WORD WEDNESDAY - A tale of two synonyms
Our puzzle today comes from NPR’s Sunday Puzzle segment, in which New York Times puzzle editor Will Shortz challenges a listener to solve a puzzle live on air. Fortunately, you have no such time pressure!
Today’s Puzzle
Name a major U.S. city that’s 10 letters long. If you have the right one, you can rearrange its letters to get two 5-letter words that are synonyms. What’s the city, and what are the two synonyms?
Monday’s Puzzle - Lockers in Lockdown
A high school has 1,000 students and 1,000 lockers. On the first day of school, the principal asks the first student to go along and open every single locker. He asks the second student to go to every second locker and close it. Then he asks the third student to go to every third locker and close it if it is open or open it if it is closed, the fourth to go to every fourth locker and so on for all 1,000 students. In the end, how many lockers are open?
[SPOILER] Answer to Monday’s Puzzle
31.
Explanation: Whew, this one was tricky! Don’t worry, they won’t all be this hard.
It helps to start small, so let’s see what happens to locker 6 with each student.
Student 1 opens all the lockers, so after this first pass locker 6 is open along with all the others.
Student 2 opens every other locker (lockers 2, 4, 6, 8, and so on), so 6 is closed.
Student 3 opens every third locker (3, 6, 9, etc.), so 6 is opened again.
Student 4 opens every fourth locker (4, 8, 12, etc.), so 6 stays open.
Student 5 opens every fifth locker (5, 10, 15, etc.), so 6 stays open once again.
Student 6 opens every sixth locker (6, 12, 18, etc.), so 6 gets closed.
After student 6, no other student will ever touch locker 6 again, because they will all start further down the hall. So, at the end of all 1,000 students, locker 6 is closed.
You’ll notice, of course, that some students changed locker 6 but others left it alone. Specifically, locker 6 was changed by students 1, 2, 3, and 6. What do these numbers have in common? Six is divisible by all of them (in other words, they are all factors of 6). This makes sense if you think about it from the perspective of an individual student. Student 4, for example, will toggle lockers 4, 8, 12, 16, and so on, hitting all of the multiples of 4 up to 1,000. So every locker that Student 4 toggles will be divisible by 4.
Now we know that every locker will get toggled once for every factor that it has. But what does that tell us about whether the locker will be open or closed at the end? Well, lockers alternate between only two states: open and closed. That means every two changes will leave the locker in the same state it was in before. All lockers started out closed, so if a locker gets toggled an even number of times, it will end up closed. If it gets toggled an odd number of times, it will end up open. Locker 6, for example, ends up closed because it gets toggled 4 times (once for each of its four factors). Locker 4, on the other hand, ends up open because it only gets toggled three times (for its three factors: 1, 2, and 4).
Are we done yet? No. But at least now we know what we’re looking for: we want to find all numbers between 1 and 1,000 that have an odd number of factors. But what kind of number has that property? Usually, factors come in pairs. By definition, for a number to be a factor of, say, x, that number has to be multiplied by something else to make x, and that something else is usually a different number. This means that most numbers have an even number of factors. For example, you can make 10 by multiplying 110 or 25, and that’s the full list of 10’s factors (1, 2, 5, 10). Similarly you can make 20 with 120, 210, or 45 for 6 total factors: 1, 2, 4, 5, 10, and 20. The only number that would have an odd number of factors is one where a single factor somehow got used twice. In other words, we’re looking for an x that has some factor a such that aa = x. What numbers behave like this? Can you see it? The only numbers that fulfill this property are perfect squares!
Let’s test this theory out by looking at a perfect square like locker 9:
Student 1: open (1st change).
Student 2: open.
Student 3: closed (2nd change).
Student 4: closed.
Student 5: closed.
Student 6: closed.
Student 7: closed.
Student 8: closed.
Student 9: open (3rd change).
It works! Now we just have to count how many perfect squares there are between 1 and 1,000. This is easy enough: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961, for a total of 31 open lockers!
Or, if you’ve recently completed a PhD in mathematics (looking at you, Raphaël):