Comment by kazinator | Hacker News Reader

Comment by kazinator | Hacker News Reader

Parent Full thread

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.

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.

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.

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.

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.

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?

This was my attempt:

https://news.ycombinator.com/item?id=28646279

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.

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.

Reply on news.ycombinator.com