Answer
Call the astronauts A, B, and C, labeling them from left to right, and suppose they start at rest relative to a nearby space station.
When A throws B, they move away from each other at equal speeds by conservation of momentum.
Say that A has velocity -V, and B has velocity +V.
When C catches B, they move to the right at velocity V/2. (see Figure 6.14 on page 99).
When C then throws B toward the left, she recoils at speed V, in her own reference frame. Remember, the astronauts are equally strong. Her velocity relative to the nearby space station is V + V /2 = 1.5V, to the right.
Meanwhile, the velocity of B is the V/2 she initially had while coming toward C, plus the - V from the second throw, resulting in a velocity of -V/2, directed to the left.
This is slower than astronaut A, still moving at -V by Newton's first law, so B will never catch up to A.
There will be no more throws, the game has ended, and someone had better rescue these astronauts!
