Ross–Littlewood paradox
abstract mathematics problem
The Ross–Littlewood paradox (also known as the balls and vase problem or the ping pong ball problem) is a hypothetical problem in abstract mathematics and logic designed to illustrate the paradoxical, or at least non-intuitive, nature of infinity. More specifically, like the Thomson's lamp paradox, the Ross–Littlewood paradox tries to illustrate the conceptual difficulties with the notion of a supertask, in which an infinite number of tasks are completed sequentially. The problem was originally described by mathematician John E. Littlewood in his 1953 book Littlewood's Miscellany, and was later expanded upon by Sheldon Ross in his 1988 book A First Course in Probability.
The problem starts with an empty vase and an infinite supply of balls. An infinite number of steps are then performed, such that at each step 10 balls are added to the vase and 1 ball removed from it. The question is then posed: How many balls are in the vase when the task is finished?
To complete an infinite number of steps, it is assumed that the vase is empty at one minute before noon, and that the following steps are performed:
The first step is performed at 30 seconds before noon.
The second step is performed at 15 seconds before noon.
Each subsequent step is performed in half the time of the previous step, i.e., step n is performed at 2−n minutes before noon.
This guarantees that a countably infinite number of steps is performed by noon.
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