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Comment by bediger4000 | Hacker News Reader

The official answer to this puzzle makes a huge assumption: that there is one correct answer. There is not one correct answer. (x, 2x, 4x) gives you a "yes" every time, therefore it is a correct answer, at least as automatically checkable, and there's an infinite number of such tuples. To find a "no" you're reduced to random guessing. That's not a puzzle, that's crap. The confirmation bias material might be true, but the puzzle does not illustrate it.

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It's amusing that you went to a website on confirmation bias, did the puzzle incorrectly, presumably read the material on confirmation bias, but still suffer from the effects of confirmation bias.

Amusingly human, if I may add. Reading "Thinking, fast and slow", by Daniel Kahneman, one key idea that I got was that even knowing against biases you are very, very likely to suffer from those biases. Disheartening results were gathered from studies done on well-trained psychologists and people prepared for the experiment, to no avail. Can't remember the details right now, but just read the book, it's awesome. Another good one was "Influence" by Cialdini, but they gave you tips on trying to avoid those biases that, upon reading Kahneman, I don't think anymore that are very useful.

You can greatly improve your performance by setting up internal and external reminders to check your assumptions and commo falacies.

Related: read The Checklist Manifesto.

The puzzle is not "Find a rule that matches everything you tested". It's figure out what rule they are using. It may be true that there are multiple ways to state the rule, such as "x_2 >= x_1 + 1 and x_3 is >= x_1 + 2" but they are equivalent.

Try the sequence: 0,0,0. It would give "yes" for (x,2x,4x), but the actual rule gives it "no".

Or make x negative - the rule gives it "no".

Your task is to learn a class of objects by example. Without seeing any negative examples, you're unlikely to learn the right class (which is unique, it is precisely the class of increasing sequences of length 3). (x, 2x, 4x) is not "a correct answer", as it does not describe the class you were supposed to learn to distinguish.

It's not an assumption; the article tells you up front that there is one correct answer.

It is an analog for how science works. When it comes to a natural phenomenon, humans can come up with multiple explanations that fit a given set of observations, but presumably (I mean, this is a basic tenet of science) nature only works in one consistent way.

Thus, the importance of a falsifying test. You form a hypothesis based on the initial observations (in this case, the number sequence 2, 4, 8), and then you propose a test that could falsify your hypothesis.

The trick is that a hypothesis can fail in several ways. It can be outright wrong, like saying "the rule is that the numbers decrease from left to right." That's obviously just wrong.

But it can also be too specific, like saying "the rule is that the exponent increments by one with each step to the right." That matches the given evidence, and tests with other base numbers will succeed too. But it's over-fitting.

Here's a concrete example: a man wearing a red shirt drops a weight and measures gravitational acceleration as 9.8 m/s^2. So he formulates a hypothesis that gravity always produces an acceleration of 9.8 m/s^2 in the presence of a red shirt.

And if he always wears a red shirt, and always tests gravity on the surface of the Earth, he'll always find supporting evidence for that hypothesis.

But of course we know that gravitational acceleration varies depending on mass and distance, and that it's the same no matter what color your shirt is. But he would only find that out if he varied his experiment beyond what his hypothesis predicts.

Well, yes, in this situation random guessing would be very likely to provide an immediate check on the solution.

(1x, 2x, 4x), as indicated in the video below, is not sufficient. It represents a subset of the values that are valid.

Think of it this way. When asked to write a unit test, do you only test the positive outcomes? No, you test to make sure the failures are as you expect as well. Otherwise, you are likely to have what you think is a failure end up as a success.

You got the problem completely wrong.

The idea isn't to come up with tuples that satisfy the predicate. The idea is to figure out what the predicate is in the first place.

Also, you're not in any way, shape, or form reduced to random guessing. If you have an idea of what the rule might be, you build a counterexample. There's a ton of value in _trying_ to get a no but getting a yes instead.

From a Computational Learning Theory perspective, we are faced with an infinite hypothesis space, with an infinite VC dimension. So yes, there's not much strategy that can be employed here.

But, the fact that there is one correct answer is not really an assumption that the puzzle makes, it is information we have been given:

> We've chosen a rule that some sequences of three numbers obey -- and some do not.

This simply means that the solution is realisable. No matter how many ways (in English) we have to describe that solution it is still the same solution.

Consider the sequence

1, 3, 5, 7

what comes next? 9 right? Or is the sequence generated by 2n − 1 + (n − 1)(n − 2)(n − 3)(n − 4) for n ∈ N. Then we've got 33.

"among all hypotheses consistent with the observations, the simplest is the most likely"

33 is correct, but it's less likely to be the basis for the generation of the sequence.

Your answer of (x, 2x, 4x) proves the puzzle illustrates the confirmation bias, at least in your case.

Does the unit test that confirms your function returns the expected result given one set of arguments prove it correct?

(x,2x,4x) does not actually give a "yes" every time. In particular, it won't work if x is negative. But, in that case, (4x,2x,x) will work.

> (x, 2x, 4x) gives you a "yes" every time, therefore it is a correct answer

I think this logic is a bit wonky - if there are sequences that get a "yes", but don't match (x, 2x, 4x) then the correct rule cannot be (x, 2x, 4x), can it?

I've read this comment three times and I still can't figure out what you're trying to say. It sounds like you're arguing that all "puzzles" should exist on finite domains so that a brute force solution exists.

"(x, 2x, 4x) gives you a "yes" every time, therefore it is a correct answer, at least as automatically checkable."

Well, no: you can type 1, 2, 3 into the system and it will tell you "yes", but your rule says that it should tell you "no".

It is crucial to the definition of "correct answer" here that your rule should not just say "yes" only for tuples which the widget also says yes to, but also your rule should say "no" only for tuples which the widget also says no to. That is what the puzzle means when it's asking, "can you guess the rule that we've created?"

This makes it very, very different from what I think you're thinking about, which is situations where someone tells you, "what is the next number in this sequence? 4, 7, 13, 25, ...?" where technically there are an infinite number of rules which will generate those 4 numbers first and an arbitrary number afterwards. Technically one of them is "simplest" in the sense that it can be expressed in 7 symbols, but in general it's a complicated problem and there is no best solution.

"To find a 'no' you're reduced to random guessing. That's not a puzzle, that's crap."

In many ways it still is a puzzle but the space that it lives in is richer. If you think about typical "puzzles" they're things like: "here's a grid with some spaces filled in with numbers,

    2 . . 2 . 2 .
    . . . . . . .
    1 . 3 . . 2 .
    . . . . . . .
    3 . . . 2 . 3
    . . 2 . . . .
    . . . . . . .

Each number is a block in a block wall. We want you to turn this into a block maze so that each 'block wall' (set of blocks connected by adjacency) contains exactly one numbered-block whose number says how many total blocks are in the wall. Furthermore the path (non-block space) of the block maze should be connected and should not contain any 'rooms' -- that is, any 2x2 or larger segments of open space."

This 7x7 grid has 10 spaces which are known to be blocks and exactly 12 more blocks scattered in the remaining 39 spaces, so just by those factors alone we're searching only (39 choose 12) ~= 3.91 billion possibilities; we can also use a quick heuristic to identify 6 places which must be "space" to break apart adjacent numbered walls, removing 91% of that search space.

The puzzles, "I have a set of integers where inclusion in the set is governed by a short rule, you can ask me any integer and I will tell you whether it is in my set", by contrast, have an infinite search space. This means that any solution is going to be more interesting, as will the means for checking that solution's validity. You could require, for instance, a Haskell expression of 140 characters or fewer which turns a nonnegative Int named `n` into a Bool, to be judged as "valid" or "invalid" if it properly filters `[1..10000000] :: [Int]`. You could even give the first 100 numbers in the set, e.g.:

    ghci> take 100 $ filter trueFn [0..]
    [2,5,8,9,13,14,18,19,20,25,26,27,32,33,34,35,41,42,43,44,50,51,52,53,54,61,62,63,64,65,
    72,73,74,75,76,77,85,86,87,88,89,90,98,99,100,101,102,103,104,113,114,115,116,117,118,
    119,128,129,130,131,132,133,134,135,145,146,147,148,149,150,151,152,162,163,164,165,
    166,167,168,169,170,181,182,183,184,185,186,187,188,189,200,201,202,203,204,205,206,
    207,208,209]

In this case that's pretty much enough to see the general pattern; the verification covers 10 million bits while the 140-character limit probably limits your search space to 1000 bits or so, so it's going to be hard to get an "incorrect" answer which agrees on that subspace of the whole.

> There is not one correct answer. (x, 2x, 4x) gives you a "yes" every time

Not true. What if x is negative?

My second guess was 0, 0, 0. My third was -1, -2, -4.

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