February 2022
—————————
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?
—————————
• Solution
Faced with a sequence of whole numbers like that, the seasoned brainteasee's thoughts fly to differences. Before tackling the actual problem here, just a word about differences. Don't worry, this has nothing to do with Diversity.
Differences: Start with a single-variable polynomial that has whole-number coefficients. At random I'll take the degree 3 (i.e. cubic) polynomial x3−4x+7. What are the values of this polynomial when x = 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 take differences.
Go along the sequence subtracting each term from the one following:
3 15 33 57 87 …
Those are the first 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 the second differences. Repeat to get third 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 the n-th differences when the polynomial is of degree n — in this case, n = 3.
Once you've reached the n-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 the next term 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.
Actual solution: The differences for the sequence 242, 682, 1562, 3110, 5602, 9362, 14762, … are:
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 104+103+102+10+1. Could it be that the (simplified) generating polynomial is just x4+x3+x2+x+1? Let's check.
94+93+92+9+1 = 7381
84+83+82+8+1 = 4681
Yep, looks like it.
Consider now the number — the number in all its Platonic abstraction — which, if written to base 9, is 22222. What would be the decimal expression of that number? Why, it would be 14762. Likewise the number which, if written to base 8, is 22222. Its decimal 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.)