Ruining the sequence game

This is an old puzzle-type question: “I give you three numbers, a, b, c. What is next?”

This is nice brain exercise, but as a mathematician I feel duty bound to tell you you can break this game in about five seconds, if you so wish. (Well, five seconds and a bit of calculation time.)

Let $f$ be a function so that $f(n)$ gives the $n$:th number in the sequence; in the above example,

$f(1) = a$,

$f(2) = b$

and

$f(3) = c$.

Suppose you want the next number to be $d$. That’s one more condition for $f$,

$f(4) = d$.

The trick now is that these are four fixed points for a function; and it is trivial to find any number of functions that give those four values, and thus are “the rule that gives the sequence”.

That is to say:

A+ Math Student: “The sequence starts 1, 2, 3. What’s the next one?”

Mathematician: “The next one is 666.”

A+MS: “What? Don’t be silly, the next one is 4!”

M: “Huh? What perverse logic is that? Your sequence consists obviously of the integer values of the function $f(x) = \frac{331}{3}x^3 -662x^2 + \frac{3644}{3}x - 662$. You’re just changing the answer because I got it right.”

Best of all, the trick can be used to go from any given number of sequence points into any further number of points you may want to insist on. It’s a bit prohibitively bothersome to calculate — but it is always possible.

* * *

Oh well, the calculation. Oi, the calculation. If you have four points, a third-degree (four-minus-one-th degree) polynomial is probably the easiest guess, that is, a function $f$,

$f(x) = Ax^3 + Bx^2 + Cx + D$

for some constants A, B, C and D, so that it holds that

$A + B + C + D = a$

$8A + 4B + 2C + D = b$

$27A + 9B + 3C + D = c$

$64A + 16B + 4C + D = d$.

Just run a Gauss-Jordan elimination in your head and… what?

$\displaystyle A = -\frac{1}{6}\,a+\frac{1}{2}\,b-\frac{1}{2}\,c+\frac{1}{6}\,d$

$\displaystyle B = \frac{3}{2}\,a-4b+\frac{7}{2}\,c-d$

$\displaystyle C = -\frac{13}{3}\,a+\frac{19}{2}\,b-7c+\frac{11}{6}\,d$

$\displaystyle D = 4a-6b+4c-d$.

For $a=1, b=2, c=3, d=666$, that gives

$\displaystyle f(x) = \frac{331}{3}\,x^3 -662x^2 + \frac{3644}{3}\,x - 662$;

clearly and obviously the rule asked for.

That’s still a lot of numbers, but a person with quick wits (not me!) could easily memorize that, and answer any what-is-the-fourth question with a horribly misguided rules-lawyering technically correct answer.

And technically correct is for a mathematician the only kind of correct that matters.

* * *

It will be a tad easier to forget the fourth number, and just slap down the second-degree polynomial that fits the given three —

$f(x) = Ax^2 + Bx + C$

for

$\displaystyle A + B + C = a$

$\displaystyle 4A + 2B + C = b$

$\displaystyle 9A + 3B + C = c$

or

$\displaystyle A = \frac{1}{2}\,a-b+\frac{1}{2}\,c$

$\displaystyle B = -\frac{5}{2}\,a+4b-\frac{3}{2}\,c$

$\displaystyle C = 3a-3b+c$

— and proclaim: “Here’s your bloody rule; as for the fourth number calculate it yourself! I don’t have time for your silly games! Ha ha ha!”

Alternatively, proclaim: “Here you go. Trolled by maths.”

(As for that $f$ giving the intended fourth point, that’s infinitesimally unlikely. Most clever sequences aren’t second-degree polynomials.)

* * *

Of course if one doesn’t feel bound to finding an explicit numerical rule, the possibilities are endless.

Less than 666 reasons the next number is 666

• The rule is my rule. The next number is 666.
• No, you’re doing it wrong. Trust me, I’ve heard this one before; the next one is 666.
• No, it’s the medals that are awarded in Tour de France: gold, silver, bronze and hamstrung. One, two, three, six-six-six.
• Obviously it is integers ordered according to the frequency of their appearances in Western literature. Those wacky Christian mystics, right? All about 666!
• “The Beast comes, all of a sudden! One! Two! Three! Six hundred and sixty six, the Number of the Beast! In medias res, Lupus Magnus Innominandum, Lucifer Deovore Daalek Satanas!” is the rule.
• What do you mean, the next one can’t be 666? What happened to respecting the other guy’s religion? Huh?
• Four? I’m so quoting that on Facebook. (This does not actually produce a sense of conviction in the other party, but rather a sense of crippling self-doubt with pretty much the same results.)
• Four? I never pegged you as a racist before.
• What do you need a rule like that for? It’s as obvious as elementary, high, university and Satan on horseback!
• “Wait, no.” (Repeat after each attempt to disagree. Each time increase the time between “wait” and “no”.)
• No no, I got this. This is much more elegant than yours. See? “One, two, three, six hundred and sixty six, blood, blood, blood, blood, blood—” (repeat with slowly increasing volume until you win)
• Is too! Look, I can point you at one guy on the Internet that agrees with me and not you—

One Response to “Ruining the sequence game”

1. http://support.floridachristian.org/entries/74601533-Reduced-Great-Collections-Of-Little-Tikes-Toddler-Bed-Rails Says:

culla belly