Test Your IQ With These Puzzles! (Not So Easy!)

Here in this Fun Fourth run up are some puzzles to tease out your “IQ“. Only those with high “IQ“s will get them, so beware! Try them all first before peeking at the answers. Scoring will be after. They start easy and grow harder.

The Questions

1. What is the next number in this sequence? 1, 2, 3, 4, 5, _____?
2. What is the next number in this sequence? 2, 4, 6, 8, 10, 12, 14, ______?
3. What is the next number in this sequence? 1, 2, 4, 8, 16, ______?
4. What is the next number in this sequence (less information is harder)? 2, 4, 6, ______?
5. What are the next two numbers in this sequence? 1, 1, 2, 2, 3, 3, _____, _____?
6. See the picture below: What number goes in the noted tile?

The Answers

1. 7. This is obviously the start of the powers of prime series, for p^k, p prime, and k ? 1 (an integer). There is no 6 because there is no prime p for which p^k = 6, for any k?1. There is a rich literature on this series, as every schoolboy with high “IQ” knows. For instance, R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge 1986, Theorem 2.5, p. 45.
2. 15. These are, as you guessed, the numbers k such that the greatest common divisor of the indices j for which the j-th prime prime(j) divides k is 1. See this explanation.
3. 31. From D.J. Schreffler’s well known write up, the “rule is to find the maximum number of regions a circle can be split in to, if you have 1, 2, 3, 4, …, n points on the circle (none of which are opposites)…The nth term is (n Choose 4) + (n Choose 2) + 1. It’s just that this happens to coincide with 2^(n-1) for the first 5 terms.”
4. 30. These are, as all surely recognized, the ordered numbers that when you spell them out in English do not contain the letter ‘e’.
5. 3, 3. I know this one wasn’t hard, but I had to give a shout out to computer scientists reading the article. These are, of course, the length of the binary representation of n, where n represents the integers.
6. Any of these: 1, 5, 25. If the circle is read as the central 5 times the lesser or equal to numbers, then the answer is 25. If the circle is read as the numbers larger than or equal to 5 divided by the central 5, then the answer is 1. If the circle is read to be symmetric, then the answer is 5.

The Explanation

We did the last puzzle in Class, “Billions of Models“, but because, shockingly, not all are following along, I thought it worthwhile to highlight it and further examples.

There are two points from this I want you to carry with you always:

1. IQ tests can have biases;
2. Any set of observations has an infinite number of explanations.

It’s my experience that many become angry when shown questions like this. They will say for instance it
“Briggs you fool! It’s obvious that the answer to #1 is ‘6’!”

The “obvious” reflects their bias, which they scarcely consider to even be a bias. Yet it is.

My answer was correct. This series, as I chose it, was indeed the powers of prime series. I picked it: I created the test question. It is the answer I had in mind. I cannot stress this too highly: my answers are correct.

It’s true I might have picked another series. I didn’t, but might have. For instance, I could have picked the digital sum series (add all digits in a number until left with one 1), which also starts like question #1. Or maybe the numeric palindromes in base 10 (after ‘9’ the next entry is ’11’, then ’22’ and so on), which also starts the same way. Or I could have picked the unary-encoded compressed factorization of natural numbers, where instead of the answer being ‘7’ it would be ‘8’.

If I wanted to choose from the catalog of interesting series that contained “1, 2, 3, 4, 5”, I had 8,032 possibilities. Mathematicians have a page (at those links) which catalogs interesting series of all kinds, and is where I got most of my examples.

But, really, eight thousand is trivial. In fact, there are an infinite number of series that contain “1, 2, 3, 4, 5”, with any number of possible successors, some of which will be ‘6’, but most will not.

If you thought the answer was ‘6’, it was because you guessed I would use a culturally familiar rule to generate the series, i.e. simple succession, or “add 1”. But maybe you guessed that because I was a mathematician, I might have set my scope wider, and since (as you know) all mathematicians love primes, I would have picked a series which made use of them in a simple and fun way, which is what I did.

I stress ‘6’ was and is and remains the wrong answer. Just as ’16’, ’32’, ‘8’, ‘4, 4’, were the wrong answers, and so are any numbers besides those stated in the last questions. They are right answers, it’s true, but to questions I did not ask.

Which can only mean that these questions had hidden or tacit information in them, information not spelled out, but which you assumed. This goes for all questions like these. Not just these here, I mean, but all of them.

It’s true the rules that led to the wrong answers just given are likely more common to you, meaning you likely chose that tacit information. Which means you went with what you were culturally most familiar with.

That move on your part is a “bias”. Not in any purposeful or harmful connotation; I mean “bias” in a pure technical sense. So used to these series are you that you might have a difficult time—and for some, impossible time—seeing your choices reflect any kind of bias. But they do; they always do.

This is easiest to see in the last question, where the bias you pick makes large differences. If you naturally had a multiplication bias, which is likely most, you went with ’25’; if you naturally had a division bias, which isn’t as many I perhaps, you went with ‘1’. Or if you went with a symmetric bias, which is very few I imagine, you went with ‘5’. All theses answers are correct assuming one bias, but false if another bias is held.

Which is all another way of saying all logic, including all probability, is conditional on the evidence you assume, tacit, implicit, felt or guessed. And that is the entire point of the Class. The one real lesson of all of it.

Now since there are an infinite number of possible sequences to answer question, even the last, to get anywhere some kind of bias is needed! Bias, in this way, is not a bad but is necessary. You have to have something to go on or no answer is possible. The chance you hit a correct one assuming I picked the series from one out of infinity is 0—exactly 0.

All this has deep implications for knowledge, logic, rationality, probability, everything. That’s what we do in the Class, so I won’t belabor it here. But it also falls in with the articles on IQ, which I write in the vain hope that we learn to say “intelligence” and not “IQ” when we mean intelligence. Today is yet more evidence that more is going on in questions like this than simple mathematical ability or rationality. Which means the questions asked on these tests cannot be all about intelligence, which means “IQ” is not intelligence: intelligence is.

Scoring

You should be able to guess by now. What’s your IQ? You don’t have one. You have intelligence.

Video

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