To continue my previous post, we did not have to wait until tomorrow - in the evening (something I quite forgot about) I had to drive my boy to an art class. The Art Academy is not far away, still we managed to round up our discussion.
To cut a square from a corner into five parts of equal area, two opposite sides (thought of as a single - combined - length) had to be cut into five equal parts.
The idea easily extends to cutting a triangle with cuts from the incenter: cut the perimeter into as many as required equal pieces and join the points to the incenter. This will also work for any inscriptible polygon - my boy came up with the extension pretty fast, still on the way to the academy, so that on the way back we had to tackle a different problem. There were a couple from M. Gardner's The Colossal Book of Short Puzzles and Problems I had in mind: 9.9 and 9.6.
An infinity of nontouching points lie inside a bounded area. Assume that a million of those points are selected at random. Will it always be possible to place a straight line on the plane so that it cuts across the area, misses every point in the set of a million, and divides the set exactly in half so that 500,000 points lie on each side of the line? The answer is yes; prove it.
This is a simple problem but much removed from the common curriculum. The boy did have an intuitive idea that the points had to be counted somehow until one half of them were "on one side" but found it difficult to choose the right direction for the line; and the direction is what matters here most.
Draw all possible lines through 2 (or more points). The number of such lines, however large, is finite. Associate with every line its slope and a line with the same slope emanating from the origin (an arbitrary point). Since the number of lines is finite, it is always possible to choose a direction with the slope different from all the present ones. Take a line with this slop away from the region where the given points are located so that all of hem are on the same side from this line. Move the line in the perpendicular direction (so that it remains parallel to itself). In no position, the line may pass through more than 1 point from the given set. Therefore, eventually it will divide the point set into two equal parts.
We had time to look into another problem (9.6):
The figure below depicts a deep circular lake, 300 yards in diameter, with a small island at the center. The two small spots are trees. A man who cannot swim has a rope a few yards longer than 300 yards. How does he use it as a means of getting to the island?
This one the boy solved almost right away - to my great surprise because I initially misinterpreted the problem and had difficulty solving it. My first thought was that the problem requires laying the rope along the shore, but, of course, all the point of the problem was that this is not necessary.
I tried to modify the problem by changing the shape of the lake. For a triangle, we arrived at the notions of circumcircle and circumcenter, and the ideas extended to the cyclic shapes with a tree at the circumcenter.