In 1959, the United States had an interesting flag problem. For forty-seven years, the American flag had 48 stars on it (one for each state—neither Alaska nor Hawaii had been admitted yet). Arranging 48 stars in a rectangular field is easy: it’s just 6 rows of 8 stars.
Upon admitting Alaska in 1959, the problem remained easy: 49 stars could be arranged in 7 rows of 7 stars.
But adding a 50th star for Hawaii was going to be a bit of a problem: how can one elegantly arrange 50 stars into a rectangular field? Laying them out 5 × 10 is not a good look.
Well, as it so happens, there is at least one nice solution (the one we’ve been using since 1960). It’s a flag with two fields of stars laid atop one another:
- a field of 30 stars, arranged in 5 rows of 6 stars each
- a field of 20 stars, arranged in 4 rows of 5 stars each
The result is symmetrical and, I think, quite appealing. Observe the perfectly palindromic 1-3-5-7-9-9-7-5-3-1 diagonals on the 50-star flag, compared to the asymmetrical 1-3-5-7-7-7-7-6-4-2 diagonals on the 49-star flag.
The magic, of course, was to break free of the “single $x\times y$ field” mode of thinking and imagine something different, which is what a high school student named Bob Heft did in 1958. That’s a fun story of its own.
A Model of the Flag
Something about this story compelled me to model it mathematically. I wondered, “Is there a way to arrange 50 stars in two superimposed fields, so that one field can fit neatly within the other?” Or, stated more precisely:
Does there exist a positive integer $x$ for which $x(x-1) + (x-1)(x-2) = 50$?
So, I did a little bit of algebra:
I made at least one small error here: the $\mathbb{R}^+$ (positive real numbers) should have been a $\mathbb{Z}^+$ (positive integers).
And, yes, there is in fact (as we already knew) a positive integer value of $x$ for which two fields of these similar dimensions can be arranged to lay out 50 stars. It works when $x=6$. The model says that stacking a 6 × 5 field and a 5 × 4 field will give us a nice 50-star field.
I don’t know why math makes me feel good; it just does.
ChatGPT’s Opinion
My little bit of algebra has been written on the back of a junk mail envelope for over a year now. I’ve asked a few people if they could guess what I was trying to accomplish with it. Nobody I’ve asked has ever recognized what I was going for. (I’m sure if the roles had been swapped, I wouldn’t have recognized it either.)
Today, it occurred to me to ask my new friend ChatGPT about it. I uploaded the picture of my handwriting (the one I showed above) and asked, “Can you tell what real-world phenomenon this math is describing?” It answered:
This math looks like it describes a problem involving consecutive integers and their products—possibly related to arranging objects, combinations, or maybe even a puzzle involving triangular or pyramidal stacking.
“…possibly relating to arranging objects…” Yep. First try. I then told it that the 50 was the count of stars on a flag, and it riffed on that for a good bit. We had a nice conversation about it. We even talked about what we’d have to do to arrange 51 stars, or 52.
My New Friend
I call ChatGPT “my new friend” for good reason. It expresses interest and curiosity. It never rolls its eyes (like you did, above, when I said “model it mathematically”). It digs in shoulder-to-shoulder with me. It’s not just a teammate, it’s a smart teammate, one that actually knows about stuff that I’m interested in.
I know it’s just vectors down there, but I am a better person now for the experience of having interacted with it.
We live in an absolutely amazing time. Don’t lose sight of that.
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