A special kind of knot

Imagine a ladder with one rung, which is not really a ladder, but imagine it. If you loop a piece of string around the rung exactly once and tie it, you can't remove it from the ladder (assuming the sides of the ladder are too long to do so). But if you remove the one rung, you can.

Now suppose the ladder has two rungs, and you want to loop a piece of string around the two rungs so that you can't remove it, but removing any one rung allows you to do so. Again, the sides of the ladder are too long to pull the string over.
How do you loop a piece of string around two rungs so removing either rung frees it?

If you can do two rungs, try for three rungs.
How do you loop a piece of string around three rungs so removing any rung frees it?

The general case is left for those seeking a challenge.

• What is your question? Nov 17 '16 at 21:41
• @BeastlyGerbil I think it's pretty obviously 'how do you tie such a knot' or 'what does this knot look like' or equivilent Nov 17 '16 at 21:43
• Yes. I guess a good answer would give visualizations of the two and three rung case, as well as an intuitive method to come upon those loops.
– cats
Nov 17 '16 at 22:24
• Did you compose this puzzle yourself? I seem to recall having seen something similar a few years ago. (I realize that you could have created it independently.) Nov 18 '16 at 5:07
• Here is the same question: math.stackexchange.com/questions/786284/… Nov 18 '16 at 16:15

like this.

you need to loop the rope so that any rung can release it

The general case

To add a rung seems double the string (so there are two parallel strings) following the old path. then cut the loops and knot the parallel ends around the new rung such that the same strings pass over and under the new rung. I would not like to attempt that on a ladder with more than 6 rungs.

• this is as far as I can tell correct, though the description for the general case is a little confusing
– cats
Nov 18 '16 at 22:40

The two rung case is relatively simple:

Loop the string around each rung like this:

and then tie the string ends together to complete the loop.
Removing either rung frees the string to pass through loop around the other ring.
This will release the string entirely.

This would be easy to explain if I had any artistic skills.

Tie the ends of the string, making it into a loop. Drape the loop over one rung. Take one of the ends that hang down and pass it through the other. You now cannot remove the string without removing that rung or letting go of the string.

Take the end of the loop you are holding and drape it over the other rung. Pull it around the rung back up, and wrap it around the strings connecting the two rungs. You'll need to untie the knot and retie it again, but you can do that.

Now, if you remove either rung, then you can just pull the loop through itself and remove it from the ladder. Otherwise, you can't remove it without untying or cutting the string.