Part of the reason you've never heard a convincing argument is that there isn't a classical way of describing this.

Imagine Alice has some "magic" ink.

She can write in blue, yellow, magenta or green or combinations of them.

However, these inks are "magic" in that blue and yellow exactly cancel each other out. If you mix the blue and yellow inks together you're left with an ink in which ever colour there was the most of. It's not paler. If you mix in exactly the same quantities then the ink disappears completely.

Ditto magenta and green.

Alice mixes an ink colour together from these four inks. As some inks cancel each other out we can define the final colour by one number being the angle the final colour makes on a circle with magenta at the top, green at the bottom, blue on the left and yellow on the right.

These magic inks have another peculiar property. You can only ask "yes/no" questions to tell what colour they are and the act of asking the question forces the ink to the appropriate colour.

So, if you have a yellow ink and ask "is it yellow" then the answer is yes - with 100% probability.

If you ask "is it blue" then the answer is no - with 100% probability.

Things get weirder when you ask "is it green". If the answer is yes then the ink is green - so it changes from yellow.

Remember that these magic inks disappear if you mix eactly the same quantity of magenta and green. So if you have an ink it must be either magenta or green. Otherwise the ink would have vanished completely and there wouldn't be any ink to be asking questions of.

If the answer to "is it green" is no then that means the ink must be magenta instead - so a no answer to "is it green" turns the ink magenta.

Because a yellow ink is neither green nor magenta in reality but the "is it green" question requires an answer, the answer will be randomly yes or no with 50-50 probability.

N.B. Because the ink actually changes colour after the first "is it green" question repeatedly asking the same question will get the same answer.

But you don't have to limit yourself to these two axes - instead we can ask "is it a 50/50 magenta/yellow mix. If the answer is yes then it really is a 50/50 magenta yellow mix. If the answer is no then it's a 50/50 blue/green mix instead.

If the ink started out as yellow (we asked "is it yellow" and the answer was yes) and then we ask "is it a 50/50 magenta/yellow mix" then we're more likely to get a yes than a no answer.

If we make the measurement we find that the probabilty of getting a yes answer is the square of the cosine of half the angle between the initial colour and the question we ask.

Now to Alice, Bob and Carol's problem. Does the ink "know" the answer to any question you can ask of it.

Let's say that Alice used yellow ink.

If Bob and Carol both do the "is it yellow" test then they both get a yes answer.

But instead, lets assume that they both do the "is it green" test. We've already established that the answer to this is yes or no with 50/50 probability. So we can get yes/yes, yes/no, no/yes and no/no as the four possible combinations of answer. But... where it gets weird is where the ink on the two letters is entangled. Bob and Carol both get the same answer - i.e. the only answers are "yes, yes" and "no, no"

This is what entangled means - if you make the same measurement you get the same answer (where same actually means opposite in most experiments)

(in practice you have to repeat the experiment over and over again to establish that B&C always get the same answer to whatever question they ask. To avoid problems where the ink "knows" what test will be done in advance when Alice mixed it they randomly decide what test to do at the last possible moment and then compare answers and only use the cases where they randomly chose the same test to do)

Lets assume that Alice writes in yellow ink that is entangled between Bob and Carol's letters.

If you actually work out the maths for different angles between the sender and the receivers you discover that there's a difference in the probability of getting certain results depending on whether there is a hidden variable.

If Bob asks "is it yellow" and Carol asks "is it yellow with 1% green[1]" then they get the same answer in all but 8 times out of 100000 when Carol gets a no answer. If there's a hidden variable then it will have a no recorded 8 times out of 100000 that Alice mixes the ink for this test.

If Bob asks "is it yellow with 1%magenta" and Carol asks "is it yellow" then they also get the same answer in all but 8 times out of 100000. Ditto - a hidden variable encodes a no 8 times out of 100000 for this test.

If Bob asks "is it yellow with 1%magenta" and Carol asks "is it yellow with 1% green" then assuming that the ink knows what the "right answer" for any question is then the MAXIMUM number of times B&C will get different answers is 16 (8+8) per 100000 tests.

However. if we start with "yellow with 1% magenta" and then ask "is it yellow with 1% green" we get a no answer 30 times out of 100000.

When Alice, Bob and Carol do this test they find that the times that Bob and Carol get different answers when they both measure "off yellow" is around 30 times per 100000. This shows that the ink cannot "know" the right answer to give to any question in advance.

[1] When I did these calculations just now I assumed that yellow with 1% magenta was 1 degree from yellow.