How to solve any number sequence puzzle

I hate these dumb “only 5% of people know the answer” posts in Facebook. Those who actually know mathematics also know you can continue that number sequence with any real number.

Suppose you’re presented with the sequence¹:

\[ n_0, n_1, n_2, n_3 \]

First, decide what number you want to be next in the sequence. If possible, let it be a small number. We’ll call this number \(n_4\)

Write the following:

\[ (x + n_0) \times (x + n_1) \times (x + n_2) \times (x + n_3) \times (x + n_4) = 0 \]

Carry out the multiplications to arrive to a polynomial in standard form, in this particular example it will be a 4th degree polynomial²:

\[ Ax^4 + Bx^3 + Cx^2 + Dx + E = 0 \]

With this polynomial, announce that the answer is your chosen number. It must be the next in the sequence because it’s the only one that completes the pattern to satisfy the equation.

If confronted, ask them to prove it. If you’re particularly vindictive, you could argue that they need to prove both a) your answer is wrong, and b) their answer is right.

Protip: they can’t.

Corollary

This kind of stupid posts irate me to no end because \(99.\overline{9}...\%\) of the time it’s not presented to actually present a puzzle to entertain the mathematically inclined, but to demonstrate a fake sense of superiority based on presenting a question to which the answer is already known.

Communicating badly and then acting smug when you’re misunderstood is not cleverness.

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