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Disappearing Bicyclist – Sam Loyd (1906)

geogebra.org

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Re: Disappearing Bicyclist – Sam Loyd (1906)

#61

Post by rhn_mk1 »

kazinator wrote: post #42
rhn_mk1 wrote: post #31

Earlier quoted context omitted.

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.

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 flag…

> 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.

Re: Disappearing Bicyclist – Sam Loyd (1906)

#62

Post by kazinator »

kazinator wrote: post #27

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…

OK, so that's a good explanation as far as the head counting is concerned. But the extra head has a body. Well 3/4 of a body: the Siamese twin has his own torso, two arms and one leg. Where is that from?

the 3/4 of a body come from to produce the Siamese twin, who has his own torso, two arms and a leg?

This is kind of like, imagine we have a slide rule with two moving parts:

    . . , | | | |
    | | | | ; ' '
Here I have six vertical features, placed side by side. Now I slide the rows relative to each other:

      . . , | | | |
    | | | | ; ' '
wee, now I have seven vertical features.

Let's do it with circles:

https://i.imgur.com/9tuNkbT.png

Just by chopping and sliding, I turned 10 circles into 11. Apparently. But of course, the 11 objects are not really circles; they don't have the full area.

Re: Disappearing Bicyclist – Sam Loyd (1906)

#63

Post by kazinator »

kazinator wrote: post #62
kazinator wrote: post #27

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…

OK, so that's a good explanation as far as the head counting is concerned. But the extra head has a body. Well 3/4 of a body: the Siamese twin has his own torso, two arms and one leg. Where is that from? the 3/4 of a body come from to produce the Siamese twin, who has his own torso, two arms and a leg? This is kind of like, imagine we have a slide rule with two moving parts: . . , | | | | | | | | ; ' ' Here I have si…

Here it is with stick figures.

https://i.imgur.com/MbrCthN.png

We can do the "head examination" here too, but it's actually irrelevant and "wrong-headed" because that situation is a consequence of what I did, rather than the basis.

We can ask, where did the extra head come from for the new stick figure on the far right? And here it is actually plain to see: the previous four figures have different slices of their head removed, which precisely add up to a complete head. There is nothing special about the head; the same holds for all the entire figure.

Sam Loyd's bicycle wheel drawing conceals the principle by having the boy figures in different positions, and around a wheel rather than a linear slider.

Re: Disappearing Bicyclist – Sam Loyd (1906)

#64

Post by jmkd »

kazinator wrote: post #55
jmkd wrote: post #54

Earlier quoted context omitted.

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?

In one sentence: Two-headed body at A+2, in B config, wrongly counted as one boy, corresponds with hip-joined Siamese twins at A+7, in A config. 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 mi…

Spent another 30 mins on all these generous explanations, some of which made some sort of sense, some of the time.

I've effectively given up, and can only conclude the following:

State A has under 13 boys and state B has over 12 boys and we round up or down visually. This explanation doesn't satisfy me at all but is enough for me to move on, defeated.

Re: Disappearing Bicyclist – Sam Loyd (1906)

#65

Post by kazinator »

jmkd wrote: post #64
kazinator wrote: post #55

Earlier quoted context omitted.

In one sentence: Two-headed body at A+2, in B config, wrongly counted as one boy, corresponds with hip-joined Siamese twins at A+7, in A config. 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 mi…

Spent another 30 mins on all these generous explanations, some of which made some sort of sense, some of the time. I've effectively given up, and can only conclude the following: State A has under 13 boys and state B has over 12 boys and we round up or down visually. This explanation doesn't satisfy me at all but is enough for me to move on, defeated.

See if this helps: https://news.ycombinator.com/item?id=28650420

Including the follow-up child comment.

I've reproduced the effect using equally spaced, identical geometric figures, linearly arranged.

Re: Disappearing Bicyclist – Sam Loyd (1906)

#66

Post by mannykannot »

thehappypm wrote: post #11
Chinjut wrote: post #5

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 hal…

Which boy is gone? :)

It is clearer if you think of it as partial pictures of 24 boys, 12 on the inside and 12 on the outside. In either position, each picture on the inside lines up with one on the outside, and vice-versa (each position is a bijection.) In position B, each of the pairs form a picture of one boy, leading us to count 12. The same is true for position A, except for the one case where each part-picture is well more than 50% (and, in particular, there are two complete heads.) Consequently, this cannot be seen as a picture of one boy, and we count it as two.

In this view, the answer to your question is 'one of each pair, except in one case in position A only.'

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