How to coil a line so that it has a good chance of running freely when heaved?

Surely it depends on the rope construction.
The twist as you coil technique is / was important for traditional 3 strand rope (and must be done the correct way round). But it is wrong and should be avoided for many modern ropes.
And for throwing I was always taught never to try to throw a single coil, but create two loose coils, one in each hand, throwing the first and then following with the second.
 
Surely it depends on the rope construction.
The twist as you coil technique is / was important for traditional 3 strand rope (and must be done the correct way round). But it is wrong and should be avoided for many modern ropes.
And for throwing I was always taught never to try to throw a single coil, but create two loose coils, one in each hand, throwing the first and then following with the second.
I'm surprised it took this long to mention the diferences in construction between 3 strand and braided.

When done correctly a braided line will naturally fall in to a figure 8. Problems arise when it is forced with a twist to lay 'neatly' with no natural twist. 3 strand should however be coiled neatly in a plain loop. Generally for shorter liners, I'll coil but longer lines, I'll do a butterfly flake which I learnt from climbing, much quicker and easier.

As to throwing agree with Dunedin, few loos coils in each hand with the non throwing hand held open to allow it to pay out easily.
 
We are talking about two different situations here:

When you want a line (say a halyard) to run through a clutch and then some mast sheaves you don't want any twists in it. If you start with a conventional coil you will be taking line off the top and so will end up with a twisted line. This is where flaking or fancy coils are useful.

When you are heaving a line you throw (some of) the coils and so the line unwinds as if it was on a reel. Thus a conventional coil will not introduce any twists.

Winding it round your elbow introduces exactly the same amount of twist as a conventional coil so makes no difference to heaving in that respect. The difference is that when coiling in hand it is easier to keep the coils in order rather than doing later ones over the top of earlier ones. That is what results in coils passing through each other which is how tangles occur when heaving. You wouldn't want to do the sort of coil where you put a backwards twist in every other coil for heaving: if you throw it wrong you will end up with an overhand knot for every two coils.

My preference for heaving is to coil into my throwing hand, then move half the coils over into my other hand in order (bitter end closest to the heel of my hand, working end closest to the fingertips) so they will come off nicely. This is not the fastest option, but eliminates any risk of tangles which means that every throw has the greatest chance of success.

For me it's bitter end at heel of one hand, working end at heel of other hand. Otherwise half is inside out?

I coil down about half, then put a finger over the top of the coils and then keep coiling. Transfer the 'top' half to my right hand, and then throw. This splitting then keeps the working end at the far right, at the heel of my right hand.
 
For me it's bitter end at heel of one hand, working end at heel of other hand. Otherwise half is inside out?
Sorry yes I was only describing the arrangement in the non-throwing hand. I don't tend to worry so much about the throwing hand, because the spinning of the coils as they fly through the air makes them less likely to tangle.
 
I don’t believe there is a ‘perfect’ way of coiling a line, all lines have an inherent evil which ensures a tangle if left alone for more than 30 seconds. I vary the method depending upon what the line is, halyards, sheets etc are usually flaked, dock lines tend to be coiled as per lifeboatman, unless for instant heaving across to someone in which case definitely flaked.
I learned last year after c.30k miles and thirrty years.....that lines have different evils. 3 strand tends to like a "proper coil" with a half twist. Stiff modern braid doesn't like to be coaxed into any particular pattern.

So as I was advised, and became convinced: do try to do a coil with 1/2 twist using 3 strand. For all other ropes, pass it through your hand "as it comes"; some handfuls come naturally as a figure 8 flake, others as a round coil...and that's fine. I have found it's much better this way.
 
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