| C1 | C2 | C3 | C4 | C5 | |
|---|---|---|---|---|---|
| R1 | |||||
| R2 | |||||
| R3 | |||||
| R4 | |||||
| R5 |
What is the fewest number of presses needed?
Make a prediction before opening the reasoning.
Compare the predictions
- Three presses.
Not supported. The complete first-row enumeration contains no solution with fewer than five presses.
- Five presses.
Supported. The diagonal plan works, and all four possible press sets have lengths 5, 13, 13 or 21.
- A working plan cannot ever be proved shortest.
Not supported. Finite exhaustive coverage can prove a minimum for this exact board.
Predictions stay on this page; they are not saved or sent.
The position and the question
Pressing the five lit diagonal squares clears this particular board. That gives an upper bound of five presses: we know a five-press finish exists. To call it shortest, we must also exclude every shorter possibility.
This investigation uses the ordinary 5 × 5 cross rule and the goal of all lights off. Press order does not alter the final pattern, and pressing a square twice cancels its effect. Therefore a shortest plan presses each chosen square once.
Show the reasoning
Verify the five-press candidate
Press R1C1, R2C2, R3C3, R4C4 and R5C5. Each diagonal square is toggled once. Every side-adjacent square between neighbouring diagonal presses is toggled twice, so it returns to off. All other squares are untouched. The whole board is dark.
Cross presses toggle side neighbours; ● on, ○ off C1 C2 C3 C4 C5 R1 R2 R3 R4 R5 All five diagonal buttons pressed once: every light is off. Reduce all plans to 32 cases
There are 2⁵ = 32 subsets of buttons in the first row. Once a subset is chosen, each lit square in row 1 forces the press directly below it. Repeat down the board. A completed first row cannot be affected by presses below row 2, so the chase makes every later row of presses compulsory for that first-row choice.
Keep only chases that finish
Exactly four first-row subsets yield an all-dark board. They are {C1}, {C3,C5}, {C2,C4,C5} and {C1,C2,C3,C4}. Their complete chase plans contain 5, 13, 13 and 21 presses respectively. The other 28 first-row choices leave at least one light in the final row.
Turn enumeration into a minimum proof
Every press set has one of those 32 first rows, and the rest of a successful set is determined by chasing. We have therefore covered every parity-distinct solution, not just a sample of routes. The smallest successful set contains five presses. Repeated presses cannot improve it because deleting a repeated pair leaves the same final board with fewer presses.
The tempting shortcut
The first working chase must be the shortest one.
Where the shortcut breaks
A solver that stops at the first solution does not automatically compare lengths. On other boards, a later first-row choice may use fewer presses.
A better decision
Count every successful chase when claiming a minimum, and explain why the enumeration covers all press sets.
Changed conditions
In a mathematical toggle experiment, append two presses of R3C3 to the five-press solution. Is the resulting seven-press sequence shortest? (The live game may stop accepting presses when the board is solved.)
Check the changed-condition answer
The added pair leaves the final mathematical light pattern unchanged: two identical toggles cancel. Removing the pair gives the original five-press solution, so the seven-press sequence is not shortest. This compares toggle sequences, not controls available after a game has ended.
Take this back to your next game
A constructive route proves that a goal is reachable. A complete covering argument is what turns a good route into an optimality proof.
How the claim was checked
Enumerate all 32 first-row masks, chase each, and apply every returned press plan independently.
- 4 successful masks: 1, 20, 26, 15
- Solution lengths: 5, 13, 13, 21
- All 4 plans finish dark
Read the fixed positions and supporting proof data. Computational checks establish only the claim and position stated here. They do not establish a player difficulty rating or make an unproved strategy optimal.
Playfield Arcade · Updated 4 October 2026 · Original case design, AI-assisted writing and code, with reproducible position checks. Our method · Report a correction