2
1. The Four Numbers Problem
The Fou r Number s Proble m i s introduced i n Sectio n 2 as the Fou r Num -
bers Gam e and , a s mentione d above , onl y simpl e arithmeti c i s used . Th e
symmetry o f the gam e i s observed i n an elementar y wa y in Sectio n 3 , except
for th e las t tw o page s wher e th e observation s ar e restate d i n mor e algebrai c
terms. Section s 4 and 5 discuss the questio n o f the finitenes s o f the lengt h of
the game . Thes e section s ar e accessibl e t o thos e comfortabl e workin g i n th e
slightly mor e abstrac t settin g wher e th e number s ar e replace d b y letters .
"Long games " ar e constructe d i n Sectio n 6 . Mor e forma l notatio n i s in-
troduced, bu t thi s sectio n a s well a s Sectio n 7 , where bound s fo r th e lengt h
of a gam e ar e considered , ar e suitabl e fo r middl e grad e an d hig h schoo l stu -
dents wh o hav e a n interes t i n mathematics . Th e mai n resul t i n Sectio n 8
on uniquenes s o f games o f infinit e lengt h i s a beautifu l applicatio n o f eigen -
values; consequently , a n understandin g o f basi c linea r algebr a i s required .
Probabilities fo r game s o f various length s ar e calculate d i n Sectio n 9 . Som e
familiarity wit h elementar y probabilit y distribution s i s needed here .
The generalizatio n o f th e gam e t o th e /c-Number s Proble m i s studie d
in Sectio n 10. Th e mai n resul t o f tha t sectio n i s a characterizatio n o f th e
positive integer s k suc h that th e fc-Numbers Game , playe d wit h nonnegativ e
integers, alway s ha s finite length . It s proo f i s a nea t applicatio n o f polyno -
mials ove r finite fields, bu t th e entir e sectio n ca n b e adapte d fo r student s a t
any level .
2. Th e Fou r Number s Gam e Rul e
At th e mos t elementar y level , the Fou r Number s Gam e start s wit h a squar e
and fou r nonnegativ e integer s placed , i n an y order , a t th e vertice s o f th e
square. Her e i s a n exampl e o f a "star t square. "
9 5
1 7
The first ste p o f the gam e i s the constructio n o f a new numbered squar e
by takin g a s it s vertice s th e midpoint s o f the side s o f th e star t square . Th e
number place d a t a midpoin t i s th e absolut e valu e o f th e differenc e o f th e
two numbers a t th e adjacen t vertices . A new numbere d squar e i s formed b y
joining th e midpoint s o f th e star t square .
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