Chapter 1
The Fou r Number s
Problem
1. Introductio n
In this preparatory chapter , w e illustrate the concept o f vertical developmen t
of mathematic s wit h a simpl e problem . Th e Fou r Number s Problem , als o
known as the Four Numbers Game , furnishes a wealth of interesting materia l
for teacher s an d student s fro m th e elementar y grade s throug h college . Th e
statement o f th e proble m use s onl y ordinar y arithmetic . Analysi s o f th e
problem, however , bring s t o ligh t mathematic s tha t include s application s o f
characteristic polynomial s an d eigenvalue s i n linea r algebra , th e algebr a o f
polynomials ove r a finite field, an d elementar y probabilit y distributions .
While the remainin g chapters , th e hear t o f our book , hav e a scope which
extends mor e broadl y an d deepl y t o problem s tha t ar e th e focu s o f curren t
research, thi s shor t chapte r demonstrate s ho w a n ide a ca n b e develope d t o
reveal interestin g mathematic s a t man y level s fo r a wide variety o f student s
and teachers .
The Fou r Number s Proble m appear s t o hav e bee n "discovered " severa l
times. Th e earlies t reference s i n th e literatur e tha t w e could find giv e date s
in the 1930's. See , for example, [CM37] , and remarks in [Fre48 ] and [Won82] .
Other introduction s t o th e gam e includ e [Hon70 ] an d [Spr63] .
A brie f outlin e o f th e content s an d level s o f difficult y o f th e section s
follows. However , i t i s th e authors ' hope , an d th e poin t o f th e book , tha t
readers wil l us e th e vertica l evolutio n o f th e proble m a s motivatio n fo r ex -
panding th e breadt h an d dept h o f thei r mathematica l knowledge .
1
http://dx.doi.org/10.1090/mbk/048/01
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