Even if both players choose their number of fingers randomly, this still isn’t fair. The other numbers all have two ways to come up, but a product of 4 (result of 3 in your mapping) can come up three ways: 14, 41, or 2*2.
You’re also neglecting 16, but I guess that’s just a re-roll.
I like this solution the best. I propose a better variant. Throw a coin four times. Six times out of 16 you will get two heads and two tails. Assign each one of the six a number 1 to 6. Ignore the other 10 possibilities. So you get a number 37.5% of the time. I haven’t figured it out, but I guess you will need more throws to get 3 binary digits than you will here to get a number between 1 and 6.
You missed the fact that he said re-roll if both players put out the same number of fingers. So it doesn’t suffer from overcounting. But it does still suffer from a player being able to influence the outcome unilaterally.
Find six pebbles of similar, but clearly different sizes (representing 1-6). Player one buries them in the sand in a random no pattern area at the same depth about the size of a dinner plate with out player two looking and smooth’s out the top of the sand. Player two would then poke a finger in the sand and with a swirling motion pulls out the first rock that gets exposed.
You may need a flat surface to place the rocks under the sand so that they are all the same depth.
Have 6 distinct objects, and 6 numbered areas of sand. Have one person put one of the objects on each of the areas of sand while the other is looking away, then the former person moves into the sight of the latter but the latter remains without sight of the objects and names an object. The person positioning the objects would be unable to influence which object was on which area while in view of the person making the selection, and so could not change things once he knows what the other person is going to pick. The person without sight of the objects that picks one has no knowledge of where each of the objects are. Thus neither player can meaningfully physically influence the outcome, but might through psychologically outwitting the other person. That might be seen as a virtue to this system, not a drawback, adding to the game that one is playing. So not random, but just as good.
I think the OP is flawed - I don’t think it’s that hard to make dice which are fair enough to play backgammon with. They don’t have to be cubes, barrel dice or top dice are probably easier to make with primitive supplies, as you can turn them - make a bow lathe.
Growing up back in India in the 70s, there were many traditional board games that were played entirely with natural supplies. This is how we did it :
Take tamarind seeds or any relatively flat and hard dicotyledon seed. Rub (grind) one side of the seed on a hard surface (concrete floor or flat rock) or split the seed into two. Now you have the equivalent of a small coin with a head and a tail.
So for 6, you will use 5 seeds. Each seed (or half of it) will have two sides, and you can assign them 0 and 1. You take the seeds in your hand, close your fist, jiggle them a bit and gently throw them on the ground. You will now get any random number between 0 and 5.
You can always designate 0 as 1, and 5 as 6 and this will work just as good as a dice.
Also in olden times, Cowry (or Cowrie ) shells were used as dice too. Cowrie - Wikipedia See picture at bottom of the linked wiki page.
Find 6 seashells as nearly identical as possible - pick smallish sea-snail type shells where the opening is too small for a finger
Using a piece of grit stuck in the end of a stick, drill a number of dots on the inside of the opening
Construct a bag from available materials (weave it from grasses or make a small round basket from palm fronds) - with a large enough opening to admit a human hand
Put the shells in the bag and shake it - then reach in and take one out - count the dots and put it back.
I think it would be fairly easy to find small sticks/reeds that could be cut/shortened to look nearly identical. They don’t have to be long and if they start to wear replacements are readily available. If one player insists on cheating by trying to study the sticks, then they can always be made to draw with eyes closed or looking away. This really seems to me to strike the best balance between being fair/random enough, easy to find/prep resources, and not impact the speed of the game with needlessly complicated “rolling” schemes.
The trouble with fingers is that it easily leads to cheating or accusations of cheating depending on how fast one of the players can move their fingers or believes the other party can. Palming one of 6 easily distinguishable pebbles and revealing it doesn’t have that problem and finding 6 distinctive pebbles is easy.
My post#17 is really no trouble at all, and works perfectly as a die if either party accepts their own choice as random.