One of the joys of writing this column is the opportunity to play with unusual branches of mathematics. This puzzle mixes geometry, graph theory and a little topology yet is simple enough to explain to a 10-year-old (I have done it).
The idea is simple. You will be told about individual lines and other lines that they cross. You will be given certain crossing information about segments between line intersections and then asked to figure out information about other segments. Sometimes there will be many possibilities.
Let's try a simple example:
Start with a line L1.
Line L2 crosses L1.
Line C1 crosses L1 and L2, like so:
So too do lines C2 and C3.
Denote the intersection of L2 and C3 by (L2, C3). Use the same labeling for the intersections (L2, C2) and (L1, C3).
How many lines does the segment between (L2, C3) and (L2, C2) cross? You'll see that there is no unique answer. In this figure the segment crosses L1 and C1:
In this figure the segment crosses no lines at all:
Warm-Up Problem: Can you draw a situation in which the segment from (L2, C3) to (L2, C2) crosses just one line?
Solution to Warm-Up:
Here is a new specification:
L1 crosses L2.
C1, C2, C3, C4, C5, and C6 all cross both L1 and L2 but do not cross one another. So far, then, the C crossing lines may have any relationship to the intersection point of L1 and L2.
Suppose however, I also tell you that:
(L1, C6) to (L2, C2) crosses C1, C3, and C4 (although perhaps not in that order).
(L1, C5) to (L1, C1) crosses L2 and C6 but perhaps not in that order.
(L1, C4) to (L1, C2) crosses C3.
Finally, C2 is below all other crossing lines (so the highest point of C2 is below the lowest point of any other line) as well as the intersection (L1, L2).
1. Can you order the crossing lines from bottom to top?
2. Which crossing lines must be above the intersection (L1, L2) and which must be below it?
3. Suppose we change the lines to segments in the above specification. That is, L1, L2, C1 and the rest have finite lengths, and in some cases they can avoid intersecting by falling beyond one another's end points. In that case, which C segments must intersect L1 and L2 below the intersection point (L1, L2) and which C segments must intersect L1 and L2 above that point?
4. How would your answer to the previous question change if the specification "(L1, C6) to (L2, C2) crosses C1, C3, and C4" changed to "(L1, C6) to (L1, C2) crosses C1, C3, and C4." That is, (L2, C2) changes to (L1, C2).
I conclude this puzzle with two challenges that I have not solved. These are generalizations of the simple example and the warm-up problem above.
5. Suppose there is some specification for a set of lines that is satisfied by at least two figures: In the first, segment S from some point (L1, L2) to other point (L3, L4) crosses m lines. In the second, segment S from (L1, L2) to (L3, L4) crosses m+k lines (k is greater than 1). Then must it be the case that for any integer i between 1 and k, there is a figure satisfying the specification such that segment S from (L1, L2) to (L3, L4) crosses m+i lines? In the warm-up problem, we identified one figure with two crossings, and one figure with zero crossings. There was also a figure with one crossing.
6. How does your answer to question 5 change when the figures involve line segments rather than lines, and where crossing includes crossing at an endpoint?