# A Not So Boring 2024

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Image by Barry Evans and Ed Pegg

The first three tetrahedral numbers (1, 4 and 10) are represented by the number of balls making up a three-sided pyramid having one, two and three layers, respectively. The 22nd tetrahedral number is the number of balls in a 22-layer pyramid, that is, 2024

At first blush, the number of the year we're living in, 2024, sounds pretty uninteresting. First off, it's obviously not prime (a prime number being only divisible by 1 and itself), unlike, for instance, prime years 2017 and 2027. I like prime years for their individuality. Same with my age: Two years ago, 79 was a good year and I'm optimistic that 83 will be equally good, 79 and 83 being prime. Meanwhile, I'm having to bravely put up with my current age: 3⁴, which has its own numerical charm.

As far as 2024 goes, it too has its charm. OK, being even, it's obviously divisible by 2. Actually, it's divisible by 2 three times, that is, 8. 8 x 11 x 23 = 2024.

It's also the sum of a series of consecutive cubes: 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ = 2024.

(Next year will be even more satisfying, starting the string of cubes with 1³ .)

But wait, there's more! 2024 is the 22nd tetrahedral number. "Wozzat?" you ask. Tetrahedral numbers are best described visually. Imagine a ping-pong ball perfectly surrounded by three more balls, making a three-sided pyramid on a triangular base: a tetrahedron (Greek tetra four, hedra base/face). Now add more balls to the next level, so now you've got a total of 10 balls. Each level adds more balls in the series 1, 4, 10, 20, 35, 56, 84, and so on. This may look like an odd series but it's easy to show (by induction) that the nth tetrahedral number (that is, the number of balls in a pyramid having n layers) is 1/6 of n(n+1)(n+2). So a single layer has 1/6 of 1 x 2 x 3 balls = 1; two layers have 1/6 of 2 x 3 x 4 = 4; three layers have 1/6 of 3 x 4 x 5 = 10. So the first three tetrahedral numbers are 1, 4, 10 ... carrying on until we get to the 22nd tetrahedral number.

That 22nd tetrahedral number is 1/6 of 22 x 23 x 24 = 2024, this Year of Our Lord. Imagine having 2024 ping-pong balls or oranges; they can be perfectly stacked, with none left over, into a 22-layer, three-sided pyramid. Curiously, this stacking arrangement is far from the most efficient way to pack oranges, a topic that is of endless interest to math nerds.

There's much more to 2024, but it starts getting a bit arcane. For instance, 10³ + 2¹⁰ = 2024. And if you're into the number two: 2²(22 + 22²) = 2024. And 2024 is a dodecahedral number.

Hmm, maybe it's time to get out and actually experience this year rather than write about it.

Barry Evans (he/him, [email protected]) has to thank the late Martin Gardner, author of Scientific American's Mathematical Games column for 25 years, for his late-blooming love of numbers.

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#### Barry Evans

Bio:
Barry Evans lives in Old Town Eureka with his girlfriend (and wife) Louisa Rogers, several kayaks and bikes, and a stuffed gorilla named “Nameless.” A recovering civil engineer, he is the author of two McGraw-Hill popular science books and has taught science and history. His Field Notes anthologies are available... more

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