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:
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.
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.