More Lights

Something is fascinating about lights being part of brainteasers that I still cannot grasp. Maybe it is the fact that a lightened lightbulb is symbolic of a Eureka! moment, a discovery or unlock of sorts.

Nevertheless, even without knowing why, I love them, and here is one more problem of those:

The circuit breaker box in your new house is in an inconvenient corner of your basement. To your chagrin, you discover none of the 100 circuit breakers is labeled, and you face the daunting prospect of matching each circuit breaker to its respective light. (Suppose each circuit breaker maps to only one light.)

First, you switch all 100 lights in the house to “on,” then head down to your basement to begin the demanding mapping process. You can switch any circuit breakers on or off every trip to your basement. You can then roam the hallways of your house to discover which lights are on and which are off.

What is the minimum number of trips you need to make to the basement to map every circuit breaker to every light?

Curious to find a solution that doesn’t involve either switching on or off the light switches in your house or feeling how hot the lightbulbs are?

The solution I present here is fantastic, all because it’s the right strategy. To contextualize, the most straightforward strategy would be to switch each circuit breaker off one at a time. This would take 99 trips to the basement; the process of elimination would map the last circuit breaker. We can do much, much better than this marathon.

I will give you a number: seven. Believe it or not, you can map all 100 circuit breakers to their respective lights in only seven trips! Here’s the binary strategy you can use:

For ease of keeping track of things, put a piece of masking tape on each circuit breaker and each light. On the first trip to the basement, flip 50 circuit breakers off, mark these circuit breakers with a “0,” and mark the 50 circuit breakers that are on with a “1.” Accordingly, as you roam around the house to tally the lights, mark the 50 lights off with a “0” and the other 50 lamps with a “1.”

On the second trip to the basement, keep off half of the circuit breakers that are marked with a “0,” turn off half of the circuit breakers that are marked with a “1,” and mark all of these circuit breakers with a second number of “0.” If they're not already on, flip on all other circuit breakers and keep their second number as “1.” Now go around the house, and again mark the lights off with a “0” and those on with a “1.”

You’ll continue this process: In the third trip, flip half (or actually, 13 since 25 is an odd number) of all of the circuit breakers in each group ( “00,” “11,” “10,” and “01”) to off, and mark them with an additional “0.” Mark each group's 12 “on” circuit breakers with a “1.” Go around the house and mark all lights off with a “0” and all lights on with a “1.” Do this same procedure a forth time, and then a fifth. After the sixth trip, you’ll have one or two lights or circuit breakers in each group. For those groups that each has one light and one circuit breaker, you’ve successfully mapped those circuit breakers to their lights! The rest takes only one more trip (the seventh and last trip!) to map them to their respective lights.

The right strategy to solve problems can make a massive difference in the work needed to accomplish a task.