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# playing with markers pt. 2

## Question

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 MEDIAN marker length of all three

for example:

1st marker group: 17/23

2nd marker group: 2/23

3rd marker group: 4/23

Median marker group: 4/23

17/23 * 23/2 * 4/23

what combination would give me the smallest value? largest value? Is there a process or means to generalize this for n markers?

## 2 answers to this question

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This one is easy. Suppose there are N markers and M broken ones.

Then, divide the markers into M-1 groups of 1, and one group of N-M+1. Multiplying the length of any of the small groups (1/N) by the reciprocal of length of the big group (N/(N-M+1)) and the median length (1/N) gives you the smallest value, which turns out to be 1/[N(N-M+1)].

In this case it is 1/483.

Edited by gavinksong

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Divide the markers into lengths 1, 11, and 11.

The largest value is the reciprocal of the smallest length multiplied by the remaining lengths. This gives 121/23.

For different values of N (number of markers), we just divide the markers into lengths 1, floor((N-1)/2), and ceil((N-1)/2). The largest value in these cases is again the reciprocal of the smallest length multiplied by the remaining lengths.

Edited by gavinksong

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