George Pólya (18871985)
George Pólya was born in Budapest on 13 December 1887. His father
Jakab (who died in 1897) had been born Jakab Pollák, of Jewish
parents, and with a surname which suggested Polish origin. It is
likely that ancestors had emigrated from Poland to Hungary, where a
lesser degree of antiSemitism existed. However Jakab converted to
Catholicism believing that this would help him advancing in a career
and changed his name to the more Hungarian Pólya. George’s mother
had also been of Jewish background with similar history. Her paternal
grandfather, Max Deutsch, had in fact converted to Presbyterianism
and worshipped with Greek Orthodox Romanians.
George’s father Jakab had been a solicitor with a great mind,
but one who was prepared to pursue a case in which he believed
with no fees. He was not financially successful despite the time
he lived in being considered a golden age for Hungary.
As a student George attended a state run high school with a good
academic reputation. He was physically strong and participated
in various sports. His school had a strong emphasis on learning
from memory, a technique which he found tedious at the time
but later found useful. He was not particularly interested
in mathematics in the younger years. Whereas he knew about the
Eötvös Competition and apparently wrote it he also apparently
failed to hand in his paper.
He graduated from Marko Street Gymnasium in 1905, ranking
among the top four students and earning a scholarship to
the University of Budapest, which he entered in 1905. He
commenced studying law, emulating his father, but found this
study boring and changed to language and literature. He had
become particularly interested in Latin and Hungarian,
where he had had good teachers. He also began studying
physics, mathematics and philosophy. His development was
greatly influenced by the legendary mathematician Lipót
Fejér, a man also of wit and humour, who also taught Riesz,
Szegö and Erdös. Fejér had discovered his theorem on the
arithmetic mean of Fourier Series at the age of 20.
Pólya soon concentrated his studies on mathematics and in 1910
finished his doctorate studies, except for his dissertation.
He took a year in Vienna and returned to Budapest in 191112 to
give his doctoral dissertation and met Gábor Szegö, seven years
younger, who was to become one of his major collaborators.
In the fall of 1912 he went to Göttingen for postdoctoral
study and met David Hilbert, Richard Courant, Felix Klein and
Hermann Weyl. In 1913 he was offered a position in Frankfurt,
but was discouraged from staying in Germany and
turned the job down after being told he was a “bloody Jew”
by a ruffian on a train and went on to the University of Paris
for further postdoctoral work.
In 1914 he took up a position at the Eidgenössische
Technische Hochschule (ETH) in Zurich, an institution which
boasted the names of the physicists Röentgen and Einstein
(1900) among its graduates. This position was arranged by the
mathematician Adolf Hurwitz
(1859 to 1919), who had studied at various
times under Kronecker, Klein and Weierstrass and was the other
great influence on Pólya. The ETH was next door, and closely
associated with, the University of Zurich and Polya had joint
teaching rights with the University.
In 1914 Pólya was called up by Hungary to fight in the
war, but by this time he had adopted Russell’s pacifism and
refused to go. The fear that he might be arrested for being
unpatriotic meant that he did not return to his native country
until after World War II. In Zurich he met his future wife,
Stella Weber. They married in 1918 and were still together
67 years later when Pólya died. They had no children.
Inspired by walks in the woods near Zurich, Pólya in 1912
published one of his major results, the solution of the random
walk problem. In this problem one walks in an infinite rectangular
grid system, at each node having an equal probability of walking
to each of the adjoining nodes on his next leg. Pólya was able
to show that in the two dimensional case it was almost certain
(but with probability 1) that one
would eventually return to the original position, but
one would almost never (with probabilty 0) return
to the origin in the case of three or more dimensions.
Pólya was interested in chemical structure, which led him
in 1924 to publishing the classification of seventeen
planesymmetry groups, a result which was later to inspire
the Dutch artist M.C. Escher.
In 1924 he spent a year in England, working with G.H.
Hardy and J.E. Littlewood at Oxford and Cambridge. This
collaboration led to publication in 1934 of the book
Inequalities, which included a new proof by Pólya of
the AMGM inequality based on the Maclaurin expansion of the
exponential function.
In 1925 Pólya, with Szegö, published arguably one of his
most influential books, Aufgaben und Lehrsätze aus der
Analysis, volumes 19 and 20 of the series Die Grundlehren
der Mathematischen Wissenschaften published by J. Springer,
Berlin. A whole generation  the generation of Erdös,
Szekeres and their circle, and later, learned their
mathematics not so much from the lectures they attended
but by trying to solve the problems of this book one after another
and debating their solutions with each other. Problem solving
as a method of teaching and learning may never have been
practiced on such a scale, and with such success,
before (or since).
One of Pólya’s most famous results, the Pólya Enumeration
Theorem, was published in 1937. This also arose from his
interest in chemical structure and looking at possible
configurations of the benzene ring and other figures
with 6 vertices. Generalising a theorem by Burnside in
Group Theory, Pólya showed how one can determine the number
of different assignments of atoms, or colours, to vertices
as sides of geometrical figures.
A special case of this is the Necklace Theorem, which
shows how many necklaces of n beads can be constructed with
k colours available, assuming there is an infinite supply
of each colour.
In the case where n is prime and necklaces are regarded
as unchanged by rotation, the number of
configurations is k+(k^nk)/n.
Thus if there are 5 beads and 3 colours the number of
necklaces is
3+(3^53)/5=3+240/5=3+48=51.
In the case where n is composite there is also a
formula but it is a little more complicated. However more details of this and another version of
the theorem can be found here.
In 1940 the Pólyas became increasingly concerned, with
George’s Jewish background, of the possibility of a German
invasion of Switzerland, and decided to leave for the
United States.
He was offered a research position by his old collaborator,
Gábor Szegö, now at Stanford, but he did not initially accept
it, going instead to Brown University. In 1942 he did move
to Stanford, however, where he stayed until his retirement
from teaching in 1953. After 1953 he stayed at Stanford,
living at Palo Alto until his death, as a Professor Emeritus.
In 1945, Pólya published one of his most famous books,
How to Solve It. Then in 1951 he published, with
Gábor Szegö, Isoperimetric Inequalities in Mathematical
Physics.
Gábor Szegö had been a winner of the Eötvös Competition
in 1912 and Pólya saw the value in competitions. In 1946
Pólya and Szegö founded the Stanford University Competitive
Examination in Mathematics. In its first year 322 students
from 60 schools in California entered. The competition
grew to having typically 1200 students from 150 schools
in 3 western states. However the competition was terminated
in 1965 when Stanford shifted its emphasis to postgraduate
study. Pólya however continued his activity in this area by
publishing problem material in books and journals.
Pólya was particularly interested in the high school
curriculum and was concerned about the new maths curriculum.
He eventually saw the curriculum change back to basics and was
not happy with the way this happened either.
In 1954 he published the two volume book Mathematics and
Plausible Reasoning and in 1962 and 1965 a further
two volume set entitled Mathematical Discovery.
From his retirement in 1953 Pólya took an active interest
in improving the standard of teaching and took steps to
establish, with NSF funding, eightweek Summer Institutes
for mathematics teachers, first at the college level
(19531960), then later for teachers of high school
and eventually moving the Institutes to Switzerland.
George Pólya died on 7 September 1985.
References
 George Pólya, Master of Discovery,
18871985, Harold and Loretta Taylor, Dale Seymour
Publications, 1993.
 The Random Walks of George Pólya, Gerald L Alexanderson, MAA, 2000
With thanks to a useful suggestion from Laci Kovács.
Written by Peter Taylor, June 2000. Adapted slightly January 2008 with
help from Kevin McAvaney.
This T Shirt, which celebrates the Necklace Theorem, is available from the
AMT Publishing.
