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This is a puzzle from one popular book called "The Man Who Counted: A Collection of Mathematical Adventures",author is Malba Tahan. How to arrange ten soldiers in five lines in such a way that each line contains four soldiers exactly?

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    Try this one: (1) Arrange ten soldiers into ten lines in such a way that each soldier is in three of the ten lines and each line contains three soldiers; (2) Do this in two different ways that are not incidence-isomorphic to each other. (Lots of people know an answer to #1. But #2 is also possible!)2011-09-07

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Like this:

$\hskip1.7in$ enter image description here

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    [pentagram and golden ratio](http://mathworld.wolfram.com/Pentagram.html)2011-09-11
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Five lines ten points

This is an alternative (sorry diagram is clunky)

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    @pedja: The golden ratio is not inherent in the pentagrammatic solution to the puzzle either. It appears in what one might call a regular pentagram, but [a pentagram that is not regular](http://en.wikipedia.org/wiki/File:Haykal2.gif) still solves the puzzle despite having little to do with the golden ratio.2011-09-11
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5 lines times 4 soldiers on a line equals 20 = two times 10 soldiers available. This suggests that every soldier belongs to two lines.

Draw $n$ lines on the plane such that no two are parallel, and no three intersect in one point.

You can always do that: if you already have $n-1$ lines then there are finite number of slopes of those lines and finite number of points of intersection -- choose a new slope not equal to any previous and draw the line with this slope not going through any previous points of intersection.

  1. Each line contains exactly $n-1$ points of intersection with other lines.

  2. There are $\frac{n(n-1)}{2}$ intersection points in total.

Now, if you put a soldier at every point of intersection, then there are $\frac{n(n-1)}{2}$ soldiers arranged in $n$ lines, each containing $n-1$ soldier. For $n=5$ you get the answer: any such configuration of 5 lines would work.

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A more irregular looking solution.

enter image description here

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    Well, I don't know in what precise sense they are the same either. All I meant is that the figure has been rotated 90 degrees and the lines have been moved around a little, but the structure is the same. One could generate a unlimited number of answers that way.2012-11-11