[HN Gopher] Disappearing Bicyclist - Sam Loyd (1906)
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Disappearing Bicyclist - Sam Loyd (1906)
Author : alizauf
Score : 86 points
Date : 2021-09-24 15:51 UTC (7 hours ago)
(HTM) web link (www.geogebra.org)
(TXT) w3m dump (www.geogebra.org)
| DoreenMichele wrote:
| Part 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 wrote:
| To 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 wrote:
| Yeah 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 wrote:
| Looked 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)
| fsckboy wrote:
| in 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 wrote:
| Yeah, 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.
| samcheng wrote:
| There's a good Numberphile video explaining this illusion:
| https://www.youtube.com/watch?v=cE44nr4d3iY
|
| (Hint: look at the heads)
| rhn_mk1 wrote:
| This one includes flags as an easy hint.
| kazinator wrote:
| The 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 wrote:
| That'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 wrote:
| None 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.)
| kazinator wrote:
| You 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 wrote:
| Much 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?
| kazinator wrote:
| Maybe this can help. Look at this cropped image:
|
| https://i.imgur.com/egAouqW.png
|
| It looks like three boys; but there are four heads here!
|
| The top boy has about 1/3 of a head coming from the inner
| disc.
|
| The bottom boy has about 1/3 of a head coming from the outer
| disc.
|
| The middle boy has a 2/3rds portion from each disc: basically
| two fused heads.
|
| This is in the B position. When the inner circle rotates
| clockwise to A, these pieces are redistributed. The top boy
| still gets enough of a fractional head from the (cropped
| away) previous boy. (Not quite. More precisely, in position
| A, the top boy gets no head material at all from the previous
| boy, yet has enough head material from the outer disc that he
| still has a complete head.)
|
| The top boy's 1/3 of a head goes to the middle boy, where it
| combines with the outer 2/3rds to make one more or less
| normal head now: no more double head here. The middle boy's
| previous inner 2/3rds moves to the bottom boy, where it also
| makes a normal head.
|
| So, the double head being gone in configuration A, we now
| count 3 heads rather than 4.
|
| Since 4 heads became 3: an extra head shifted out of here in
| the direction of rotation, and that's what supplies the extra
| head for the other boy in the bottom left, who now gets
| joined at the hip with a twin, sharing a leg with him.
|
| In one sentence:
|
| _Two-headed body at A+2, wrongly counted as one boy,
| corresponds with hip-joined Siamese twins at A+7._
| erikerikson wrote:
| Consider that in state A there is a spot with two boys.
| Consider that in state B no such spot exists. Why is that?
|
| Broadly, in state A from that position, the boys are on the
| outside AND inside. As you progress around the circle, they
| transition from inside to outside more or less smoothly. By
| rotating the most inside boy to be aligned with the most
| outside boy, you haven't mismatched any of the other boys
| enough to invalidate them as part of your count. Consider
| that in state B you discount the boy part in the position
| where in state A you counted 2 boys.
|
| If that doesn't do it for you, can you describe what helped
| and where I added or maintained confusion?
| Igelau wrote:
| > Where does he go?
|
| 1906. Typhoid probably got him.
| pagade wrote:
| Counting clockwise from A, 5th bicyclist head disappears into
| hand.
| bondarchuk wrote:
| A similar puzzle is Magic Microbes (ctrl-f it) from this page:
|
| www.karlsims.com/puzzles.html
| Luc wrote:
| See also the Missing Square puzzle:
| https://psychology.wikia.org/wiki/Missing_square_puzzle
|
| Don't read the text if you want to figure it out yourself.
| zem wrote:
| i 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.
| Chinjut wrote:
| Put 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 wrote:
| Which boy is gone? :)
| lowbloodsugar wrote:
| My 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 wrote:
| I 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"
| crdrost wrote:
| Something like that, yeah.
|
| If done right, the idea is that in the one configuration
| you have thirteen 24/25ths of a boy, so thirteen
| 96%-boys. You hide this as each one being slightly
| skinny.
|
| In the other configuration you have twelve 26/25ths of a
| boy, so twelve 104%-boys. You hide this as each one being
| slightly fat.
|
| The danger is that if you make the drawing too detailed
| and gorgeous, someone will be able to look at a little
| detail like eyes or so, if one of these 4% slices
| contains an eye then you may end up with boys with 1 or 3
| eyes, something that any looker would say spoils the
| illusion.
|
| You have a couple options there.
|
| - People will accept a "dead zone" where the two moving
| parts interact, you can try to locate an eye inside the
| "dead zone" so that you don't trigger this W-T-F moment.
|
| - Use cartoonishness/ambiguity. So maybe this dot means
| "eye" on this cartoon but "freckle" on that cartoon,
| similarly by locating the exact eye on the border between
| the two, maybe half a line goes from being "long
| eyelashes" to being "the middle of a winking eye" or so.
| Igelau wrote:
| Jack, in the red shirt.
| kazinator wrote:
| Waldo, in the striped sweater.
| quartz wrote:
| The inside boy at 8 o'clock. In position A he's there, in
| position B he's gone. Think of each position as a discrete
| state rather than thinking of it as "moving boys".
| larrydag wrote:
| Also the 2 o'clock boy has no flag in A then he has a flag
| in B.
| nixpulvis wrote:
| I was pretty confused until I noticed the bottom right head
| turning into an arm.
| ypcx wrote:
| This is the most elegant explanation.
| dang wrote:
| Anybody want to figure out the year?
| kthejoker2 wrote:
| The puzzle was originally conceived in 1906.
|
| Love me some Sam Loyd, there is no greater chess puzzler in my
| mind.
| egypturnash wrote:
| It'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&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/
| acomjean wrote:
| Video 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
| dang wrote:
| Added above. Thanks!
| nicwolff wrote:
| Simple, the "B" boys each average 1/13 bigger.
| tommoor wrote:
| For anyone else that missed it - there is a green slider at the
| top to rotate the inner disk.
| Igelau wrote:
| The 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.
| jmkd wrote:
| Can only count 13 boys whether from point A or B..what am I
| missing?
| kthejoker2 wrote:
| The 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.
| erikerikson wrote:
| Moving 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 wrote:
| Thanks. Doesn't help me understand it (nor do any of the
| above comments) but does help me see it.
| bloak wrote:
| Unfortunately 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.
| Grustaf wrote:
| That sounds like something from Martin Gardener, I definitely
| heard of this trick!
| [deleted]
| schoen wrote:
| Yes, I believe he has a Dr. Matrix story in which Dr. Matrix
| attempts to do this and gets arrested.
|
| Edit: "Sing Sing" in _The Magic Numbers of Dr. Matrix_.
|
| > In desperation he did a foolish thing. He tried to make
| some twenty-dollar bills. His method was bizarre and
| surprising. With a paper cutter he sliced each of fourteen
| bills into two parts, cutting them neatly along the broken
| vertical lines on each of the schematic bills shown on the
| left side of Figure 4. [...]
|
| > Unfortunately--or rather, fortunately--the United States
| government places duplicate serial numbers at opposite comers
| of every bill, and most of the numerologist's new bills
| therefore bore pairs of serial numbers that did not match.
| True, Dr. Matrix's method of making new bills was not exactly
| counterfeiting--he merely "rearranged" the parts of genuine
| bills. Nevertheless, the Treasury Department took a dim view
| of his work and it was not long until he found himself firmly
| confined within the matrix of cells at Sing Sing.
|
| (note that although Gardner is best known as a nonfiction
| writer, the Dr. Matrix stories are fictional)
| Someone wrote:
| '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=...
| (1804)
|
| 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-... (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 wrote:
| If I had to guess they were using the five ask a test to a)
| see if it would pass a cursory inspection, which it appears
| it didn't, and b) refine the technique. People are generally
| more skeptical of larger denominations (especially 100 in the
| US which you can sometimes get a bit of grief over when
| trying to use), but if someone were to notice the 5 is
| counterfeit somehow it's not quite as suspicious just to pay
| with a different (legitimate) note.
| spiderice wrote:
| Regarding that story about the Canadian notes, it seems like
| an unnecessary risk to me to go spend the altered $5 bill
| with the foil. It sounds like you can turn $5 in to $100, and
| only have to risk getting the $100 in to circulation. Or you
| can turn $5 in to $105, but then you have to risk getting 2
| bills in to circulation. It's significantly more risk for a
| very tiny increase, no?
| dane-pgp wrote:
| It does work with chocolate, though...
|
| http://mathandmultimedia.com/2014/07/22/explanation-infinite...
| kypro wrote:
| Reminds me of the infinite chocolate trick,
| https://www.youtube.com/watch?v=qnpugKVitl0
| scollet wrote:
| Ah, an early compression algorithm.
| bigmattystyles wrote:
| Is this just a visual version of the 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 wrote:
| No... 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 .
|
| 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.
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(page generated 2021-09-24 23:01 UTC)