While my students are taking their final exam, i found myself playing with my whiteboard markers. I have 23 markers in all. One of my many habits is that I enjoy snapping my markers together to make long line segments. Unfortunately 3 of my markers will not snap together properly meaning that at best I have three line segments. therefore each line segment is considered a set marker length out of the total markers. E.g. a marker group of 17 markers has length 17/23.
Now the question: If i create three marker lengths using all of the markers, and multiply the length of one line segment by the reciprocal of another by the average marker length of all three
meaning for example:
1st marker group: 17/23
2nd marker group: 2/23
3rd marker group: 4/23
average marker group: 23/69
17/23 * 23/2 * 23/69 = 391/138, approx 2.83
what combination would give me the smallest value? largest value? Is there a process or means to generalize this for n markers?