Ah, an early compression algorithm.
Disappearing Bicyclist – Sam Loyd (1906)
51–60 of 66 posts
Re: Disappearing Bicyclist – Sam Loyd (1906)
#52Re: Disappearing Bicyclist – Sam Loyd (1906)
#53Can only count 13 boys whether from point A or B..what am I missing?
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.
Thanks. Doesn't help me understand it (nor do any of the above comments) but does help me see it.
Re: Disappearing Bicyclist – Sam Loyd (1906)
#54You 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…
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?
Re: Disappearing Bicyclist – Sam Loyd (1906)
#55You 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…
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 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.
Looking at the cropped image again, consider what would happen if the middle double-headed figure did not have a big 2/3rds of a head from the inner disc, but only 1/3rd, making just one normal head. Upon rotating to the A position, the bottom boy would now get that inner 1/3rd and therefore would not have a complete head!
I believe that there is justification in relating this to the Tangram Paradox:
https://mathworld.wolfram.com/TangramParadox.html
Here, a subterfuge involving subtle fractional differences is also taking place. Very similar, corresponding body shapes actually have a different area, and that difference corresponds to the extra piece.
In the bicycle wheel problem, we ignore that two heads are two individuals if they have the same body and are fused at the head, yet we do not ignore that two heads are two individuals if they share only a hip and leg.
In the Tangram paradox, we cannot ignore the glaringly obvious extra area of the small rectangle forming the foot, but find it hard to perceive the extra area of the larger body.
Hard to see extra area versus hard to see two heads that are partially merged.
Re: Disappearing Bicyclist – Sam Loyd (1906)
#56You 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…
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?
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?
Re: Disappearing Bicyclist – Sam Loyd (1906)
#57You 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…
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?
This was my attempt:
Re: Disappearing Bicyclist – Sam Loyd (1906)
#58You 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…
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?
Here's a picture with side-by-side comparison and things numbered: https://i.imgur.com/zw6n7C7.png
EDIT: I just left this without comment because I had to go eat dinner with family, but notice how in the first picture there are four faces that are right along the split. But in the second picture, there are only three.
Re: Disappearing Bicyclist – Sam Loyd (1906)
#59I 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?"
Re: Disappearing Bicyclist – Sam Loyd (1906)
#60You 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…
Bravo for the explanation. These types of visual games are often convincing because they are reliant on the observer just glancing for a few seconds and not prolonged study.