4 1. SQUARES , HEXAGONS, AND TRIANGLE S
Those Lega l Move s mov e th e shap e t o thes e places :
^
jt'


The shape with dashe d edge s fits nicely , but i t doesn' t com e from on e of th e
moves a , b , c , o r d . A bi t o f experimentin g gives tha t th e dashe d shap e come s
from th e mov e U p 1 then Righ t 2 . Th e differen t copie s of the shap e fi t togethe r
perfectly, leadin g t o thes e answers :
Answer 1: Yes , the shap e cover s th e whol e plane .
Answer 2 : No , the shap e neve r overlap s itself .
In som e sense , thes e ar e th e 'good ' answers . Answe r 1 says tha t th e shap e
can b e use d fo r tilin g th e floor. Answe r 2 say s tha t th e shap e doesn' t hav e
any extr a bit s whic h caus e th e tile s t o overlap , o r whic h preven t th e differen t
copies of the shape from fittin g togethe r properly . No t al l shapes have these nic e
properties. Fo r example , i f we start wit h thi s triangle :

Applyin g al l Le -
j / \ ga l Move s yield s
/ thi s pattern :

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For thi s shape , th e answer s are :
Answer 1: No , th e shap e doe s no t cove r th e whol e plane .
Answer 2 : No , th e shap e neve r overlap s itself .
Task 1.1.2: Answe r Question s 1 and 2 for variou s shapes , an d fin d interestin g
shapes whic h cove r th e plan e wit h n o overla p whe n yo u appl y al l Lega l Moves .
Use th e example s t o devis e rule s whic h allo w yo u t o quickl y answe r Questio n 1
or Questio n 2 . Som e sampl e shape s ar e give n o n th e nex t page .
In additio n t o th e exampl e shape s o n th e nex t page , yo u shoul d als o inven t
your own . Fo r example , non e o f th e give n shape s ha s 'curved ' edges , s o yo u
should tr y t o com e u p wit h som e that do . Th e mos t importan t goa l is to devis e
rules t o hel p yo u answe r th e Question s 1 and 2 . You r rule s don' t hav e t o appl y
to al l shapes : i t i s useful t o hav e rule s whic h onl y wor k som e o f the time .

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