1. Cross Lights study 4
3 steps in the included checked route. Try the board first, then compare the reasoning.
PRACTICE COLLECTION
Compare sequences by their effect
Trace overlapping crosses and reorder planned presses. Use cancellation and odd-or-even toggle counts to explain why different sequences can finish at the same pattern.
Print these three studies with answers
Start with a decision
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-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.
Apply the ideas to a complete board
These are fixed original studies, not new daily puzzles. The three study links below and this collection’s printable pack use identical starting positions. Their checked solution routes are included on each practice page. A collection’s technique drills may also appear in its lessons.
2. Cross Lights study 5
4 steps in the included checked route. Try the board first, then compare the reasoning.
3. Cross Lights study 6
5 steps in the included checked route. Try the board first, then compare the reasoning.