TECHNIQUE 2 OF 4
Two identical toggles cancel
Repeated presses can be simplified before you perform them. What matters for each button is whether it is pressed an odd or even number of times.
The rule
Pressing the same square twice toggles every affected cell twice, returning those cells to their earlier states. Three presses have the same net effect as one; four have the same effect as none. In a planned sequence, identical presses can be removed in pairs. This is a statement about the resulting light pattern, not the recorded move count.
A common mistake
A cancelling pair does not erase two moves from the game's counter. The live round also ends as soon as all lights are off, so use the practice comparison if an intermediate press would finish the round.
TRY IT 1 · lights-cancellation-1
| C1 | C2 | C3 | C4 | C5 | |
|---|---|---|---|---|---|
| R1 | |||||
| R2 | |||||
| R3 | |||||
| R4 | |||||
| R5 |
What can you justify?
In the toggle sandbox, apply R3C3 then R3C3. Will R3C3 be on or off afterwards?
Written explanation
R3C3 starts on and is toggled 2 times. An even number leaves it as it began, so it finishes on. The two identical presses cancel as light operations, even if an intermediate sandbox board is dark.
TRY IT 2 · lights-cancellation-2
| C1 | C2 | C3 | C4 | C5 | |
|---|---|---|---|---|---|
| R1 | |||||
| R2 | |||||
| R3 | |||||
| R4 | |||||
| R5 |
What can you justify?
In the toggle sandbox, apply R3C3 then R3C3. Will R2C3 be on or off afterwards?
Written explanation
R2C3 starts off and is toggled 2 times. An even number leaves it as it began, so it finishes off. The two identical presses cancel as light operations, even if an intermediate sandbox board is dark.
Different buttons do not cancel completely
In a practice sandbox, compare pressing R1C1 once and R1C2 once on the same five-by-five starting pattern.
The tempting move
Predict no net change because the two presses are next to each other and their crosses overlap.
Why it fails
Both presses affect R1C1 and R1C2, so those cells are toggled twice. But R2C1 is affected only by the first press, and R1C3 and R2C2 only by the second. Those three cells toggle once.
What is actually justified
Only the shared part cancels. Removing a pair of identical button presses is valid; removing two different presses solely because their crosses overlap is not.
Transfer the idea
Ignoring the move counter, what is the net effect of pressing R3C3 seven times? What if you press it eight times?
Check your reasoning
Seven presses have the same effect as one because six presses cancel in three pairs. Eight presses cancel completely and leave the light pattern unchanged. The actual number of button presses is still seven or eight.