Riddles with morals

Last week's six trick questions prompted numerous interesting comments. Since I can't transcribe them, I'll summarize them in the answers below:

1. The first trick question is doubly so. In principle, one in four years is a leap year, but not always. If, in addition to being a multiple of 4, the year is secular—that is, a multiple of 100—it is only a leap year if, in addition, it is divisible by 400; thus, the year 2000 was a leap year, but 1900 was not, and neither will 2100. And since the statement doesn't specify which four consecutive years it refers to, both possibilities would have to be considered: a four-year period with a leap year and one without.

The second trap has to do with the well-known assumption bias. If someone asks, "Do you have a cigarette?" and the person questioned has a pack, they will answer yes (unless they don't want to share their tobacco); but if they ask us what month has 28 days, we assume "exactly 28 days." Because, in reality, all months have 28 days, and some have 1, 2, or 3 more days.

It's a riddle with a moral, and the moral is that we often, without realizing it, impose more conditions on ourselves than required. There are numerous very interesting problems of this type (one of the best-known, which we've discussed more than once, is the 9-point problem to be connected with 4 straight lines without lifting the pencil from the paper or crossing the same line twice).

2. “Are you asleep?” is a question whose only possible answer we already know, yet it’s not necessarily superfluous. “Are you dead?” is another possibility. Someone talking in their sleep or a vampire might surprise us with an unexpected answer; but, in general, the only possible answer is “no.”

3. The highest mountain in the world, before it was known to be Mount Everest, was Mount Everest.

4. Most dogs —and humans—sleep more hours in January than in February because January has three more days (or two if it's a leap year). Again, we're confused by the assumption bias: where it says "more hours," we mean—assuming—"more hours per day."

5. The cat . In this case, a subtle feminist claim could be the moral.

6. An irresistible force and an immovable object are mutually exclusive concepts; both cannot exist at the same time (though, incidentally, some people believe that an omniscient being who knows in advance what we are going to do is compatible with free will).

The princess's hair

And when we talk about riddles with morals, we inevitably think of fables and traditional tales. Here's a puzzle based on a classic tale:

Once upon a time, there was a princess who was so proud of her abundant and beautiful golden hair that she never cut it, not even the ends. And when she saw with concern that some hairs remained tangled in her silver comb every day, she had her maids count them every night for fear of losing her golden attribute.

To the princess's relief, the count always remained around 150,000 hairs, even though about 50 were falling out every day. Knowing that human hair—even princely hair—grows about a centimeter a month, how long was the princess's hair?