1.1 TH E SQUAR E GRI D 3
all bee n done . I f w e shade th e plac e wher e th e origina l squar e land s eac h time ,
then a littl e bi t o f experiment ing show s that eventuall y everythin g get s colored .
That is :
Observation 1: A s we apply al l Lega l Moves , th e origina l squar e cover s th e
whole plane .
That is , just b y using Today's Legal Moves, we can move the origina l squar e
to cove r any point o n the plane. Th e result look s like square tiles covering a wall
or floor. Continuin g t o thin k abou t tiles , w e se e tha t a s w e mov e th e origina l
square, i t neve r land s o n to p o f a previousl y place d square . Tha t is :
Observation 2 : Differen t Lega l Move s neve r giv e overlapping squares .
Actually, tha t statemen t need s t o b e mad e mor e precise . Fo r example ,
Left 2 then U p 1 is a Lega l Move , an d U p 1 then Lef t 2 i s als o a Lega l Move ,
and bot h o f the m mov e th e origina l squar e t o th e sam e place .
Task 1.1.1 : Fin d a n iron-cla d descriptio n o f the Lega l Move s so that Observa -
tion 2 is true .
This proble m i s ver y commo n i n mathematics : yo u notic e something , an d
you have the idea firmly in your mind, but i t takes a bit more thought t o correctl y
put you r idea s t o paper . Thi s proces s i s much lik e a legislato r tryin g t o draf t a
law withou t an y loopholes . S o your firs t Tas k i s t o plu g thos e loopholes ! Afte r
you d o this , tr y t o giv e a clea r explanatio n o f why th e square s neve r overlap .
There isn' t anythin g specia l abou t th e squar e w e shaded , s o w e ca n tr y
shading somethin g else :
The shape is a parallelogram, bu t we will refer to it as 'the shape.' Motivate d
by the fact s w e discovered abou t th e Lega l Moves applied t o the origina l square ,
we have th e followin g questions :
Question 1: A s we appl y al l Lega l Moves , doe s th e shap e cove r th e plane ?
Question 2 : A s we apply Lega l Moves , doe s th e shap e eve r overla p itself ?
To answe r th e questions , w e start b y lookin g a t th e simples t Lega l Moves :
a) Right 1
b) Upl
c) Up 1 then Right 1
d) Left 1
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