If I write 2 then 4 then 6, then we feel good because we know that next comes 8. We can foresee it. We are not in the hands of destiny. Unfortunately, however, this has nothing to do with truth.– Arthur Seldom, The Oxford Murders (movie)

The series 2, 4, 8, could obviously be followed by 16, but also by 10 or 7004.– Arthur Seldom, The Oxford Murders (movie)

It's always possible to find a rule, a justification which allows a series to be continued by any number. It all depends on how complicated the rule is.

I remember getting
questions at school of the form “which number comes next?” At the time I
thought these questions were perfectly normal. I now think they are
nonsensical. As such it troubles me to see similar questions (with diagrams
rather than numbers) are being used in psychometrics assessments. For instance
here are ones called diagrammatic reasoning tests. Whether numbers or diagrams, the
idea is the same and it should be put to an end.

These questions expect you to extrapolate
from a finite set of data. The problem is, as with the above quotes, there are infinitely
many ways to do this. The only difference between them is that some “feel” more
right than others. They are intuitive, they are “simple”. But both of these
things are in fact rather subjective. And so while these questions pretend to have
only one right answer, they really do not.

Here is an example. The series 1 2 3 5 …

This could be “all integers with at most
one factor”, i.e. all the primes and the number 1 – then the next number is 7.
It could also be the Fibonacci sequence, but starting at 1 2 instead of 1 1 –
then the next number is 8. Of course one
could think up infinitely many rules for completing this sequence. Another
simple rule is to assume it is periodic 1 2 3 5 1 2 3 5…. - then the next
number is 1. Of course if you looked only at the first three elements in the
series you would probably guess the next number is 4.

The question, then is not really “what is
the next number?” it is: Find a function from the natural numbers to the
natural numbers which has the given sequence as its first mappings. The function
should be “simple”, meaning it should be described (possibly as a recurrence
relation) only with addition, subtract, exponentiation, etc. and should be the
one function that whoever is marking the question would think is the simplest.

The problem is that the problem is never
actually stated like this. The ways in which you are allowed to describe your function
are not enumerated and there is no objective means of determining what is

“simple”. Thus for any above mediocre mind, the problem is not to find the next number, but to determine how far beyond the standard set of descriptions for functions they should allow their mind to search.

“simple”. Thus for any above mediocre mind, the problem is not to find the next number, but to determine how far beyond the standard set of descriptions for functions they should allow their mind to search.

Thus the question really only does the
following: it forces you to confine your search to what is expected already. It
hinders the ability to think beyond this and it penalises anyone who happens to
think differently from the standard. It creates a false impression of truth and
limits human creativity. The only way to ask these questions (if you have to
ask them at all) is to give a precise description of the form of function
allowed and then make sure only one function in this set satisfies the
requirements. The same is true for diagrammatic questions.

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