In reply to maxsmith:
"Can you explain this again in a different way?"
> Each time you put the rope over you head you put a twist in the rope in the coil. <
This is key - the rope in the coil doesn't look twisted but if you straighten it out without uncoiling you will see that the coils=twists [try wrapping a bit of ribbon around a rolling pin and then removing the rolling pin and pulling straight, one twist per turn].
So if you make 10 coils, drop on the ground then tie in and pull the rope straight you have 10 twists. Repeat a few times (bearing in mind you're probably "handed" so will probably coil with the same added twist each time) and you'll soon have loads of twists and the rope with tangle like a bastard.
If you both tie in first you cannot have a net number of twists unless one of you turns around so the +10 twists in the coil must have -10 twists in the rope between the two of you*. When you uncoil these all go away. If you drop the coils and straighten you'll end up with a mix of + and - twists which will cancel each other out as the rope is pulled through and the twists won't accumulate a make tangles.
Hence the 2 top tips: both tie in first (ensure no accumulation of twists) and uncoil rather than drop (ensures twists are undone).
* there is a tendency to let the rope rotate in your hand as you coil so the -ve twists in the loose rope tend to get carried through to the coil so in practice you end up with something like 5-ve twists in the loose rope, -5 twists in the coils and 10 coils which would make +10 twists if you pull it out. Net twists 0 assuming the ends can't rotate (ie you are tied on).
Do the ribbon test - it's much easier to try it than explain it!