By the way, I'd like to note that it will likely be a formalization of not the last result in that conversation, but of one of the initial proofs that relied on the Langford sequences and theorems of their existence.

The "wide ring" gadget is an interesting idea. I think it will be linear pyramids and not gaussian. A set of these gadgets can also form a basis, and in such a different basis the potential fields may look very different - almost flat, perhaps?..

The potential fields are a curious phenomenon. I feel like - and I haven't verified this in any way yet - the smoothness is dictated by the fact that all values are within a not-so-broad range. Then as you "peel" the hexagon from the outer layer using those 6-rings, building the potential field, each next layer inwards shouldn't changed too much.

I had also explored a case of n->inf where the sums turn into integrals, and consecutiveness turns into the uniform distribution. Then it's not so hard to find a solution. It's not in the article, because it turned out to be a dead end for solving the discrete case, and I didn't want to make the article too heavy - but if I remember correctly, there I also observed the smoothness.

Upon a quick research, there are many variations, and the closest to what you're describing is the "pandiagonal" magic square, where additional diagonals are considered (except they wrap around the edges of the square in a slightly funny way, so that every diagonal still contains exactly N numbers): https://en.wikipedia.org/wiki/Pandiagonal_magic_square

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The VSCode integration is the cherry on top. All files with the correct extension (such as xxx.drawio.png) will automatically open for edit in the draw.io UI embedded in the IDE.

I use this feature often to build on top of the previously made diagrams.

> "It's not just that the top hypothesis they provide was the right one," he said.

> "It's that they provide another four, and all of them made sense.

> "And for one of them, we never thought about it, and we're now working on that."

The Google's co-scientist still seems to make a useful assistant.

The initial set of strategies wasn't very diverse and compensated for the binary search "weaknesses" on the ends of the spectrum by sometimes guessing 1 and 98.

But after adding some more pure strategies to the set, we've got a far better mixed strategy that prefers the numbers between 28-70 as the first pick: https://github.com/gukoff/ballmer_puzzle#winning-strategy

> the spread on that surely goes over the 0 line.

Do you imagine starting with $1 or $1000? :)

Let's add a condition that Ballmer has infinite money, we start with a specific budget, and we can't continue playing if we exceed budget randomly changes after each game,

In the game where you start with $N, win $1 with probability p > 0.5 and lose $1 otherwise, the chance of eventually losing all your money is (p/(1-p))^N. [1]

So, the ruin chance actually becomes exponentially lower the more money you have at the start.

The steps in the random walk above belong to a simple, Bernoulli-like random distribution. Meanwhile the mixed strategy is a more complex discrete random variable because it can do more steps than just +1 and -1.

However, I believe that the same principle applies for the mixed strategy.

If you zoom out and consider "batches" of steps, you can apply the Central limit theorem and see that all these random walks work roughly the same. The caveat being that you need a large enough starting budget to "zoom out" :)

Granted, the standard deviation for the mixed strategy is ~$1. I would guesstimate that if you start with ~$1000, there's no way you will ever lose your money.

> What would be more interesting is to monte carlo simulate this strategy and look at the win/loss distribution. Presumably the choice is then not so clear cut.

Agree, this would be a nice demonstration! I will think about doing this next time I get a couple of hours of free time.

[1] https://math.stackexchange.com/a/153141/65143

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