Next number in this sequence

I found this video on Youtube, where Harvard students are asked five questions of varying difficulty. At 2m55s they ask the question: which number logically follows this series?

4, 6, 9, 6, 14, 6, … ?

The answer they give is 19

Can anyone explain the logic here?

In questions like this, somebody always points out that any sequence of numbers can be justified, so any number would fit. That aside, what is the justification that they are most likely to be thinking of?

After a given number, say”6” for some reason and then go up by five.

I came up with 19 just from your OP, didn’t watch the video. My guess at the logic is there are 2 sequences, one goes up by 5 and the other by 0, which are then interleaved. Interleaving sequences are standard for a “guess the sequence” puzzle.

I got it just from reading the O.P.

Me too, just from the OP

I got 19 by saying the sequence out loud and going with the number that fit the rhythm. But that isn’t “logically.”

IMO that’s a bit of an overstatement. There are often multiple sequences that can be justified, but implicit in the question is “Find the simplest rule that explains the given sequence. Now apply that rule to give the next entry.”

Simplest is always a smidgen subjective, but any well-formed puzzle will have a clear best answer.

Not all puzzles are well-formed. We had quite a discussion awhile ago where somewhere along the way someone tweaked part of the puzzle, unwittingly changing the correct answer. But in a way that a sloppy thinker would get the wrong answer the asker mislabeled as correct. It was the malformed version which took flight virally on TwitFace, not the correct original version we eventually found.

One of my favorite observations about tests:

You’ll struggle to correctly answer a question written by somebody much dumber than you are. You’ll see all the mistakes and ambiguities they never noticed.

That was the bane of my scholastic career. And still comes up at work with some regularity 40 years later.

I clearly remember this happening in elementary school. There was a puzzle along the lines of “which metal doesn’t belong.” Something like Gold/Zinc/Aluminum/Iron.

They expected the answer to be “gold,” because it was the only colored metal, but I said it could easily be Iron, because that was the only magnetic metal.

This is why I was never any good at those “lateral thinking” puzzles. To me, they just always seemed like “make up an answer, then justify it to fit the question.”

The correct answer is aluminum because the others are all four letters.

For any (increasing) sequence a, b, c, d,…, you can justify continuing it with y by pointing out that there exists a polynomial P(x) such that a, b, c, d,…, y are its roots. The polynomial will then of course just be P(x) = (x - a)(x - b)(x - c)(x - d)…(x - y).

So if you have the sequence 3, 6, 9, you can extend it by 7325, and point out that 3, 6, 9, and 7325 are the roots of P(x) = x^4 - 7343 x^3 + 131949 x^2 - 725337 x + 1186650.

Plus, all of these numbers have never been in Cliff Clavin’s kitchen.

I got 19, as well, right away by basically ignoring the 6s and seeing the progression of +5 for the other numbers. There may very well be a neater answer that justifies the 6s, but at least I got to it with little effort.

Thanks, everyone. I should have seen that pattern. I must be getting old.

The correct answer is ‘gold’ because it doesn’t have an ‘n’ in it’s name.

No, the correct answer is zinc because the other three start with letters in the first half of the alphabet. Truly a fatuous question.

As for the OP, I looked at and said 19 but it is still pretty dumb. What is the missing entry in 2, 5, 8, 11, 13, 15, 31, 36, 40, 46, 52, 56, 60, 63, ?, 69? This actually appears in Neal Sloane’s atlas of numerical sequences.

Millbourne. These are the stops on the Market St. Subway/El in Philadelphia.

To be more formal about the logic here: The usual way to analyze sequences like this is to look at arithmetic and geometric differences, then iterate. Generally a simple additive or multiplicative rule is considered the “simplest” way to complete a sequence, so if you find one of those, you’ve found it, nth-root polynomials notwithstanding.

Arithmetically, the first differences in the sequence (difference between neighboring numbers) is:
The second difference (difference between neighboring first differences) is:
You can keep going, but this one is the first one that really stands out as an obviously regular pattern, and the next number is obviously 21!
Applying that back to the first difference, you’d get
And back to the original sequence, you add 13 to the last number and get 19 as the next.

You can also look at first differences between alternating elements, which you’ll do if you realize that this is two sequences woven together, and get:

Also, the answer to the metals question is clearly Zinc because it’s the only one currently used to mint coins in the US.

As far as I am concerned, if you can come up with a(ny) correct, logically reasonable answer, and justify it, off the cuff then you deserve the prize.

(Why is a raven like a writing-desk?)

Obviously what they’re looking for is the simplest formula for generating the next element, or at least one of several relatively simple ones.

There are lots of things that naturally increase arithmetically or geometrically or by some other simple formula and being able to recognize those is an actual skill that’s worth developing.

Overfitting a high-degree polynomial to sparse data on the other hand rarely solves any real problems.

There are cases where there are legitimately several relatively simple and concise formulas that could be used to generate the next element in a sequence, but this isn’t one of them.