The 4 squares problem – extended

May 1st, 2010 admin Posted in Family, Tidbits | No Comments »

A “4 Squares Puzzle” is doing the rounds over the internet nowadays. here are the original Microsoft Powerpoint slides which I received via email from my father. (Not sure who the original author of these slides is, but the slides report the author to be Nakit Yonetimi.) It may be useful to look over the pdf file once before proceeding. The question posted in the slides is different from the one I am going to pose, but going through the slides helps build context and helps get mentally warmed up.

The question I posed to myself after thinking through the puzzle was, “How can we divide a square into 7 equal parts with only a straight edge and a compass available?” Note that the question implies that we do not have a ruler or a scale. We have a straight edge, but without any markings on it to indicate inches or centimeters. Even if it did have the markings, such markings can only measure accurately up to a certain level. For example, say you have a scale with markings at the granularity of a millimeter. Say the square had a side equal to some irrational number, say or , or, even a simple integer which is not a multiple of 7, such as, 8 millimeters. There is no way to measure or or millimeters using such a scale.

There are at least 2 approaches to dividing this square into 7 parts. The first is a simpler approach and the second is slightly more involved. Let me talk about the second one first. The first one will then become easy to see. The main intuition behind the first idea is that a triangle’s area depends only on its base and height. If we can mark out 7 equidistant points along the square’s border, thus creating 7 equal bases, we can join the bases to the center of the square to create 7 regions with equal areas. The heights of these shapes will be equal, and the bases are equal by construction.

As shown in the figure to the far left, the corners of the square are joined, and the intersection gives us the center of the square. Then, assuming we already have the 7 equidistant points of the square’s edge marked out,  joining the center to the 7 points creates 7 regions with equal area, as shown in the figure to the immediate left. It may not be obvious why these regions are equal in area. If you divide up the quadrilateral regions (regions with 4 edges) into 2 triangles, you should start seeing that the area of the quadrilateral piece is the same as the area of a triangular piece since they both have the same height (, where is the length of the square’s side) and same base ().

The question then is, how do we make those 7 markings on the square? Let us first imagine that we peel off the edge and lay it out along a straight line. If there were some simple way to then divide the peeled out, straight line edge into 7 equal parts, we can visualize rolling the laid out edge back on to the square. Of course, the actual process would involve measuring the widths of each piece using the compass and then laying them back on the square’s border. At the corners these segments may have to be further broken down into two pieces. How to do that it is relatively intuitive with a compass and I will not go into explaining that. The next two figures illustrate the process of peeling off the edges of the square and laying them onto a straight line, and the process of rolling the laid out edge back on to the square once the 7 equidistant markings have been made.

The question now remains, how do we divide the laid out border into 7 equal pieces. More generally, this question can be posed as, how can we divide any line segment into n equal segments.

The figure to the left illustrates the steps. The steps are labeled (a) through (e). In Step (a), we draw a line using the straight edge at some angle to the line segment to be divided. This edge is shown in a light color and its length does not really matter. Then in Step (b) make 7 equidistant markings on the newly drawn line segment using the compass. The length between the markings does not matter. Just pick something that seems to fit in the space. Then, in Step (c), the 7th such marking and the end of the original line segment are joined. In Step (d), using the compass, identify a point corresponding to the 7th marking, on the other side of the base line segment. Connect the ends of the base line segment to that point and complete the paralellogram shown in the figure in Step (d). Using the compass mark out the other 6 markings on the line on the other side of the base line segment. This is shown in Step (e). Join the corresponding intermediate markings using the straight edge. Voila! The line segment we wanted to divide into 7 pieces just got divided into 7 equal pieces.

Now, by retracing the steps laid out in the figures above, you can divide the square into 7 equal regions. This question is also posed as, “How can you divide a square cake into n equal sized pieces?”. Now that we know how to divide a line into n equal parts using a straight edge and a compass, we may realize that there is a simpler way to divide up the square cake. Instead of dividing up the whole border into 7 equal-length line segments, why not just divide up one side into 7 equal length segments. Then we can cut the square into 7 long rectangles, in a fashion similar to the one proposed in the original puzzle (see the pdf slides posted at the beginning of this article).

This also brings up another interesting question, although quite off-topic, it may seem. We have been using the phrase straight edge throughout this article. But what is straight? How was the first straight edge made? My guess is that a taut string might have been used to define straightness. But then what is space itself is curved? Does that mean a taut string is also actually curved in accordance with the space it is in? What does it even mean to say that space is curved? Curved with respect to what? What is the underlying measure of straightness, against which the space appears curved? There are probably questions for another disussion. I do not know the answer. I will have to think and read a bunch before it may be clear. It may, of course, never be clear. There is no reason everything should make sense. Nature is not obligated to be understandable by humans, much less one particular human that I call me.