Questioning, Not Answering

Another Rant on “Mathematical” Puzzles

A photo of two islands shaped like a question mark.
Photo by Jules Bss on Unsplash

A few months ago, I complained about those internet memes which claim to be mathematics. My complaint about them is about the presentation, not the underlying problems: “Only 1% of people get this right!” The questions are framed to encourage people to feel stupid about math.

Please don’t feel stupid about math.

So recently I’ve been seeing a new one:

A brain teaser involving various symbols and equations.
The devil is in the details.

I have become convinced by several arguments that the intended answer is 333. I am calling it “intended”, even though several people insist on calling it “correct”… which I’ll explain before continuing.

I’ll get back to this puzzle, but first, a riddle.

A man and his young son are travelling in a terrible storm. They are the only two people in their car. The car’s tire skids out, sending the car into a spin that ends in a terrible crash. The driver dies on impact, and the son is thrown from the car. An ambulance arrives and takes the child to the hospital. But the doctor stops before beginning the operation, gasping and saying, “This is my son!” Why did the doctor say this?

This is an old riddle, dating at least to the surge of Second Wave Feminism in the 1970s. The intended answer is that the doctor is a woman, and she is the mother of the boy.

The original intent of the riddle is to illustrate sexist implicit bias: People think of doctors as male, even though there are plenty of woman doctors. Even now, it’s common for people to assume that the doctor is a man, and declare him to be the gay husband of the driver, or a step-father… those protestations illustrate the bias, even if they weren’t considerations when the question was originally written.

At the same time, though, this is not (to my knowledge) a historical event: There is no single “correct answer”, only the answer that the question writer intended and any of several other plausible answers. So while it’s fair to point out that it’s statistically far more likely that the doctor is a woman than that they’re a step-father or a same-sex partner, “statistically far more likely” isn’t the same thing is “correct”.

This riddle has no “correct” answer, unless we necessarily define “correct” as “what was intended by the writer of the question”. As a teacher, I welcome students giving me answers that match the parameters of a question but that differ from what I was expecting; I don’t deem them “incorrect” (although if you answer “What is five times seven?” with “a number”, I will ask you to be more specific).

So back to my first question:

The brainteaser again.
Don’t mind me, I’m just here so you don’t have to scroll back up.

There are some details here. Let’s start with the clocks.

I think most people would be inclined to see 9, 9, and 3, and realize that equals 21. So the rule for the clocks is “take the hour hand”.

There are some problems with this, from a mathematician’s standpoint. One is that “the rule for the clocks” suggests that there will be a different rule for the calculators, and yet a different rule for the light bulbs. And indeed, that does appear to be the case.

Another is that this is a conjecture, but that we don’t have enough data to support or refute that conjecture. I think it’s important that we unravel “mathematics” (whatever that is) from “mathematical notation” (which is what’s being abused most obviously here). But in either case, a conjecture with insufficient data leads to conjectures for answers, not to definitive values.

The reality is, this is not how mathematical notation works. 6 is not mathematically related to 9 even though they’re rotations of each other. We are supposed to set aside our rules for how notation works and come up with our own rules… but then, what we have at best is “what the person who came up with this intended”, which is not and should not be confused with “correct”.

The next row is the calculators. Since the three are identical, it is a fair conjecture that each calculator represents 10. I’m clearly too locked into to my perceptions of mathematical notation, because I didn’t notice that the digits add up to 10. Hence, the value of the calculator is supposed to be the sum of the displayed digits.

The rule I came up with instead was that 1234 was 10 more than some value, and that value happens to be 1224. What I couldn’t figure out was the significance of 1224.

Thinking about it now, though, there’s a perfectly valid conjecture regarding the calculator values that yields 1234 -> 10 and 1224 -> 0: The input number represents how many hours have passed since a specific midnight, and the output number represents how many hours after midnight it is (1234 hours after midnight, it’s 10; 1224 hours after midnight, it’s midnight).

I have no doubt that the person who created this puzzle intended the first one: Add the digits. But is it correct simply because that’s what the puzzle maker intended? Are any other conjectures incorrect?

Next come the light bulbs. It is apparently important that there are five rays coming out of each, while there are four coming out of each in the last row. Because these are the rules of these puzzles. And there is a separate, complex conversation to be had about “the rules of these puzzles” and how they evolve.

By this reasoning, each ray equals 3.

This leads people who have gotten this far to swap out the symbols with the corresponding numerals and get 9 + 9 * 3 * (3 * 4) = 333.

… which is what the puzzle maker probably meant.

But …

What’s interesting to me is that I haven’t seen anyone argue in discussions about the placement of the bulbs in the last line (granted, the puzzle is usually posted with the instruction to only give a number, and any discussion will be deleted).

If we do a straightforward swapping of symbols for numbers, this is what we get:

9 + 9 * 4⁴ 4 = 9225

Mathematicians are often accused of being picky about tiny details, and so I think the assumption is that we would love problems like these. But, as I tell my students repeatedly: Mathematicians are picky about different things than others are. The truth is, there’s plenty of stuff we don’t care that much about.


Some simple arithmetic equations.
No tricks here. It’s 17.

I made this up as a rebuttal to the one I’ve been discussing. I wrote this personally, and there’s one thing that bothers me more than anything else. Can you tell what it is?

The thing that bugs me most is that the 18 looks like 1⁸. That’s the kind of sloppiness that could lead a Future Me to not understand what I’ve written.

I do make my 1s, 7s, and 9s consistent, so having two forms of each in this picture does grate on me a little. But if you write all your 1s consistently, I don’t care much how you write them.

And the sans serif 1 (as in 18) does run the risk of being confused for an absolute value sign, but even so, it’s my default because my serif 1 runs the risk of looking like a 2 (especially if I’m being sloppy), and that’s a much bigger problem.

So mathematicians are picky about how their notation lines up. 2² is different than 22. The base four logarithm of 5 is not the common base logarithm of 4⁵. That’s the stuff you need to be careful about.

This stuff about rays coming off of light bulbs? That’s just not important. And in no sense can we decide that 1234 somehow “equals” ten.

Another common implication of this sort of problem is that the answer is assumed to be an integer. My first interpretation of the calculator line is that 1234x = 10, that is, = 10/1234. But that would yield a rather messy fraction for the last line, regardless of your other assumptions, so I discarded it. There’s no mathematical justification for this: -4.57112341 is no better or worse a number than 3 is.

Here is what I consider to be a better mathematics exercise:

Problem: Give a value for the question mark. Defend your answer. (Any answer with a reasonable defense will be accepted.)

The brainteaser again.
Me again! Did you miss me?

There is an excellent series of problems called “Which One Doesn’t Belong?” where the goal is not to find the single correct answer but rather to give a valid reason why one of four items is different from the other three. There is at least one answer for each item.

Which one doesn’t belong: 2, 3, 9, 11.

Possible answers: 2 is even, 3’s name has a doubled letter, 9 isn’t prime, 11 is two digits.

Mathematics at its heart is about solid reasoning, not about picayune details and certainly not about “correct answers”.

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