Problem 3

The country of Totalia has money with special values: they issue a 1 pound coin as well as coins with irrational values of $x^k$ pounds for $k=0,1,2, \ldots$, where $x$ is a certain fixed number greater than $3.3$. While highly unusual, this system has a certain benefit: any positive integer cost can be paid by using no more than 14 coins of every given denomination. What number $x$ do Totalians use? Give your answer rounded to 4 decimal points.

If you tested your solution before 06/02/2025 12:38, please test again -- correct solutions before this time were not recognised due to an error. We apologise for the inconvenience.

To deter guessing without thinking, we ask that you also solve the following simple arithmetic problem before checking your answer:

What is 8 × 8 + 2?

This problem was first solved on Thu 6th February at 3:21:23pm