## February 2022

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In the Math Corner of my February Diary I offered this:

What is the next number in the sequence 242, 682, 1562, 3110, 5602, 9362, 14762, …? Why is it topical?

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**• Solution**

Faced with a sequence of whole numbers like that, the seasoned brainteasee's thoughts fly todifferences. Before tackling the actual problem here, just a word aboutdifferences. Don't worry, this has nothing to do with Diversity.

: Start with a single-variable polynomial that has whole-number coefficients. At random I'll take the degree 3 (i.e. cubic) polynomialDifferencesx^{3}−4x+7. What are the values of this polynomial whenx= 1, 2, 3, 4, 5, 6, …? You can work them out with straightforward arithmetic:

4 7 22 55 112 199 …

That's a sequence we might want to investigate, not knowing about the generating polynomial. How do we investigate it? We takedifferences.

Go along the sequence subtracting each term from the one following:

3 15 33 57 87 …

Those are thefirst differences. You can see where this is going. Repeat what we just did on those first differences: subtract each term from the one following:

12 18 24 30

Those, you will not be astounded to learn, are thesecond differences. Repeat to getthird differences:

6 6 6

Whoa! They're all the same!

For a starting sequence generated by a polynomial, this always happens; and it happens at then-th differences when the polynomial is of degreen— in this case,n= 3.

Once you've reached then-th differences and seen they're all the same, you can reverse-engineer — climbing back up through the rows of differences by addition until you reach the original sequence. That gets you thenextterm in the sequence, without having to figure out the generating polynomial. (There's a way to do that, too, but it's beyond my scope here.)

OK, now we can proceed to the actual solution.

: The differences for the sequence 242, 682, 1562, 3110, 5602, 9362, 14762, … are:Actual solution

First: 440, 880, 1548, 2492, 3760, 5400, …

Second: 440, 668, 944, 1268, 1640, …

Third: 228, 276, 324, 372, …

Fourth: 48, 48, 48, …

So it looks as though our mystery sequence is generated by a fourth-degree polynomial.

Never mind that, though. Let's reverse-engineer to get the next value for third difference, second difference, first difference, and the sequence itself:

48+372 = 420; 420+1640 = 2060; 2060+5400 = 7460; and 7460+14762 = 22222.

Geronimo! We've solved the brainteaser. The next term in the sequence is 22222. Why is that topical? Because this is the brainteaser for February 2022, a month which contains the date 2/22/22 (or if you're British, 22/2/22.)

In this case, the generating polynomial is easy to figure.

First note that every number in the sequence is even. The generating polynomial is therefore 2 times some slightly simpler polynomial, still with whole-number coefficients. We can just halve every number in the sequence to get a slightly simpler sequence:

121, 341, 781, 3110, 2801, 4681, 7381, 11111 …

That last number is of course 10^{4}+10^{3}+10^{2}+10+1. Could it be that the (simplified) generating polynomial is justx^{4}+x^{3}+x^{2}+x+1? Let's check.

9^{4}+9^{3}+9^{2}+9+1 = 7381

8^{4}+8^{3}+8^{2}+8+1 = 4681

Yep, looks like it.

Consider now the number — thenumberin all its Platonic abstraction — which, if written to base 9, is 22222. What would be thedecimalexpression of that number? Why, it would be 14762. Likewise the number which, if written to base 8, is 22222.Itsdecimal equivalent would be 9362 … And so on, back along our mystery sequence to its first member, 242. That's the decimal expression for the number written 22222 in base 3.

(Since there is no such digit as "2" when writing numbers to base 2, and "base 1" makes no sense in this context, I left off the first two terms of this sequence. On a strictly polynomial basis they are 10 and 62.)