perplexus.info :: Shapes : Cutting the Card

Cutting the Card (Posted on 2009-09-16)

Take a card that is 6 cm by 12 cm, and mark the center point.

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?

Submitted by Charlie

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Solution: (Hide)

Two of the segments will be 5 cm long and form a straight angle with each other, ending on opposite long sides of the rectangle.
One 3-cm segment will be drawn, perpendicular to a long side of the rectangle. And one 6-cm segment, perpendicular to a short side of the rectangle.
Between one of the 5-cm segments and the 3-cm segment is an area of 6 cm². Between a 5-cm and a 6-cm segment of this type is 12 cm². Between a 3-cm and a 6-cm of this type is 18 cm². The wedge between the other 5-cm segment and the 6-cm segment is 24 cm², and between that other 5-cm and the 3-cm is 30 cm². The two 5-cm segments making the straight angle cut off two 36 cm² pieces. Larger multiples are what's left after cutting off the previous sizes.

Based on Enigma No. 1556, "Card trick", by Bob Walker, New Scientist, 1 August 2009, page 24.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Congratulations (and correction)	Steve Herman	2009-09-20 11:13:20
re: one solution	Stephanie	2009-09-20 02:42:41
one solution	Daniel	2009-09-16 12:24:01

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