The Munchkin road making authority have commissioned you to work out where their roads will end.

Using Cartesian coordinates where the first tile is placed with opposite corners on (0,0) and (1,1) show that the coordinates of the destination of a road must always be of the form

æ
ç
è 		a+bÖ3
2
 , c+dÖ3
2
 ö
÷
ø

for integers a, b, c and d
Investigate ways in which you can create roads. You may like to consider these questions:
  1. Can you make roads which end with rational values for the x coordinate of the destination?
  2. Can you make roads which end with rational values for the y coordinate of the destination?
  3. Can you create a road which ends with the x coordinate equal to any integer multiple of one half?
  4. Can you make roads for which the coordinates of the destination are both rational? Both irrational?
  5. Can many roads lead to the same destination?

You might like to experiment with this interactivity
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