## Monday, March 29, 2010

### The Catenary Chain

This week's post on Steve Strogatz's New York Times column reminded me of this puzzle.  In it he re-explains math in a creative and intuitive way, from basic counting through imaginary numbers, functions, and more.  The post reminded me of this because in it he talks about how mathematical functions can explain every-day shapes, such as how water at a drinking fountain forms a parabola.  A hanging chain forms a catenary, but its shape can be approximated with a parabola.  It is a great new one I found at Wu Riddles, and apparently it originated at a Microsoft Interview.  BUT: don't be intimidated by all the math, I have faith that you can solve this one.

You have a 6-foot long chain that is suspended at its ends, tacked to a wall. The tacks are parallel to the floor. Due to gravity, the middle part of the chain hangs down below the ends, forming a 'U'-type shape; the height of this 'U' is 3 feet from top to bottom. Find the distance in between the tacks.

Solution:
As the commenters pointed out below, the distance between the tacks is zero.  The only way for the dip in the chain to be three feet is for the chain to drop 3ft straight down, then 3ft straight up.  There is no chain left to spare for horizontal distance.

1. I'm gonna go out on a limb here and say the tacks are right next to each other. If the chain is only 6 ft long then it needs to hang half its length to reach 3 ft.

A guess but my best guess.

2. Zero, sucka! Less of a U shape, more of an I.

3. Nice. Yeah I started trying to recall geometry and parabolas and discriminants, etc. before realizing that 3+3=6

4. The sum of two sides of a triangle cannot be equal to or less than the length of the third side. So even if the chain was taut (imagine a weight is pulling down the center, if you will), no side in the resulting isosceles triangle could be 3'.

That is, of course, unless the two sides are touching and it is not a triangle at all!

Basically, I'm just agreeing with the other two people above me.

5. way to bring back the Triangle Inequality Theorem, Jon Hopper...