Then draw four straight line segments, each from the center point to the perimeter, and each an integral number of centimeters long, so that if you wish to cut off an area of any integral multiple of 6 cm², you need only cut along two of these lines. Of course cutting off a multiple of 6 cm² will also leave a multiple of 6 cm².

What are the lengths of the four segments and how can they be arranged around the center?

Daniel, I agree that all solutions are probably variations on your idea, but I sure came at it from a different angle.  I figured that the areas themselves were more important, so I figured the the desired areas before I worried about the cuts. 

What's funny to me is that apparently the areas I came up with aren't necessary.  Your solution uses different areas than mine.  Anyway, using your same notation with the corners of the card being at (0,0), (6,0), (6,12) and (0,12), the cuts I used are:

(0,6)
(0,10)
(3,12)
(6,2)

Which give areas of 6, 12, 24, and 30.  And when artfully cut, those can give any integer multiple of 6 you desire, up to 66. 

Posted by Stephanie on 2009-09-20 02:42:41