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The good, the bad and the pleasure (not pressure!) of mathematics competitions
SIU Man Keung (蕭文強 )University of Hong Kong
The “good” of mathematics competitions• clear and logical presentation
• tenacity and assiduity
• “academic sincerity”
The “good” of mathematics competitions• clear and logical presentation
• tenacity and assiduity
• “academic sincerity”
The “good” of mathematics competitions• clear and logical presentation
• tenacity and assiduity
• “academic sincerity”
The “good” of mathematics competitions• clear and logical presentation
• tenacity and assiduity
• “academic sincerity”
• arouse a passion for and pique the interest in mathematics
The “bad” of mathematics competitions
• competition problems vs research
• over-training?
Eötvös Mathematics Competition in Hungary (started in 1894)Almost all notable Hungarian mathematicians were winners in the competition when they were school pupils, such as
Lipót Fejér, Theodore von Kármán, Denes König, Alfréd Haar, Marcel Riesz, George Pólya, Gábor Szegö, Tibor Radó, Paul
Erdös, Paul Turän, …………
Note the role played by Középiskolai Matematikai és Fizikai Lapok, a journal on mathematics and physics for secondary schools founded in 1894.
IMO MedalistsBéla Bollobás (Hungary, 1959B, 1960G, 1961G) Senior Whitehead Prize, etc.
Grigori Margulis (USSR, 1962S) Fields Medalist, Wolf Prize, etc.
László Lovász (Hungary, 1962S, 1965G
+ Special Prize, 1966G + Special Prize)Fulkerson Prize, Wolf Prize, Kyoto Prize, etc.
Richard Ewen Borchards (UK, 1977S, 1978G) Fields Medalist, etc.
Peter Shor (USA, 1977S) Navanlinna Prize, MacArthur Award, etc.
Noam D. Elkies (USA, 1981G, 1982G) Putnam fellow, etc.
William Timothy Gowers (UK, 1981G) Fields Medalist, etc.
Grigori Perelman (USSR, 1982G) Fields Medalist (declined), Millennium Prize (declined), etc.
Terence Tao (Australia, 1986B, 1987S, 1988G) Fields Medalist, Crafoord Prize, MacArthur Award, etc.
● ● ● ● ● ● ● ●
Putnam Fellows in the William Lowell Putnam Mathematical CompetitionRichard Feynman (1939)
Nobel Laureate in Physics, etc.
John Milnor (1949,1950)
Fields Medalist, Wolf Prize, Abel Prize, etc.
David Mumford (1955,1956)
Fields Medalist, Shaw Prize, Wolf Prize, etc.
Daniel Grey Quillen (1959)
Fields Medalist, etc.
Peter Shor (1978)
Navanlinna Prize, MacArthur Award, etc.
●. ●. ●. ●. ●. ●. ● ●.
The “bad” of mathematics competitions
• competition problems vs research
• over-training?
Is the passion for the subject of mathematics itself genuine?
Is the interest sustained?
1988] Australia, Canberra, IMO,[29th
integer.an of square theis
1
that Show . divides 1
such that integers positive be and Let 22
22
ab
baba ab
ba
square.
perfect anot is assume and 122Let
Bulgaria] ofmember teama toprize [Special
kkab
ba
Rewrite the equation as a2 – k ab + b2 = k ----- (*).If (a, b) is an integral solution of (*), then ab 0, or else ab + 1 0 so that a2 + b2 = (ab + 1) k 0, which is a contradiction. Furthermore a > 0, b > 0, or else k is a perfect square !
Let (a, b) be an integral solution of (*) with a > 0, b > 0 and a + b smallest. May assume a > b. We shall produce from it another integral solution (a, b) of (*) with a > 0, b > 0 and a + b < a + b. This is a contradiction!
Regard (*) as a quadratic equation with roots a, a , then a + a = kb and aa = b2 – k.
Show that a is a positive integer and (a, b) is a solution of (*). Note that a = (b2 k) / a (b2 1) / a (a2 1) / a < a,
so a + b < a + b .
1988] Australia, Canberra, IMO,[29th
integer.an of square theis
1
that Show . divides 1
such that integers positive be and Let 22
22
ab
baba ab
ba
My false ‘insight’: a = N3, b = N.
a2 + b2 = N2 (N4 + 1) , ab + 1 = N4 +1,
hence
My ill-fated strategy: Try to deduce from a2 + b2 = (ab + 1) k the equality [a (3b2 3b + 1)]2 + [b 1]2
= {[a (3b2 3b + 1)] [b 1] + 1} { k [2b 1]}.
. 1
222 Nab
ba
Failed!
Brute-force search:a 1 8 27 30 64 112 125 216 240 343 418 512 …
b 1 2 3 8 4 30 5 6 27 7 112 8 …
k 1 4 9 4 16 4 25 36 9 49 4 64 …
ALL IS CLEAR !The solutions of a2 + b2 = (ab + 1) k consist of
(A) (N3, N, N2) where N {1, 2, 3, …} ,
which is called the Nth basic solution;
(B) (a, b, k) obtained from a basic solution
via a sequence of transformations of
the form
ai + 1 = ai ki bi, bi + 1 = ai, ki + 1 = ki.
Barker sequence (R.H. Barker, 1953)
Each off-phase shift and the original sequence differ by at most one place of coinciding entries and non-coinciding entries in their overlapping part, i.e. its aperiodic autocorrelation function has absolute value 0 or 1 at all off-phase values.
Barker sequence (R.H. Barker, 1953)
Examples and non-examples:1 1 1 0 1 is a Barker sequence of length 5.
1 1 1 0 1 0 is not a Barker sequence of length 6.
There are Barker sequences of length 2, 3, 4, 5, 7, 11, 13:
s = 2 1 1
s = 3 1 1 0
s = 4 1 1 1 0
s = 5 1 1 1 0 1
s = 7 1 1 1 0 0 1 0
s = 11 1 1 1 0 0 0 1 0 0 1 0
s = 13 1 1 1 1 1 0 0 1 1 0 1 0 1
Theorem (R. Turyn, J. Storer, 1961)
There is NO Barker sequence of odd length s larger than 13.
Conjecture (unsettled for over half a century)
There is NO Barker sequence of length s larger than 13.
The problem stimulates a lot of research in combinatorial design. Researchers change the problem with various variations and obtain fruitful results.
• Jonathan Jedwab, What can be used instead of a Barker sequence? Contemporary Mathematics, 461 (2008), pp. 153-178.
“Olympiad mathematics”
¾ elementary number theory
¾ algebra
¾ combinatorics
¾ sequences
¾ inequalities
¾ functional equations
¾ plane and solid geometry
¾ ……
“Olympiad mathematics”
¾ elementary number theory
¾ algebra
¾ combinatorics
¾ sequences
¾ inequalities
¾ functional equations
¾ plane and solid geometry
¾ ……
¾ Why can’t this type of so-called “Olympiad mathematics” be made good use of in the classroom of school mathematics as well?
John Baylis and Rod Haggarty, Alice in Numberland: A Students’ Guide to the Enjoyment of Higher Mathematics (1988)
The professional mathematician will be familiar with the idea that entertainment and serious intent are not incompatible: the problem for us is to ensure that our readers will enjoy the entertainment but not miss the mathematical point, ….
Von Neumann with the first Institute computer
John Von Neumann (1903-1957)
馮諾依曼
Two cyclist A and B at a distance 20 miles apart approached each other, each going at a speed of 10 m.p.h. A bee flew back and forth between A and B at a speed of 15 m.p.h. When A and B met, how far had the bee travelled?
L
w BA v
Bee b
etc.
)(
)(
33
2323
22
1212
11
11
btL
Lttwbt
btL
Lttvbt
btL
Lwtbt
etc.
22
5
2
4
3
2
1
wbbL
wbwb
vbvbL
wbbL
wbwb
vbvbL
wbbL
wbwb
vbvbL
wbbL
vbvbL
wbbLL
wvbL
wbwb
vbvbvb
b-vwb
bLLLL
1
1 1 321
(t1)(t1)
(t2)
(t3)(t2 + t3)
(t1+ t2)
John von Neumann
(1903 – 1957)
Positional warfare
Guerilla warfare Each approach has
its separate merit. They complement and supplement each other.
Positional warfare
Guerilla warfare Each approach has
its separate merit. They complement and supplement each other.
Each approach calls for day-to-day preparation and solid basic knowledge.
Positional warfare
Guerilla warfare Flexibility and
spontaneity are called for in positional warfare. Careful prior preparation and groundwork are needed in guerrilla warfare.
( 陣而後戰，兵法之常。 運用之妙，存乎一心。 )”
History of Song Dynasty Biography of Yue Fei
( 《宋史 岳飛傳》․ )
“Setting up the battle formation is the routine of art of war. Manœuvring the battle formation skillfully rest solely with the mind.
Yue Fei (岳飛 )
(1103-1142)
Archimedes(287 B.C. – 212B.C.)
LIU Hui 劉徽(3rd century)
ArchimedesMeasurement of Circle (3rd century B.C.)
《九章算術》(100 B.C. – 100 A.D.)Commentaries by LIU Hui(3rd century)
「割之彌細。所失彌少。割之又割。以至於不可割。則與圓周合體而無所失矣。(The finer one divides, the smaller is the left over; divide and again divide until one cannot divide further, then the regular polygon coincides with the circle with nothing being left out.)」Commentary of LIU Hui to Problem 31, 32 in
Chapter 1 of Jiu Zhang Suan Shu (3rd century) 《九章算術》劉徽註
Archimedes
On the Quadrature of the Parabola (3rd century B.C.)
“… the IMO should be seen as just the tip of a very large, more interesting, iceberg, for it should provide an incentive for each country to establish a pyramid of activities for masses of interested students.”
[what I learnt from my friend Tony Gardiner, an experienced UK IMO leader]
Postscript in M.K. Siu, Some reflections of a coordinator on the IMO , Mathematics Competitions, 8 (1) (1995), 73-77.
Other activities:― mathematics club
― mathematics magazine
― problem session
― contest in doing projects
(at various levels, to
various depth)
― contest in writing book
reports and essays,
producing cartoons,
videos, softwares, toys,
games, puzzles, …
The good, the bad and the pleasure of mathematics competitions
Are to which we should pay our attention.
Benefit from the good; avoid the bad;
And soak in the pleasure.
Then we will find for ourselves satisfaction!
to the audience for coming to the talk,
to the organizers of TAIMC 2012 for inviting me to give the talk,
to Ms. Mimi Lui of the HKU Department of Mathematics for assisting me in the preparation of slides.
Thank you
