Using your Head is Permitted

August 2008 riddle

Suppose that you want to divide the unit square into three equal-area pieces. The unit square is the square whose corners are (0,0), (0,1), (1,0) and (1,1), and one way of dividing it into three equal-area pieces is by two cut-lines, one from (0,1/3) to (1,1/3), the other from (0,2/3) to (1,2/3).

This means you will need cut-lines with a total length of 1+1=2. Suppose, however, that the tool you are using to cut the square can not be "lifted" or "stopped", but rather cuts continuously until it finishes dividing the square. The same cutting scheme as before can no longer be applied as-is. It must be modified, perhaps to the following cut sequence:

Using straight lines, cut from (0,1/3) to (1,1/3) to (1,2/3) to (0,2/3).

This description shows how to cut the unit square using a single cut line, but the total length of the cut-line is now relatively high: 1+1/3+1=2 1/3.

This month's riddle: describe a single-cut division of the unit square to three equal-area parts, such that the total cut length is minimal. As part of your description, you are asked to specify what the total cut-length of your division will be, writing it to six decimal places.

List of solvers:

Oded Margalit (18 August 21:03)
Itsik Horovitz (28 August 02:37)

Elegant solutions can be submitted to the puzzlemaster at Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.


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