A math puzzle about splitting land

A father is splitting up land among his two sons in estate planning. How can he divide the land fairly?

One approach is to split the land evenly. But even this method can get complicated if we add some realistic assumptions. Today’s puzzle illustrates why splitting land can be a mind-boggling exercise.

The problem

“Your father owns a rectangular field, from which the city has appropriated a smaller rectangular patch. He wants to split the remainder between you and your brother so that each of you two gets equal area.

How does he do this?”

And let’s add one more restriction: how can he split the land using a single straight line?

(Source: Janko Gravner, UC Davis, Advanced problem solving, Problem set 1)

Example shapes

Here are a few shapes to get you started in brainstorming. The big rectangle is the original land and the shaded grey area is the land appropriated (i.e. removed) by the city. How can you divide the remaining land equally using  a single line?

The solution is a general algorithm that should cover all of these cases:

Corner

Side

Middle, parallel

Middle, skewed

Hints

As a bit of history, this puzzle is sometimes used as an interview brain teaser or technical question when testing job seekers on their problem solving ability.

It is sometimes stated in the following terms: how can you split in half a rectangular piece of cake, with a small rectangular piece removed, using a single cut from a knife?

Can you figure out an answer?

The difficulty in this puzzle is the smaller rectangular piece can be in any place and in any orientation. Notice the removed piece does not have to be in a corner, on a side, or even be parallel to the sides. The removed piece can be angled and skewed and placed anywhere.

And while you’re brainstorming, remember the solution has to come from a single straight line division.

Solution:

Normally I have included answers in the post. At a reader request to avoid spoilers, I have moved the solution to the comment section

Alternately you can read another solution writeup here.



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  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    The elegant mathematical solution requires a small trick about geometry. The trick is that any line passing through the center of a rectangle bisects its area.

    (A line through the center of a rectangle either creates two equal triangles–if it is a diagonal–or it creates two equal trapezoids or rectangles)

    The original rectangular plot of land has infinitely many lines passing through the center that bisect its area. But once you remove a small rectangular plot, there is only one line that bisects the area–namely, the line that passes through the centers of both rectangles. This line bisects both the original plot and the removed rectangular plot, and consequently splits the land evenly.

    rectangle_skew_divide

    Source: http://w-uh.com/articles/030524-moving_Mount_Fuji.html#answer2

  • http://romwell.newgrounds.com Roman Kogan

    The linked write-up has a very incorrect statement:

    After you remove the rectangular piece, there is only one – the line which passes through both the center of the cake, and the center of the removed rectangular piece.

    In fact, there are still infinitely many solutions. One proves this as follows: pick line at any slope and start moving it continuously across the cake. Initially, the cake is completely below the line, and at some point, it is completely above the line. By Intermediate Value Theorem, at some time the line bisects the cake.

    There are thus as many solutions as there are directions of lines (uncountably many).

  • Jon

    Nice solution. But,
    “But once you remove a small rectangular plot, there is only one line that bisects the area–namely, the line that passes through the centers of both rectangles.”
    This is wrong. There are many other lines that bisect the area, they just don’t go through the center of the original rectangle. You should say “there is only one SUCH line that..”

  • Seth

    I agree, you answer is not unique, though it is a clever way to get one answer, if you only have a picture (and no measurements) A simpler algorithm that is more general:

    Create a line. Move it across the rectangle until it divides the area evenly.

  • Nathan

    Thanks for that, it’s a nice puzzle with an enjoyable ‘aha’ moment. Cheers,

    Nathan

  • Pingback: Simoleon Sense » Blog Archive » Weekly Wisdom Roundup #53 (Weekend Reading For The Smarter Types)

  • eye5600

    It’s also true the for any point in the interior, there is a line through that point which divides the area. Proof: Pick a point and a line. Note the resulting split. Rotate the line around the point. The split changes in a continuous way, and by the time you have rotated 180 degrees, the spit has reversed. By the mean value theorem, there is one angle which is an even division.

  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    Roman, Jon and Seth: Thanks for the correction to the solution.

  • Dave

    Here’s a different take on the question.

    The solution provided divides the land equally in terms of area, but realistically there are probably more or less desirable patches on the estate (e.g. fertile soil for farmers, or perhaps more underground oil for oil companies).

    A simple and fair (equitable) distribution method would be to have one son divide the land up into 2 pieces, and have the other son choose which piece of land he wants.

  • Joe

    I was asked this question during a recent job interview. My way of coming up with a solution was to rephrase the original puzzle by replacing rectangles with circles – i.e., a circle within a circle. When looking at the puzzle in this way, it’s more intuitive to see a line connecting the two centers being the best answer, and then you can extend the analogy to the rectangles.

  • Sam

    @ Dave: That is actually Celtic inheritance tradition. The youngest son divides the property, then the oldest son selects his section, and subsequent sons take their choices in birth order until the youngest son is left with whichever piece remains.





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