This IQ test question was making the rounds the other day. See if you can answer is before peeking.

Don’t Peek

The Answer

The answer is obviously 84.

This is from the formula, which many of you spotted right away (I’m always proud of regular readers):

$$n\left(p_n – (-1)^{n+1}\right),$$

where p_n is the n-th prime number and n the top number(s) in the triangle (the first few primes, you recall, are p_1 = 2, p_2 = 3, p_3 = 5, p_4 = 7, p_5 = 11, p_6 = 13, …). Since in the first example there are two 2s, you can think of the first n in the formula as the first 2, and the second n as the second 2. Et cetera.

To confirm, p_2 = 3. Thus

$$2\left(3 – (-1)^{3}\right) = 2\left(3+1\right)=8.$$

And with the 5s we have, recalling p_5 = 11,

$$5\left(11 – (-1)^{6}\right) = 5\left(10\right)=50.$$

Finally, with 6s, recalling p_6 = 13,

$$6\left(13 – (-1)^{7}\right) = 6\left(14\right)=84.$$

Neat math, n’est-ce pas?

“No, Briggs! No! No! No!“

You object?

“The answer is obviously 72! You used the wrong formula, Mr Sophistication. Everybody knows you’re wrong.”

I am? What’s the right formula?

“Look, genius: 2^2 = 4, and 2 x 4 = 8. Yes?”

Usually.

“And 5^5 = 25, and 2 x 25 = 50.”

That’s true.

“That means the clear-to-anybody-but-you formula is 2 x n^2. Which means 2 x 6^2 = 72. That’s the right answer.”

I had no idea my formula was wrong. But now I can see that yours is wrong, too.

“It is not.”

Is it. Look, 2 x 7 + 2 x 7 – 20 = 28 – 20 = 8, does it not?

“So what.”

And 5 x 7 + 5 x 7 – 20 = 70 – 20 = 50, yes?

“I don’t like where this is going.”

That means the real formula is 2 x n x 7 – 20. Which means 2 x 6 x 7 – 20 = 64. That’s the real right answer: 64!

“No! No! No! Must you be so obtuse? It’s clearly 72.”

Wait. I want you understand I take your, and all, criticisms seriously. I see the errors I’ve made. And I can now plainly see the true correct formula.

“Well it’s about time.”

The correct formula (let’s call the answer y) is obviously:

$$y = 2n^2 + k(n-2)(n-5),$$

where k does not equal 0. That is the obvious natural extension, or generalization, to your formula. The question of k is a curious one, but we can easily deduce it’s value must be 3, since we have triangles above, and everybody, even I, know triangles have three sides. That means for n = 6 we get

$$y = 2 \times 6^2 + 3 \times 4 \times 1 = 84.$$

Aha! So the right answer was 84 all along, and you were just teasing me. Good joke. You know, it’s not often we mathematicians—

“You can’t do that!”

Hold up, hold up. I see what you mean. My guess for k was silly. Since these are triangles, and all triangles have 180 interior degrees, k = 180. That means y = 792.

“No!”

You mean k is the number of lines in pictures? I didn’t see it at first. I have it now: k = 6. Thus y = 96.

“I’m done with you.”

Too bad. I was just about to change the subject.

Author! Author!

The right answer to any and all questions much always include this: the mind of the author of the questions.

I tried making this point in an earlier post, but I don’t think it stuck. I was too abrupt, the math too obscure. So here I expanded on the idea to make everything (I hope) clear.

My interlocutor above was convinced the author of the puzzle had a very specific and not especially difficult formula in his (the puzzle author’s) mind. Given everything we know about these kinds of puzzles, our wide experience with them, seeing what forms they take and their correct answers, my friend is likely right, too. His formula was very probably the right answer, and all mine were wrong.

I cannot say this with certainty, because I don’t know with certainty precisely what the puzzle author was thinking. He may have been like me, a fan of prime numbers. But of all the scenarios I think possible, I judge my friend’s “72” hypothesis almost certainly—but not certainly certain—the correct one.

I can therefore guess that the 72-hypothesis holds. I make this decision based on the likelihoods of various hypotheses I formed conditional on the evidence I assumed. And as everybody following the Class knows, decisions are not probabilities. Decisions, guesses, come after the uncertainty, and they depend on different evidence or premises which we use to form the probabilities. Evidence or conditions like what will happen if I’m wrong, or if I’m right, and how important are these things to me?

It turns out I don’t care much either way about being right or wrong, and therefore decide the 72-hypothesis is the best guess: I make that guess, then I do the math and write the answer 72. Then I await the author, or his representative (like a teacher), to tell me if I was right.

For puzzles like these, we combine the uncertainty and decision steps so fast that they seem one. We’re so good at this in ordinary situations that the acts seem inseparable. But they are not: they are entirely different.

And so it remains that in judging IQ test questions, or as we’ll see any questions, we are not just working out the math, or logic, we are also making a decision about the goals of the questions, about the authors’ intentions. This must be clear for the example which we use, because all the formulae I gave were entirely consistent with the stated available evidence. They were also consistent will all the unstated evidence we considered.

And there is lots of unstated evidence. For instance, you assumed what the “2”, “5”, “6”, and “12” meant. You assumed the triangles encompassed different examples, that they were boundaries for separate examples. You assumed because of the form of the question that some kind of mathematical manipulation was called for. You assumed you needed the rules of math and logic (rules which were also assumptions). All these things and more, you assumed, were on the author’s mind.

And you didn’t assume that it wasn’t math. Like, say, you didn’t assume the “2” was a shape and not number, and that, again for instance, the “8” was something like two “2”s stacked, one inverted, with the ends closed. You didn’t assume because the “5” (also not a number) had two straight sides, the rule was t0 repeat the figure and put a “0” next to it. That means, for “6”, which has no straight sides, the answer would be stacked, one inverted, something like this:

Put your newfound realization to work on this one. Hint: it also has different possible interpretations. There is more than one possible right answer, depending on your assumptions.

Explanations here and here, if you need them. But do try first. There can be more than one “correct” answer in the given set, A–F, but there can also be more consistent answers than just this set.

A Very Large Number

There is lots, then, going on in even the simplest questions, but you’re so good at thinking (obviously true, since you are here reading this) that you no longer see the steps involved, nor give much consideration to their assumptions.

The first kicker is this: for our starting puzzle I gave several possible formulas consistent with my assumptions about what the evidence meant. How many different formulas are consistent with evidence provided?

Infinity.

Yes, really: here’s a small subset of that Infinity:

$$y = 2n^2 + i(n-2)(n-5), i = 1, 2, \cdots .$$

This is obviously infinite; each formula, one per i, is entirely consistent with the known observable data. It is clear (well, to those used to playing wth equations) many other variations can be found.

This is always so. That’s the second kicker, and what I promised at the beginning. All questions have an infinity of answers consistent with the observed data. Which is why the observed data is never enough.

“Let the data speak” leads to the worst schizophrenia, an infinite cacophony. “Show me the data” is like being presented with a picture of a falling grain of salt in a blizzard. The data is not sufficient; it never can be its own explanation.

This applies everywhere. You must always bring outside assumptions. And those must relate to the mind of the question writer, whether that be a man, Nature, or God himself. This is inescapable.

Answers being infinite, how can we ever be, perhaps not certain, but at least reasonably sure we’ve made good guesses, at least most of the time?

The 6 February 2022 NYT Sunday crossword was titled “Sci-fi Showdown.” The clue for 70-across was “The better of two major sci-fi film franchises” (8 letters). What would you answer? Here’s the relevant section of the grid (the blue means nothing):

The down clues that overlapped the last four letters of 70-across were in order: “It’s a ______!” (71 down), “Body part that precedes band” (67 down), “Ones involved in a transaction” (47 down), and “Let out, in a way” (55 down).

Let’s work through this. How many possible answers are there for “It’s a ______!”?

Infinite.

One possibility is “It’s a five!”, as those fond of (early) primes are wont to say. Or maybe the guy reaching into his pocket expecting a one. “It’s a teen!”, as readers of the news might say. A proud father will exclaim “It’s a girl!” .

Many English words fit. But who’s to say the answer must be an English word? That’s an assumption about the creator’s intention. I’ve been doing these puzzles for a long time, and sometimes squares can be whole words themselves, or even symbols. Maybe that’s so here. Then we might have, say, “It’s a Smiley-Face Smiley-Face Smiley-Face Smiley-Face!” to imitate a laughing text (I don’t know how to get pictures of smiley faces in place of letters on these posts). Or maybe just “It’s a LMAO!”

Since anything might do for a symbol, the possibilities are endless.

We are in the same situation we started with, but with a new wrinkle. The answer to 71-down does not exist alone. It must fit with all the other words it touches directly, and fit indirectly with all the words (or symbols) in the entire puzzle. They are all connected. Each other word has also has infinite possibilities, and indeed so does the puzzle as a whole, though the explanation for the joint mass of clues will surely become exceedingly complex the greater we deviate from the author’s mind. We must add epicycles upon epicycles to justify each additional entry we make.

Unless you have guessed the mind of the puzzle author correctly. Then everything fits tight.

In a way of proving that, the answers—plural—to 70-across were “Star Trek” or “Star Wars”. The down answers were “Wrap / Trap”, “Waist / Wrist”, “Payees / Payers”, and “Leased / Leaked.”

Two possible answers for each, and all fit into the whole puzzle.

And I know these, these are not a guess, because I know the mind of the author of the puzzle, he being obliged to publish it after the puzzle has run.

In other words, I know that part of the cause or explanation of the grid.

Nature’s Grid

It is no different in Science. In her Defending Science — Within Reason, Susan Haack likened scientific explanations to a giant multi-dimensional crossword puzzle. All the answers must fit together.

Picture a scientist as working on part of an enormous crossword puzzle: making an informed guess about some entry, checking and double-checking its fit with the clue and already-completed intersecting entries, of those with their clues and yet other entries, weighing the likelihood that some of them might be mistaken, trying new entries in the light of this one, and so on. Much of the crossword is blank, but many entries are already completed, some in almost-indelible ink, some in regular ink, some in pencil, some heavily, some faintly.

That some of the entries might be mistaken is usually not given hearty consideration, especially when grants are on the line. Much in Science is mistaken, as you and I have seen over the years. And a lot that is wrong precisely because that part of the cause which is the Author’s mind is ignored. We saw not long ago a “prominent” (a quote, not a scare-quote) scientist reject alternate testable theories of biological change because the Author’s Mind “would be let back in”. The widespread rejection and neglect of teleology, which reveals that Mind, has choked Science.

However, that is too much for us today. We shall return to this subject again.

Test Your IQ Test

There is no escaping the mind of the author in an IQ test (which was the point of the first time I attempted to explain this). Which means IQ tests are not quite the tests of intelligence you may have thought.

If you have a math test to give to kids, you take great pains to spell out the precise exact conditions of the questions. You try to indicate your mind in the set up of the question, so that it is plain you want numbers, and numbers of a certain kind, and got in a prescribed way.

For instance, solve for x in this equation:

$$1 + x = 0.$$

Where the answer is obviously

$$x = e^{i\pi}.$$

Which you would not do to a 5th grader. Even though this x is correct, as are an infinite number of other possibilities (x= -\int_0^{\pi/2} sin(y)dy, etc., etc.). The tacit assumption, given by any number of practice questions in class and in homework (before it was decided math was “racist”) was that the answer would be a single number itself, and nothing else. Certainly not a symbol or political statement.

And since we have just introduced negative numbers in class, and the subject of the test is negative numbers, the student rightly surmises the mind of the author (his negativity?). Only then can the student bring his math skills to bear and solve the problem. These are entirely different steps than that first necessary step of assessing then guessing the mind of the author.

The same is thus true for all IQ test questions. If you had guessed, as I first suggested, 84 as the answer, because this was obviously and truly consistent with the evidence, and the mind of the author was elsewhere, you would be marked wrong, regardless of your intellectual ability to do math. The IQ test score would reflect your inability to surmise the mind of the test writer, but many would mistakenly take your low score as indicating inability to calculate.

Of course, intelligence is more than calculation ability, too. Part of intelligence is discerning the minds of the question writers, and some of that discernment is indeed cultural can thus improve with practice. Thus it is possible to get better at IQ tests, to a degree. And so, quod erat demonstrandum, IQ test scores smooth over the various dimensions of intelligence, as one-number summaries of complex entities so often do.