# 2022-P1 Best Solutions and Final Results

Problem 1 – Magnetized ball

By 17th January 2022, there are 1060 registered participants from 78 countries. In total, 186 solutions were submitted for problem 1, out of which 79 were correct. Congratulations to everyone who managed to solve this problem!

As usual, when selecting the best solutions, preference has been given to those solutions which have been submitted before the publication of the first hint. Overall, there were 27 such solutions. To narrow down the selection, I excluded first the solutions with a brute-force mathematical approach to finding the magnetic field of a homogeneously magnetized ball, for instance solving the Laplace equation in spherical harmonics. This is not because I don’t appreciate such skills – knowing how to use a brute-force approach with appropriately chosen mathematical apparatus is definitely praiseworthy. However, this mathematical apparatus is a bit beyond the intended level of this competition – usually taught at MSc courses. Second, I excluded solutions that were documented less clearly than the majority of the submissions. Handwriting does not exclude the possibility of getting a best-solution-award automatically, but at the very least, it should be well organized and easy to read. Finally, I excluded a few submissions which would have been informative and useful, but came clearly from engineering textbooks of permanent magnets, failing to provide a solid proof of the equality $B=-2H$ for spherical balls with homogeneous magnetization.

The best solution awards (all 1/7th of the full bonus):
Victor Lequin and the team of Eddie Chen and Zhening Li – a really concise proof of $B=-2H$, based on the uniqueness of the solution.
Absur Khan Siam – using vector potential to calculate the magnetic field in an interesting way.
Nicely written solution corresponding to the hints (but submitted before the first hint):