A student who takes much more than five minutes to calculate the mean of with 10% accuracy has no mastery of mathematics, even if he has studied non-standard analysis, universal algebra, supermanifolds, or embedding theorems.
V.I. Arnold (1937-2010), Russian mathematician
One of my math professors in college said: “If you cannot do ten Arnold’s Trivium problems, you should not be given a degree in math. You simply don’t know much mathematics”. This professor used to work with Arnold at the famed Mekhmat department at the Moscow State University. Arnold, by the way, was one of the finest mathematicians of the 20th century.
The trivium is a collection of 100 undergraduate-level problems in mathematics, written by Arnold as a response to what he perceived as the declining quality of math/physics education in Russian universities at the time. Despite the name Trivium, the problems are NOT trivial at all. I found that solving the Trivium problems is a humbling, character-building experience.
Here is the original paper that contains the 100 Trivium problems (translated into English):
(I tried some of the 100 problems recently, and found that I can only do 11 of them, barely enough to justify my math degree. But I’ve forgotten a lot of math in the past few years.)