Case 28 · logic · Medium · About 5–8 minutes
The museum’s demonstration cabinet has four indicator lights: A, B, C, and D. All four were off before five scheduled staff checks. One check was canceled, but the cancellation mark has vanished from the log.
At the end, A and B are on. C and D are off. Which visit never happened?
The five scheduled checks
- Ada: toggle A and B.
- Basil: toggle B and C.
- Clio: toggle C and D.
- Dara: toggle A and D.
- Esme: toggle A and C.
The cabinet rules
A toggle reverses a light’s current state: off becomes on, and on becomes off. Each completed check toggled both named lights exactly once. Exactly one of the five checks was canceled; the other four happened once each.
No bulb failed, no switch was used by anyone else, and the initial and final states are verified. The order of completed checks is unknown, but toggles do not depend on that order. Two toggles of the same light cancel one another.
Identify the canceled visit and explain the final state. As a check, work through all four completed visits, or count how many times each light changes.
The cabinet artwork is atmospheric. Its visible switch count, bulb colors, and apparent lit states do not represent the four named indicators. The text gives the complete state record.
Hint 1 - reveal when you want a nudge
First determine what would happen if all five visits took place. Count toggles of each light.
Hint 2 - reveal when you want a nudge
With all five visits, A and C would be on. Which single omitted pair changes that predicted outcome into A and B on?
Reveal the solution and explanation
Basil’s visit was canceled.
With all five visits completed, A would toggle three times, B twice, C three times, and D twice. The predicted lit pair would therefore be A and C.
Omitting Basil’s B-and-C toggle changes that prediction: B switches from even to odd, and C from odd to even. The final lit pair becomes A and B, exactly as observed.
Check the four actual visits directly: Ada, Clio, Dara, and Esme toggle A three times, B once, C twice, and D twice. Only A and B finish on.
The five possible omissions give different outcomes: omitting Ada leaves B,C on; Basil leaves A,B; Clio leaves A,D; Dara leaves C,D; Esme leaves every light off. Only Basil’s cancellation fits.
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