Est. 1998 — 27 years of woodworking knowledge
Friday Puzzler -- six knots
Alex Y
Six lengths of rope are passed through a pipe, so they stick out each end. They are clamped at both ends of the pipe, so that it is impossible to tug at individual ends to find out which ones go together. You tie pairs of rope ends at each end of the pipe together, three knots at each end of the pipe.
How likely is it that when the clamps are removed and the ropes pulled through the pipe, you will find the six ropes to be joined in one continuous loop?
Re: Friday Puzzler -- six knots
Larry Barrett
The odds of having a loop after tying 6 ropes in pairs at each end is about 50/50.
Re: Friday Puzzler -- six knots
Alex Y
Correct, with "about" being a key word. 
Unless you are estimating without actually calculating, go ahead and give the actual along with your reasoning.
If you are estimating, how did you come up with that estimate?
Re: Friday Puzzler -- six knots
Larry Barrett
Here is my calculation:
Label the ropes at one end A, B, C, D, E, F; and at the other end a, b, c, d, e, f.
Tie the ropes in pairs at one end. It doesn't matter how you do this, so assume A is tied to B, C is tied to D, E is tied to F.
Now at the other end, select any rope, say a. To end up with a loop, a must not be tied to b. So having selected a, the probability that you select c, d, e, or f is 4/5.
Assume that you select c, so a and c are tied together and there are 4 ropes remaining.
If you select b (probability = 1/4), then to end up with a loop, b must be tied to either e or f, but not d; so this probability is 2/3. Assume you selected e. Then the only remaining ropes are d and f. Tie these together and the bundle will form a loop. Combined probability = 1/4 * 2/3 = 1/6.
If instead of selecting b you select d, then to end up with a loop d must be tied to e or f, but not b. This is same logic as above, so the combined probability = 1/6.
If you select e or f (assume e with probability= 1/4) instead of b or d, then to end up with a loop, e must be tied to b or d, but not f with probability = 2/3.
The combined probability is again = 1/6.
Same logic for selecting f, with combined probability = 1/6.
So the probability of a loop after tying ropes in pairs at each end is
4/5 * 4/6=8/15 - a little better than 50/50.
Correct!