A programmable robotic mouse is placed at an intersection on a square grid, the borders of which are extendable as needed.
From its initial location the mouse moves one cell forward.
It turns right with its next move incrementing by 1.
This incremental process continues up to a certain constraint whereby the mouse resumes the process with a move of one space until that constraint is met again; continue this process until you either return to your starting position or you evidently will never return.
What generalisations can be made about how variations of the value of the constraint affect the path forced upon the mouse? 

Note:It will be necessary to test a range of constraining values.
(In reply to
Revision by brianjn)
The following tweaking of brianjn's program uses SCREEN 12 instead of SCREEN 9 for a more square view (each increment of x is the same size on the screen as an increment of y), as the 640 x 480 pixels is the 4:3 ratio of a conventional screen. It necessitated replacing the second parameter of the SCREEN command with a PAINT command after the CLS. Also, due to the larger vertical number of pixels, L could be increased to 7 while keeping the multipleof4 values still in range.
And of course these won't work under Windows Vista, which has dropped support for fullscreen DOS graphics.
BTW, the mouse is turning left, rather than right, in this program.
DECLARE SUB startpos ()
DECLARE SUB plot ()
DECLARE SUB Keyin ()
DECLARE SUB path ()
DECLARE SUB star ()
DIM SHARED move 'counts steps
DIM SHARED a 'For..Next label
DIM SHARED f 'For..Next label
DIM SHARED L 'sides of polygon
'Since Rt angle turns, L=4
DIM SHARED s 'For..Next label
DIM SHARED v 'Maximum steps (move)
'before iteration
DIM SHARED z 'Determines move length
DIM SHARED x 'horizontal coordinate
DIM SHARED y 'vertical coordinate
DIM SHARED gr 'mod value of or$ input
DIM SHARED or$
CLS
SCREEN 12
COLOR 15
CLS
PAINT (1, 1), 9
PRINT "No. from 1 to 20",
INPUT or$
IF or$ <> "*" THEN
startpos
plot
END IF
Keyin 'holds screen display
SUB Keyin
DO
LOOP WHILE INKEY$ <> CHR$(13)
END SUB
SUB path
FOR a = 1 TO s
SELECT CASE move 'this is C equiv. of switch
CASE 1 'up
LINE (x, y)(x, y  z), 7
y = y  z
CASE 2 'left
LINE (x, y)(x  z, y), 7
x = x  z
CASE 3 'down
LINE (x, y)(x, y + z), 7
y = y + z
CASE 4
LINE (x, y)(x + z, y), 7
x = x + z
END SELECT
star
NEXT a
END SUB
SUB plot
move = 1
FOR f = 1 TO L
FOR s = 1 TO v 'v determines the shape
path
move = move + 1
IF move = 5 THEN move = 1
NEXT s
NEXT f
END SUB
SUB star
LINE (x  1, y + 1)(x + 1, y  1), 14
LINE (x  1, y  1)(x + 1, y + 1), 14
END SUB
SUB startpos
v = VAL(or$)
z = 18 'param for step size
z = INT(6 * z / v) '6 only reduces
'initial z value
'for convenience
x = 320 'centre of screen values
y = 240 'startpos may be offset.
gr = (v / 4  INT(v / 4)) * 4
'determines the "mod" value
'of the spiral category
L = 7
IF v = 1 THEN gr = 1
IF gr = 0 THEN y = 60: x = 100
IF gr = 1 THEN y = 240: x = 350
IF gr = 2 THEN L = 2
IF gr = 3 THEN x = 300: y = 150
END SUB
Edited on January 26, 2008, 11:49 am

Posted by Charlie
on 20080126 11:47:26 