01
Split the possibilities by the mandatory switch
When A is off, the gate stays closed for every B/C pair: off/off, off/on, on/off and on/on. These are four distinct combinations. When A is on, B and C both off still leaves the gate closed. The remaining three B/C pairs open it: off/on, on/off and on/on. Across the eight combinations, three open the gate and five keep it closed. This count is a summary of the complete list, not a probability statement; the worksheet makes no assumption about how often a player selects each combination.
02
Use a counterexample to catch a grouping error
Compare the intended rule with (A AND B) OR C. In that second rule, C being on is enough to open the gate even when A is off. The combination A off, B off, C on therefore separates the two rules: the intended rule stays closed, while the second opens. This single counterexample proves the rules are different. It also shows why writing the parentheses is substantive rather than cosmetic. Preserve the rule beside its response table so a later visual redesign does not accidentally change the relationship between the switches.
03
Save a compact puzzle specification
The reusable sheet records switch labels, allowed states, the exact grouped expression, the meaning of OR and whether earlier actions affect the result. Add the eight response rows and the grouping counterexample. For this puzzle, three combinations open and five close, and earlier switch positions do not matter. If a later design adds a timer, a latch or a sequence requirement, a static three-switch table is no longer a complete description. Record the additional state explicitly instead of assuming the original worksheet already explains the expanded puzzle.