*A Mathematician’s Miscellany*. It was later analysed in detail by Sheldon Ross in his 1988 book

*A First Course in Probability*.

Littlewood wrote:

Balls numbered 1, 2, … (or for a mathematician the numbers themselves) are put into a box as follows. At 1 minute to noon the numbers 1 to 10 are put in, and the number 1 is taken out. At half a minute to noon numbers 11 to 20 are put in and the number 2 is taken out. At one quarter minute before noon 21 to 30 in and 3 out; and so on. **How many are in the box at noon? The answer is none:** any selected number, e.g. 100, is absent, having been taken out at the 100th operation.

(Littlewood, 1953, Page 5).

** Simplified Version of the Paradox **

We start with an infinite number of balls, numbered sequentially. At each stage, **two** balls are placed in an urn and then **one** is removed and discarded. The stages are telescoped so that they are completed in a finite time . How many balls are in the urn after time ? We examine two strategies.

** Strategy I: Remove the first member of each pair **

We tabulate the balls in the urn, and their number, at each stage:

After stage n, there are n balls in the urn. Ultimately, all the odd-numbered balls have been discarded. Therefore, **all the even-numbered balls remain in the box.**

** Strategy II: Remove the -th ball at stage **

Now we consider Littlewood’s strategy: we remove the -th ball at stage .

Apparently, there are still balls in the urn after the -th stage. However, the numbers are missing. Thus, at the completion of the process, balls with every natural number have been removed! We must conclude, as Littlewood did, that **ultimately, there are no balls left in the box!**

**Supertasks**

Supertasks were first suggested by Hermann Weyl although he did not use that term. A *supertask* is a countably infinite sequence of operations that occur sequentially in a finite interval of time. The term was coined by the philosopher James F. Thomson, who devised the paradox of *Thomson’s lamp.*

One obvious resolution of the Ross-Littlewood paradox is to say that supertasks are impossible. Thomson emphatically denied that supertasks are possible. If they were possible, propositions in number theory, such as **Goldbach’s conjecture**, could be determined in a finite time by a search of the set of natural numbers .

** Sources **

Littlewood, John E., 1953: *A Mathematician’s Miscellany*. Methuen & Co.~Ltd., London. Reissued as *Littlewood’s Miscellany*, Be’la Bolloba’s (ed.), Cambridge University Press, (1986). ISBN: 0-521-33058-0.

Ross, Sheldon, 1988: *A First Course in Probability*. Ninth Edn 2012. Pearson Publ. ISBN: 9780-3217-9477-2.

Thomson, James F., 1954: Tasks and Super-Tasks. *Analysis*, **15**(1), 1–13. doi: 10.2307/3326643.