quadrant. When you flip configurations, an extra boy is shifted into the sector, forming the boy pair, so you can now count three boys in that sector instead of two. What is shifted out is just a fraction of a leg, so there is a net gain of one boy. The remainder of the circle is constructed so that there appears is no net change in the number of boys there. The fraction of a leg which is shifted in replaces a fraction of a leg, and the boy which is shifted out is replaced by a boy. But note that this is a paradox: you can't shift a leg into one end of a register, such that a person is shifted out of the other end, and yet have the person count in that register remain the same! That's like shifting a 0 into a bit register, such that 1 is shifted out the other end, yet the parity remains the same. The subterfuge is that in configuration B, there is a Siamese twin body with two heads (A + 2 clockwise). The casual observer glosses over this, because the heads are fused together quite well, and counts them as one boy. When the configuration is switched to A, this twin head is separated. There are no more Siamese heads sharing a body, and that separation is what keeps the apparent head count the same, even though a head is shifted out in compensation for a leg shifted in. Another noteworthy feature is that when the boy pair is formed, it is also Siamese twins, sharing one leg. So in the A configuration we have Siamese twins sharing a leg, which gets counted as two people, but in the B configuration, we count Siamese sharing an entire body as one person, because the heads are so well fused as to almost look like one. If you consistently count all instances of Siamese twins as either two people, or else as one person, then the count is consistently 13 or consistently 12. The paradox here rests in a kind of "visual equivocation fallacy". The equivocation fallacy is that, during the course of counting boys around the circle in the two configurations, the observer's definition of whether a Siamese twin is one person or two changes, due to visual deception. Well-fused heads count as one person, but joining at the hip and sharing a leg counts as two people.
- jmkd 5y agoMuch appreciate the care you took to write this but multiple readings and flipping configurations later I'm as lost as I was when I first counted the discrepancy. Can it be described in a simple sentence?
- 5y ago
- bondarchuk 5y agoA similar puzzle is Magic Microbes (ctrl-f it) from this page: www.karlsims.com/puzzles.html
- kypro 5y agoReminds me of the infinite chocolate trick, https://www.youtube.com/watch?v=qnpugKVitl0 https://www.youtube.com/watch?v=qnpugKVitl0
- bigmattystyles 5y agoIs this just a visual version of the missing dollar riddle? https://en.wikipedia.org/wiki/Missing_dollar_riddle https://en.wikipedia.org/wiki/Missing_dollar_riddle This was my grandpa's go to riddle for his grandkids (but with Francs). :-)
- crdrost 5y agoNo... the missing dollar riddle asks people to confuse cash inflows and outflows by changing the subjective interpretation of a middleman, who starts out being an outflow (he/she is "part of the business") and then is treated like an inflow (he/she is "one of the money-havers"), it's essentially a linguistic puzzle in an accounting context, one person can be referred to in two different ways. This one is closer to the missing square puzzle, https://en.wikipedia.org/wiki/Missing_square_puzzle https://en.wikipedia.org/wiki/Missing_square_puzzle . This is a calculus puzzle, you have approximately 13 boys "missing a tiny slice" of themselves versus 12 boys with "an extra little slice" of themselves... it's more clever than just "we take the slices and reassemble them", it's "the slices slowly sweep across the boys' body so that when we slide the circles we essentially accumulate a whole 'second boy' inside the internal circle." Sort of like how the missing square puzzle has a clever way to have two almost-parallel lines which are not parallel hiding a long skinny rhombus containing the extra area, but the rearrangement exposes it in a much more visually arresting form as a whole missing square.
- scollet 5y agoAh, an early compression algorithm.
- Igelau 5y agoThe boy who occupies the 4-5 o'clockish space in Configuration A appears to have a fist where half his face should be. I nominate him as the vanisher since there's no fist-face in Configuration B.
- zem 5y agoi learnt about this sort of trick from martin gardner (who possibly coined the term "geometric vanishes" to describe them). he was how i first encountered sam loyd too.
- undoware 5y agoI de-mystified this by considering the boys as a spiral. Move the wheel, the spiral increases by one node. In summary, this puzzle is puzzling because we are given a wheel and keep thinking 'circle' instead of, "is it actually a wheel, or not?"