Problem 1

We are given a positive integer

r

and a rectangular board divided into

20\times12

unit squares. A move from one square to another is permitted only when the distance between their centers is

\sqrt r

. Find a sequence of moves leading between two adjacent corners of the board that lie on the long side. (a) Show that this is impossible if

r

is divisible by

2

3

. (b) Prove that it is possible for

r=73

. (c) Can it be done for

r=97

Step 3 of 4: Construct the path for

73

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In plain words

Construct the path for $73$

8^2+3^2=73

Detailed analysis

For $r=73$ , use steps with absolute coordinate changes $8$ and $3$ . The following legal path stays on the board: $(0,0),(8,3),(16,6),(8,9),(11,1),(19,4),(11,7),(19,10),(16,2),(8,5),(16,8),(19,0)$ .