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The 1779 Euler puzzle: "Six army regiments each have six officers of six different ranks. Can the 36 officers be arranged in a 6-by-6 square so that no row or column repeats a rank or regiment?" Euler said it's impossible.

The 1960 proof by mathematicians and computers: "It's impossible."[0]

The 2022 solution(?): "In a paper posted online and submitted to Physical Review Letters, a group of quantum physicists in India and Poland demonstrates that it is possible to arrange 36 officers in a way that fulfills Euler’s criteria — so long as the officers can have a quantum mixture of ranks and regiments."[1]

My question: If we are permitting this (headline style) to be called a "quantum solution" (to a non quantum problem), should we expect and brace for a torrent of "quantum solutions" to previously impossible problems? Are these meaningful? Or just really cool and interesting?

[0] https://www.cambridge.org/core/journals/canadian-journal-of-...

[1] https://arxiv.org/abs/2104.05122

7 comments

> If we are permitting this (headline style) to be called a "quantum solution" (to a non quantum problem), should we expect and brace for a torrent of "quantum solutions" to previously impossible problems? Are these meaningful?

The paper does something meaningful with its solution - it constructs an "impossible" error-correcting code that you can use if you have a quantum channel. That's not really practical to use yet, but one could imagine one day e.g. sending messages through space slightly more efficiently.

Thank you, I can see why this could actually become practical. I was prematurely wincing at what I thought was watered down use of the word solution but that's just pedantry on my part. Cheers.

But utterly fails to address the Euler puzzle as I Euler was not asking about quantum mixture of ranks and regiments.

That's okay. Euler wasn't trying to arrange these army officers for any sort of practical purpose, the whole point of the puzzle is an exercise in mathematics. So it's mathematically interesting that the puzzle cannot be solved in a conventional manner, but it's also interesting that it can be solved by applying quantum concepts. Interesting research like this is the whole point of Euler's puzzles.

But yeah I guess it's a disappointment to all those army generals who were hoping for a way to arrange their officers in a 6x6 square according to Euler's constraints. The use of quantum ranks may have deleterious effects on battlefield effectiveness.

> So it's mathematically interesting that the puzzle cannot be solved in a conventional manner, but it's also interesting that it can be solved by applying quantum concepts.

I guess the question was: Is the "quantum officers puzzle" the same puzzle as the one proposed by Euler or a variant?

If they're not the same, it's not a "quantum solution" but "a solution to a quantum variant". That can be interesting on its own (with proposals made in the discussions here), it's just that "Yeehaw, we solved something for the first time in 243 years With Quantum[tm]" seems to be the wrong take away (although it might mean the next round of public funding)

I mean this is a common sort of thing, where you take something that has no solution, relax the setting, and find it has a solution in that relaxed setting. You're not finding a solution to the original problem, but you are finding a solution to a generalization, and that's still interesting.

I think it is worth noting that Euler conjectured there was no solution to any square of order 4n+2, but in 1960 it was proved that only orders 2 and 6 have no solutions.

Given that it was just a conjecture, why is it worth noting? Getting back to the main topic, it wasn't only conjectured, but also proven.

It's noteworthy because it's an a conjecture by Euler and because it's an example of hasty generalization. Either Euler jumped to conclusions almost laughably quickly or something interesting happens once n >= 10.

Because the conjecture for the order 6 is just a particular case of his wider conjecture that he posited, it gives a bigger picture.

What does "square of order 4n+2" mean here?

The term "order 4n+2" means a number of the form n (a positive integer) times four, then add two.

So 6 fits (6 is 1 times 4, plus 2) and 10 fits (10 is 2 times 4, plus two) but 12 does not (twelve can not be made by multiplying a positive whole number by four, then adding two to that number).

Thus a 6x6 square is permissible, as is a 10x10 square. But not a 12x12 square.

A square of x by x

for x ∈ {2, 6, 10, 14, 18, 22 ... } = {2, every uneven n multiplied with 2}

4n+2 = 2*(2n+1)

A 4n+2 by 4n+2 square.

I think it is worth noting that not everyone gets 100% right. You're point would be most accurate if Euler was a Quantum computation expert.

I haven't read his biography, so can't know for sure.

Euler was so successful and prolific that the list of discoveries named after him is huge.

So big, in fact, that a great many things are named after the second person to discover them.

He's one of - if not the - greatest mathematicians in history.

I don't know, I felt cheated when I learned about imaginary numbers in high school (wait, you can take the square root of a negative number?). It felt like you had patiently learned the rules of math and then someone just made some shit up.

Probably when I started my slow deviation away from math (certainly a part of why I never majored in it).

It's too bad though. Over the years since I have come to trust mathematicians more and more.

A co-worker explained to me how the imaginary component of complex numbers represents the phase information when performing an FFT. I think he was even trying to explain to me how it is why an FFT is not reversible, why you lose the phase information from the original (but I was already lost).

(Odd too that the human ear cannot distinguish between two audio sources for which every thing is the same but for the phase. Related?)

I am assuming now, from a position of ignorance if that is not obvious, that imaginary numbers are quite clever after all, perhaps neither a hack nor "imaginary".

Imaginary numbers simply represent a different and more powerful system like how integers gain power when you introduce 0 and negative numbers. Rational numbers (fractions) can’t represent irrational numbers like pi or e etc.

It’s often useful because you can use complex numbers as an intermediary step in many calculations (such as solving cubic equations) while still ending up with a non imaginary number at the end.

The history is actually fairly interesting: https://www.youtube.com/watch?v=cUzklzVXJwo

School teaching tends to be authoritarian, so that's a reaction I felt too. "Less than no apples? Minus times minus is plus? What the hell is this and who ordered it?"

I wish it'd been explained like: counting numbers like 1, 2, 3 obey certain laws, e.g. adding in different ways gets the same result. If you relax just a few of the laws in the right way, you can find a broader class of things obeying the shared logic. In the case of complex numbers, we're dropping ordering, and finding that takes us from 1-d sliding and stretching (adding and multiplying) to 2-d, where the stretching becomes stretching and turning.

I think the biggest problem with authoritarian thinking about math is that math is not a discovered set of authoritarian principles, but rather a colossal project to make up weird relationships that satisfy the rules of the weird relationships that were already made up.

Approaching imaginary numbers as:

> "Eventually, mathematicians got sick of not being able to achieve a negative result through multiplication, so Rafael Bombelli finally made up imaginary numbers using the work of other Italians and even some Greeks. Unfortunately for him, most people thought his idea and rule set was stupid until about two hundred years later. We will now learn what sorts of nonsense he made up--those of you who are interested in electricity had better listen close--"

will inevitably lead to a different mindset about math than:

> "Today's lesson is that the square root of negative one is i. Write that down, because it WILL be on the test."

I feel like they could have explained "minus times minus is a plus" using the distributive law.

  -1 x -1 = -1 * (0 - 1)
          = (-1 * 0) - (-1 * 1)

Presumably at this point you were happy with "anything times zero is zero" and "anything times unity is itself", so:

          = 0 - (-1)

and now we can just use the additive laws of negative numbers, which hopefully were intuitive enough:

= 1

They're used in a generalised solution to cubic equations (analogous to the quadratic solution we all learn in high school), where they represent the side length of a square with negative area.

https://www.youtube.com/watch?v=cUzklzVXJwo

Excellent video.

Imaginary numbers seem like something that shouldn’t be taught in high school. Algebra, geometry, trigonometry, and calculus are all useful even if you don’t recall the specifics of the notation and calculation. Imaginary numbers are niche.

They feature fairly prominently in quantum mechanics.

In my humble opinion, imaginary numbers are a core part of algebra, calculus, trigonometry, and physics, and all sufficiently intelligent people should learn about them. The reason exponential curves and sine waves show up all over the place in reality is because they are the solution to some simple differential equations. Imaginary numbers show you how these are fundamentally very similar things. Similarly, imaginary numbers make it clean and simple to solve cubic equations. And the idea that introducing this concept of "imaginary number" makes multiple practical applications much simpler, therefore we added it to mathematics, is core to how mathematics itself evolves.

I’m willing to believe that. But I certainly never learned that. I only briefly got into differential equations by the end of high school. For our program imaginary numbers weren’t much more than learning some rules about how they work exponentially and an awkward introduction to vectors with complex numbers that really should have started with just real numbers.

Maybe just teacher quality though. I can’t forget the frustration of both the class and our calc teacher when she realized our trig teacher from the prior year never conveyed the relationship between various trigonometric identities and the Pythagorean theorem and instantly made it vastly more intuitive.

Imaginary numbers are unfortunately named. It's not a misnomer per se, but they are far from being just figments of imagination.

I mean, I'm no scientist but the article does present it in a way that implies it can be solved by changing the problem / cheating. I mean sure, it might work if the ranks are all ranks at the same time or are in all positions at the same time, to use an overly simplified popular science take of quantum.

To be fair to the paper authors, they embrace and joke about it in their own headline '36 entangled officers'. PopSci headlines are always clickbait/sensationalised.

> PopSci headlines are always clickbait/sensationalised.

For most people this level of dishonesty would get them fired or at least a stern request to leave. If I go to my boss and tell him sure I did my job if my job was something other than what you asked, he will tell me "pls go".

This should be called what it is, lies, and it should not be allowed on HN, and it should get all quantamagazine's social media accounts suspended.

I'm not sure if we are "permitting" anything, but I don't think it would be a bad thing if mathematicians and researchers in quantum computing explored the implications of this field with regards to previously stated problems.[0] Is there a particular justification you suggest a paper in math should have before exploring a problem besides that it's interesting?

[0] https://chaos.if.uj.edu.pl/ZOA/?which=people&lang=en&who=Ada...

It’s mostly just that the solution is to a totally different problem once you remove the constraint that any given square contain only a single value (as opposed to a quantum superposition of values). AFAICT this doesn’t answer any open questions about the original puzzle.