TECHNIQUE 3 OF 4
The final pattern does not depend on press order
Overlapping crosses may look as if they interfere unpredictably. In fact, each cell only needs a count of how many selected presses affect it.
The rule
For a planned set of toggle operations, a cell changes exactly when it is toggled an odd number of times. Adding those toggles in a different order does not alter that parity. Thus pressing A then B has the same final light pattern as B then A, even if their crosses overlap. Intermediate boards can look different.
A common mistake
Order independence does not mean every choice of buttons solves the board, nor that the intermediate positions are identical. It also does not override the live game's stop-on-win behaviour.
TRY IT 1 · lights-order-1
| C1 | C2 | C3 | C4 | C5 | |
|---|---|---|---|---|---|
| R1 | |||||
| R2 | |||||
| R3 | |||||
| R4 | |||||
| R5 |
What can you justify?
In the toggle sandbox, apply R2C2 then R2C3. Will R2C2 be on or off afterwards? Reversing the order must give the same final pattern.
Written explanation
R2C2 starts off and is toggled 2 times. An even number leaves it as it began, so it finishes off. Each cell has the same odd/even toggle count in either order. This compares final patterns, not intermediate boards.
TRY IT 2 · lights-order-2
| C1 | C2 | C3 | C4 | C5 | |
|---|---|---|---|---|---|
| R1 | |||||
| R2 | |||||
| R3 | |||||
| R4 | |||||
| R5 |
What can you justify?
In the toggle sandbox, apply R2C2 then R2C3. Will R2C4 be on or off afterwards? Reversing the order must give the same final pattern.
Written explanation
R2C4 starts off and is toggled 1 time. An odd number reverses it, so it finishes on. Each cell has the same odd/even toggle count in either order. This compares final patterns, not intermediate boards.
The same finish does not mean the same intermediate board
Use a practice sandbox with all lights initially off. Compare plan A, R1C1 then R5C5, with plan B, R5C5 then R1C1.
The tempting move
Claim that the displayed board after every move must be identical because press order does not matter.
Why it fails
After the first press, plan A lights the top-left corner cross and plan B the bottom-right corner cross. The intermediate patterns differ. Only after both presses have been applied do the plans produce the same final pattern.
What is actually justified
Order independence concerns the net result of the same toggle operations. It does not identify intermediate states, and this sandbox example is not a request to continue a live round after it has already ended.
Transfer the idea
Compare two plans: press R2C2 then R2C3, or press R2C3 then R2C2. The crosses overlap. Must their final patterns match, and what happens to a cell affected by both presses?
Check your reasoning
Their final patterns match. A cell affected by both is toggled twice and returns to its starting state. A cell affected by exactly one changes, and a cell affected by neither stays as it was.