Person 4 starts at locker 4 and changes every 4th locker. Person 3 starts at locker 3 and changes every 3rd locker. Person 2 starts at locker 2 and closes every 2nd locker. Person 1 starts at locker 1 and opens every locker. Imagine there is an endless string of lockers in your school. Here’s a question from 2004: A Way to Think about the Locker Problem So every other locker means every locker whose number has what as a factor? And how many times would a switch have to be flipped to be on at the end? Bigger hints I imagine the row of lockers as a long line of off/on switches. Thinking about the influence of FACTORS in the first problem might help.
Then we change the state of every third (opening some, closing others):ĭoctor Jodi gave only a hint: Our office is overflowing with patients at the moment, so let me just try to put a band-aid on these problems for you.
Then we change the state of every second locker (opening those): To illustrate, we start with all lockers closed: Which locker doors will be open when the process is complete? You follow the same pattern for every fourth locker, every fifth locker, and so on up to every thousandth locker. Then, for every third locker, you close each opened locker and open each closed locker. Suppose you open all of the lockers, then close every other locker. Here is a question from 1996, which asked about two problem, the first of which is our subject: Word Problem Hintsġ) There are 1000 lockers numbered 1 - 1000. (I have sometimes wondered why, with all the slight variations we’ve seen, they all still involve lockers, rather than, say, pulling chains on lamps, which might be easier to state.) Tiny hints
So if you haven’t seen this problem before, be sure to read only the first answer or two below and try it yourself, before looking at the subsequent spoilers. You’ll soon see the connection to the topic of last week’s post on divisors. But I prefer to give them a reference to one of the answers in which we gave only hints, because this is a fun problem to discover the answer for yourself. I have often told people that, believe it or not, they could find the answer by searching the Ask Dr. A classic problem we’ve seen hundreds of times involves students opening and closing lockers.
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