111 divided by 3

America vs. Russia

S.Chmutov, S.Duzhin

According to V.I.Arnold [1], everybody should be able to divide 111 by 3 without a calculator. In October 2000, the present authors conducted a test with the students of mathematics in two small universities in the US and Russia (Mount Holyoke College, Mass. and University of Pereslavl, Pereslavl-Zalessky). We announced that we were going to have a public opinion poll. Every student was given a folded piece of paper with the question ``$111:3=\ $'' and asked to write the answer in secret from the others.

Results follow.

University Course/Year Number of Students Number of Answers: Correct/Blank/Wrong All Wrong Answers
Mt. Holyoke ``Linear algebra'' (2, 3, 4 years) 24 22/0/2 $111:3=33\frac{2}{3}$;
111 : 3 = 21
Mt. Holyoke ``Calculus 1'' (1st year) 9 7/0/2 $111:3=2.\overline{702}$ (periodic decimal);
Pereslavl ``General Algebra'' (3rd year) 16 10/2/4 111 : 3 = 1;
$111:3 = 3\frac{2}{99}$;
111 : 3 = 33.61;
111 : 3 = 111n : 3n, where $n\in{\mathbb Z}$.
Pereslavl '`Computational Geometry'' (4th year) 11 9/2/0  


1. The answer 111n:3n, where $n\in{\mathbb Z}$, is consonant with the famous Arnold's story [1] about the French boy who was asked to add 2 and 3 (his answer was ``2+3=3+2, because addition is commutative'').

2. A probable reason why letter `l' appeared in one of the answers: the student mistook the last `1' of 111 for `l' because of teacher's handwriting.

The table suggests that nowadays typical American students know mathematics a little better than their Russian counterparts. Is it a sign of decline of the Russian school mathematical education?


[1] ``Californian wars in Dubna''. V.I.Arnold interviewed by E.Kokurina. -- ``Obshchaya gazeta'', no. 38, September 21, 2000 (in Russian). Text and PostScript files available on-line.