Every day, a man comes to my door with a United States nickel in his hand. He asks me whether I’d prefer to examine the heads side (which is always painted either black or white) or the tails side (which is always painted either red or green). I choose each day according to my whims.
And the same thing happens to my sister. Different man, different coin, but each day he’s there with a painted nickel, offering to let her examine either the heads side or the tails side.
Sometimes we call each other to compare notes on the colors we’ve seen. Here’s what we’ve concluded:
- Our heads sides are never both white.
- Whenever one of our tail sides is green, the other one’s heads side is white.
We have thousands of observations to support these conclusions: On days when we both examine our heads sides, we never both see white. On days when we examine opposite sides and one sees a green tail, the other always sees a white head.
The Brain Teaser: Today we both chose Tails and both saw green. What colors were on our Heads sides?
Solution: By point 2) above, they were both white. But by point 1) above, that can’t happen. So….?
So what now?
Possible Resolution I. There are no such coins. You’re right. There are no such coins. But there are subatomic particles that behave exactly like these coins. It’s easy to set up an experiment where every day, my sister and I each receive an electron. We can examine the spin of our electrons in either the up/down direction or the left/right direction. And here’s what we find:
- Our electrons are never both spin down.
- Whenever one of our electrons is spin right, the other is spin down.
- There are days when both of our electrons are spin right.
By points 1) and 2) and the same simple logic we used for the coins, there can never be a day when both electrons are spin right. Nevertheless there are such days. In fact, on those days when we both choose to make left/right measurements, our electrons are both spin right about 8.3% of the time. That’s not a huge percentage, but it’s sure not zero either.
So Possible Resolution I doesn’t work, at least if we replace the coins with electrons.
Possible Resolution II: The coins are very very sneaky and they like to screw around with our minds, so they change their own colors depending on the choices we make, just to fool us. Sometimes they are both white on the heads side, but not on the days when we’ve both decided to check the heads side — so that Rule 1 is false but we get fooled into believing it.
Aside from its inherent implausibility (intelligent coins? intelligent coins with nothing better to do than to mess with our heads?), this resolution falters on the fact that my coin has no way of knowing what choice my sister is making.
Possible Resolution III: Neither side of either coin has a color until we decide to examine it, so that on a day when I examine my tails side, it makes no sense to ask about the color of the heads side in the first place.
Therefore I am not allowed to pose this brain teaser in the first place.
If that strikes you as implausible, I invite you to devise another Possible Resolution. Plenty of people have done so. None, though, has ever devised a Possible Resolution more plausible than Possible Resolution III.
Welcome to quantum entanglement.
If you liked this example, you’ll love Chapters 14 and 15 of The Big Questions.
Technical Appendix: For the cognoscenti, I wrote R=B+W, G=B-W, and assumed the coins are prepared in the initial state BB+BW+WB. Rule 1 follows. You can check that this state is equal to both 2BR+WR+WG and to 2RB+RW+GW. Rule 2 follows. You can check that this state is also equal to 3RR+RG+GR-GG, so that when both parties make R/G observations, GG comes up 1/12 (about 8.3%) of the time.
