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?
Ten soldiers puzzle
-
1Try 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
4 Answers
Like this:
$\hskip1.7in$
-
0[pentagram and golden ratio](http://mathworld.wolfram.com/Pentagram.html) – 2011-09-11
This is an alternative (sorry diagram is clunky)
-
1@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
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.
Each line contains exactly $n-1$ points of intersection with other lines.
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.
A more irregular looking solution.
-
0Well, 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