The One-Page Worksheet That Proves Reality Isn’t “Pre-Set”

Imagine three friends: Alice, Bob, and Charlie. Each has a magic box. The boxes were built together in a special “quantum” way, so the three boxes are linked even if the friends are far apart.

Each friend can choose one of two buttons to press:

• Button X
• Button Y

When someone presses a button, their box flashes either:

• +1 (think “green”)
• −1 (think “red”)

The weird part is not the individual flashes. The weird part is the pattern across all three boxes.

The GHZ setup (the rule the boxes always follow)

The boxes are linked so that these four “team rules” are always true:

1. If Alice presses X, Bob presses X, Charlie presses X then the product of their three results is −1.
2. If Alice presses X, Bob presses Y, Charlie presses Y then the product is +1.
3. If Alice presses Y, Bob presses X, Charlie presses Y then the product is +1.
4. If Alice presses Y, Bob presses Y, Charlie presses X then the product is +1.

“Product” just means multiply the three results. Example: (+1)·(−1)·(−1) = +1.

So the boxes are guaranteed to obey those four multiplication rules every time.

What “common sense” would try to say

A very normal idea is:

Each box already “knows” what it would answer for X and what it would answer for Y, before anyone presses anything.

So for each person we imagine two pre-set answers:

• Alice has A(X) and A(Y)
• Bob has B(X) and B(Y)
• Charlie has C(X) and C(Y)

Each of these is either +1 or −1.

This is the “no magic messaging” + “pre-set answers” idea.

The trap (where common sense breaks)

Look at rules 2, 3, and 4:

2. A(X) · B(Y) · C(Y) = +1

3. A(Y) · B(X) · C(Y) = +1

4. A(Y) · B(Y) · C(X) = +1

Now multiply the left sides of (2)(3)(4) together:

Left side becomes: A(X) · B(Y) · C(Y) · A(Y) · B(X) · C(Y) · A(Y) · B(Y) · C(X)

Group them by person:

• For Alice: A(X) · A(Y) · A(Y)
• For Bob: B(X) · B(Y) · B(Y)
• For Charlie: C(X) · C(Y) · C(Y)

Now use a simple fact:

• Since A(Y) is either +1 or −1, then A(Y)·A(Y) = (+1) always (because (−1)·(−1)=+1, and (+1)·(+1)=+1)

So A(Y)·A(Y)=1, same for B(Y)·B(Y) and C(Y)·C(Y).

That means the whole multiplied left side simplifies to:

A(X) · B(X) · C(X)

So multiplying rules (2)(3)(4) forces:

A(X) · B(X) · C(X) = +1

Because the right sides were +1·+1·+1 = +1.

But rule 1 says:

A(X) · B(X) · C(X) = −1

Both can’t be true.

So the “pre-set answers” idea crashes into a contradiction.

What the GHZ paradox is saying

• If you believe each box carries hidden, pre-written answers for X and Y…
• and you believe the boxes don’t send signals to each other faster than light…
• then the four rules above are impossible to satisfy.

But quantum physics says those four rules can be satisfied (and experiments agree).

So GHZ is a “no excuses” proof that nature can’t work like: “everything already had a definite value and we’re just revealing it.”

It’s not just “quantum is weird.” It’s: common-sense hidden answers cannot explain what really happens.

A simple analogy (not perfect, but helpful)

Think of it like three synchronized dice that don’t decide their numbers until you choose which kind of roll you want (X-roll or Y-roll). And the “team multiplication rule” always comes out right. If you try to pretend each die had both answers written in advance, you get stuck: the math forces two opposite outcomes at the same time.