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原始链接: https://news.ycombinator.com/item?id=43145753

这项讨论围绕数学难题的规则和“精神”展开，可能涉及使用有限的操作代表数字。参与者辩论使用功能（包括基本算术）和后继函数（S（n））的有效性，认为他们可以琐碎的解决方案。有些人将其视为一款规则发展以保持参与度的游戏，而另一些人则更喜欢更严格的预定义限制来进行竞争。质疑更先进的功能，例如阶乘，平方根和地板。唐纳德·诺斯（Donald Knuth）仅使用Digit 4和有限的操作集作为相关问题。一位参与者详细介绍了堆栈机器的实现，面临着用特定指令集编码整数的挑战，进一步强调了表达能力和最小表示之间的张力。

用四个2s制作任何整数 2025-02-25

用四个2s制作任何整数 2025-02-24

（评论） 2024-09-19

（评论） 2024-08-16

原文

I feel like the second you allow functions you've thrown the spirit of the game.

Ex, the gamma function is (n-1)! So now you're making 7 with four twos and a one. You've broken the spirit.

If I can hide numbers in a function call... It's trivially easy to always succeed.

> I feel like the second you allow functions you've thrown the spirit of the game.

+, - (both binary and unary), ×, ÷ are functions, as is raising to a power. Why would you allow them?

As always in this kind of things, one can disagree about what constitutes an elementary function, but I don’t think taking square roots should be disqualified in this puzzle.

> Ex, the gamma function is (n-1)!

And 2 is just S(S(0)) (https://en.wikipedia.org/wiki/Peano_axioms)

> If I can hide numbers in a function call... It's trivially easy to always succeed.

I wouldn’t call the construction given by Paul Dirac trivial. Do you think it is, or do you know of a simpler one?

> And 2 is just S(S(0))

This is a good example of why you need rules on which functions are allowed. Repeated application of the successor function makes the entire exercise trivial

But also, if the criteria for allowed functions is that they are "reasonable, elemental" (as per the fine article), then I think it would be quite hard to come up with a set of rules to encode that in a way that includes log() and sqrt(), but not S(). It's hard to imagine a function that is less elemental than S() (except maybe the identity function), or why its inclusion would be unreasonable when the other two aren't.

Given how the article is laid out, I think it would be more appropriate to view the game from the lens of when we teach the operations in school, as opposed to what are fundamental or elementary operations / functions in math.

Though I also think square root is cheating, it has an implicit 2 inside of it, where as raising to the power of 2 and log 2 are explicit.

You could also argue for only infix operators.

A good game must be somewhat challenging or else it is not really a game. Anything that makes the game trivial ought be omitted for it to be a game.

Yeah, thinking of the puzzle as a game, rather than a competition, allows for a different perspective.

If I think of a competition, then I'd expect the rules to be determined ahead of time according to some pre-imagined criteria. If someone manages to find a clever hack within the rules that allows for trivial "breaks", then that's good for them and they just get to beat everyone else at it.

But if I think of a game, then it's much more natural for the rules to adapt over time as people realise that some types of "play" make the game less fun, or straight-up boring. They don't have to be self-consistent, or logical. They're essentially arbitrary, and just whatever they need to be to make the game "better".

My criteria are "no letters or digits from any language" (other than 2). So you can't use the log or S() or the gamma function, but you can use sqrt, because there is accepted symbology that does not require any atom of non-mathematical language to represent.

While you are right that Succ() is as elemental as it gets (including in both Peano and set-theory construction of natural numbers), it is seldom used outside of theoretical foundations.

So perhaps the implied rule is not about it being "reasonable, elemental", but rather about "common" functions and operands (yes, it's still a can of worms, and you'd need to be explicit about what that is).

> While you are right that Succ() is as elemental as it gets (including in both Peano and set-theory construction of natural numbers), it is seldom used outside of theoretical foundations.

Well, depends on how you define seldom. What if I told you that twitter would break without the use of Succ()? :-)

This is a great point. I think what you're really responding to is that it's a game without clear rules, and so part of the "game" is thinking about creative interpretations of the rules themselves and pushing the boundary of what others originally assumed the rules to be.

Granted there is creativity in this sort of game -- indeed, most "games" in life are like this -- but it's quite a different thing from winning a game with clearly defined rules like chess, or this game with the set of allowed operations specified up front.

I think a key distinction is that those are functions of two parameters. You can't just use them as many times as you like "for free" like the square root trick at the end of the article, because you need to "spend" at least one extra 2 on the second parameter each time.

That's not the whole story of course, you still need to agree on the set of allowed operations, but I think it makes a big difference even though it seems incidental at first.

Surely we accept unary minus as one of our functions? Once we accept one unary function we're just quibbling over details.

I agree that you need to define and agree upon a finite set of allowed operations before playing the game. IMHO, square root, logarithm/exp, floor/round/ceiling, sin/cos/tan ought to be included in the list. But that's just like, my opinion, man.

> I wouldn’t call the construction given by Paul Dirac trivial. Do you think it is

Yes? It's doing exactly the thing that your parent comment complains about in the gamma function, introducing additional constants (in this case, mostly 2s) that, for no particular reason, don't count.

Why would you interpret squaring as consuming a 2, but square rooting as not consuming a 2?

That's exactly the point. What exactly, is the set of allowable functions used for the problem? I think you, and the post you reply to, are stating the same thing.

Where do you draw a line between "Functions available on a 4-function calculator" and "Functions I can make up specifically to generate a target integer"? I think you have to rigidly define this, or the game loses meaning.

Right, I invented the jrockway function which is defined as f(2, 2, 2, 2) = 7 so I made 7. Maybe the rule is "someone else has to invent the function", but I invented that so you can use it. (Please make me a the__alchemist function.)

Maybe the rule should be that the function has to be invented before the inventor has knowledge of this game. But now I'm just going through /usr/bin looking for binaries where the 2222th byte is 0x7.

Let karparov(2222) be defined as the number of whole seconds that have elapsed since this message was posted. You cover all naturals this way!

But you're all missing the point. A winning "solution" to this game is whatever the reader accepts as a legal solution which at the same time is as creative as possible. That's necessarily subjective, but that's fine. Anybody is free to argue that it's a stupid game if these are the rules, and those folks just don't need to play and can let everybody else have some fun!

That gets at what makes using additional functions like in the blog post a bit arbitrary: we don't have special notation for "+ 1" or "* 2", but we do for "^(1/2)" and "log_2". It's not hard to imagine a different world where "+ 1" or "^2" had special notation, and suddenly we'd be able to solve the question in even simpler ways.

It's still a fun puzzle, it's just based more on our shared notational conventions as much as the underlying math.

I would say that any function that implicitly favors a single number must be explicitly stated, and thus, if used for this game, be the number 2. So all uses of the radical must state which root (2). Dirac's solution then wouldn't work because the use of 2 is O(n).

Logs would also need to state the base. No implicit use of e or 10, and lg wouldn't be allowed in place of log2.

I haven't said much other than logs and roots are binary operators with one of the operands usually implicit in the notation, so if we don't have special notation for powers and exponentiation, then we shouldn't allow the same for their inverse operations.

Why not?

Why is it ok to use "22" = 2 * 10^1 + 2 (when it could be a number in base 3 — 2 * 3^1 + 2 = 8 decimal — or any other base)? This implies base 10, just like root implies base 2, or ln means e.

As I said, this is a game, and trying to imply certain artificial constraints will be really hard with how abstract maths is.

Again, mention of successor function is apt: everything else is built from 1, succ() and another axiom, definition or so. So everything else can be reduced to this.

I said that this implicit use of 10 or some other number shouldn't be allowed. So log, ln, lg (i.e base 2) shouldn't be allowed, but log_b(x) where b and x are states is OK, just as 10^x, e^x, and 2^x require you to explicitly expose the base (and for this puzzle, disallow 10 and e since neither is a 2).

Successor is essentially s(n) = n + 1, so that shouldn't be allowed either.

Yeah no they're not. If I give you the list "abc/=^" absolutely everybody agrees which ones are symbols and which ones are letters. What you meant to say is that it's really easy to be precise, if you are operating on human language instead of mathematical pedantry.

How about adding αβγδ or ℵ (aleph) to that list? Or шћдж? Or add Chinese or Arabic letters to the list.

Letters are symbols used to write down words of a natural language.

If you are unfamiliar with a language, you are more likely to call them "symbols" instead of "letters".

This was my initial thought once we got to the Gamma function.

My reasoning is (I'm pretty sure it's the same as yours), why is the gamma function allowed, but not others? I could insert arbitrary functions to make the game arbitrarily solvable.

While this hit me at the Gamma introduction, I think it leads back to the beginning: It's a poorly defined problem from the rules at the start of the article. It should instead define the set of allowable functions (or operations) explicitly. I think you could modify this to retain the intent of showing how the problem scales with knowledge level.

I think you have a point, but as others have commented "allowing functions" is not the problem, as fundamental math operations are functions. But if we limit ourselves to only functions that map (tuples of) integers to integers ((Z, Z, ...) -> Z), the spirit of the original game is retained. This disallows sqrt and log, but retains addition, subtraction and multiplication (but not division). Factorial (n!) is allowed, as is exponentiation to a non-negative power.

Wonder if someone could come up with general solution within these constraints.

I think that, if you are restricted to a finite list of n-ary functions, n>=2, each returning a single value, then you can’t do it, as you will only have finitely many valid expressions.

This may be easier to see in a stack machine / RPN model. An expression is a list of operations, drawn from a finite set, each of is either “push the number 2” or something that decreases the stack size by at least 1. And you need exactly 4 pushes. So a valid expression has four pushes and at most 3 other operations, because otherwise the stack would underflow. This gives a finite number of possible expressions, but there are an infinite number of integers, so it can’t work.

I didn't mean to say that n would need to be 2 or more, unary functions (I even mentioned factorial) should be fine. 1-tuple is valid tuple :)

And yes I think your analysis that only allowing n-ary funcs with n>=2 would make general solution very much impossible since you can only have a limited number of inputs.

Are you sure the number of possible expressions is finite? Or is it countably infinite? I could in theory make an arbitrarily long expression out of the four fundamental operators, and because I can derive -1 from a finite set of operations it is possible to derive any integer from any even integer. You can also derive zero from the rules.

> I could in theory make an arbitrarily long expression out of the four fundamental operators

Not with only four inputs you can't. You can only have three operations, because you have no way of getting another input parameter.

It’s just about having fun at the end of the day, the gamma function and square root are considered fundamental enough. But if one wants they could try to limit to different subsets of functions and prove which numbers are possible or impossible to achieve just with those.

They also say “mathematical tools” not arbitrary functions.

This is why people have brought up the successor function too: A + B, is, by definition B applications of Succ() on A:

  A + B = Succ(Succ(Succ(...Succ(A))))

So using your own argument, we could say that using '+' is simply a convention on how we can write down the above — if we insist on spelling "conventional" things out, we must be able to use the underlying elementary function[]. Or isn't a factorial n! really n(n-1)...21, so all those numbers spelled out?

The mathematical root probably first appeared as a square root and was later extended to support other exponents.

But is there any fun in this? As noted elsewhere, the game is in finding the rules, and a solution within those rules.

[]Since all the natural numbers other than 1 are defined using a Succ() functions, there's a trivial solution. But if we only limit ourselves to this most elementary operation, we can't get a 1 because that's an axiom in itself ("There exists 1" or "There is a set of cardinality 1").

Not sure what you mean with "correct", because "correct" is an equivalence class of slightly different definitions? Eg. https://en.wikipedia.org/wiki/Peano_axioms has, instead of both yours and mine:

  A + 0 = A
  A + Succ(B) = Succ(A + B)

They would all be proven in the same manner, though some might be slightly stronger in relation to commutation, making some proofs easier off the bat.

I would argue that the Gamma function is more fundamental than the factorial operation. But you are still correct that if arbitrary functions were allowed, the game would degenerate to triviality.

There's a version of this with "4"s that I have done through 20. I have used factorials and square roots but nothing more. I felt dirty.

BUT I did not use "44", which I did see in some solutions. That seemed out of bounds to me!

> I feel like...you've thrown the spirit of the game.

It's a little "brain teaser" game, to encourage kids to practice fairly basic math. Don't take it too far out of context.

Since in essence you can define your own functions f that give you any number you want from 2 (and for example not defined anywhere else). I.e. rules never said you can't use any function. They are vaguely saying "any mathematical operation".

> use any mathematical operations

Okay, then this is easy, just use the successor function.

  S(n) = n+1

  6 = 2*2*2-2
  7 = S(2*2*2-2)
  8 = S(S(2*2*2-2))

Etc.

S(N) is just S(N), it's not N+1 unless you define it that way. S(0) is not 1, it's S(0). The successor function doesn't really get mapped to numbers without an explicit additional mapping.

No, it's not. That's just the standard model of peano arithmetic. You can map the successor function to many different operations and objects that aren't even numbers.

Yes it is. "1" is just a label for what is also known as S(0) no mater what other ways you invent to refer to that element. Just like 一 is another label for the same concept.

I took a lot of math in my schooling, and have continued using math on a daily basis in my engineering career. I even subscribe to many math channels on YouTube, but this is the first time in my life that I've even heard of this function. I know there aren't real rules to this puzzle, but this function doesn't seem well-known at all.

The successor function is how natural numbers are defined in most axiomatic arithmetic systems.

A natural number is either zero or a successor of a natural number.

Addition is defined as a recursive application of successor functions.

  m + 0 = m,
  m + S(n) = S(m + n).

as an example -- 3 + 2:

S(S(S(0))) + S(S(0)) = S( S(S(S(0))) + S(0) ) = S( S( S(S(S(0))) + 0 ) ) = S(S(S(S(S(0)))))

There's nothing really in the definition of the successor function that necessarily requires that it's interpreted as n+1, though. It's just an interpretation from the context in which it's used. It could represent any operation as long as it is isomorphic to adding one -- but there's nothing special about "adding one". You could have it represent multiplying by a constant, and interpret "zero" as the number one.

So a number in this interpretation is either 1; or 2 times a natural number.

1, 2, 4, 8, 16, 32 -- you're working only with powers of 2 now.

The above "addition" rule above still works, but now it represents "multiplication" instead of addition. I'll replace all the S's in the above example with x2

2x2x2x1 "+" 2x2x1 = 2x(2x2x2x1 + 2x1) = 2x(2x(2x2x2x1 + 1) = 2x2x2x2x2x1 = 32

So now, instead of addition, we've recursively defined multiplication where the successor function is interpreted as multiplying by 2. There's an infinite number of ways that you can interpret the successor function.

So basically, I do think it's cheating, and if you do want to define it as n+1, it would be even simpler to just define a function that takes any number to the desired output.

I'm not sure what kind of math schooling you did, but did you learn to construct the natural numbers from scratch?

This may have been the fault of math education. In my college people learn real analysis (constructing the real numbers) before they learn to construct the natural numbers, which is backwards to me. I recommend learning it: constructing the natural numbers from just sets in the tradition of Zermelo–Fraenkel is mind blowing the first time you see it. Of course you could just use Peano axioms without touching set theory too.

I had the same thought. Also with square roots hiding 2s behind notation, etc. The whole project isn’t really very coherent without specifying what specific operators you can use (and how many times).

See also: "Representing numbers using only one 4" written by a 26-year-old Donald Knuth in 1964 (https://www.jstor.org/stable/2689238 reprinted as Chapter 10 of his Selected Papers on Fun and Games) — it uses the single digit 4, and the three operations √x (square root), ⌊x⌋ (floor, i.e. greater integer not greater than), and x! (factorial), and ends with a (still unsolved) conjecture about whether every integer can be represented in this way.

The appendix (written for the book in 2011) points out an earlier (1962) 1.5-page paper π in Four 4's by J. H. Conway and M. J. T. Guy, written when they were students at Cambridge, that has a similar idea: https://archive.org/details/eureka-25/page/18/mode/1up?view=...

For example,

    5 = ⌊√√√√√(4!)!⌋

because 24! lies between 5^{32} and 6^{32}.

Maybe it's just me, but writing sqrt(2+2) instead of sqrt(2*2) or sqrt(2^2) was such an odd choice. It obfuscates the reason why 2=sqrt(2+2), and unnecessarily so.

Good point and feedback, but an odd choice by the author?

It could be the phenomenon of the author's cognitive bandwidth being consumed by everything in the article, including each argument, the overall argument, the writing, the formatting, etc. etc., and with time pressures. The critic can focus at their leisure on one point, with bandwidth to spare - and so it's obvious! :)

I agree it potentially wasn't a conscious choice, but it's still interesting nonetheless.

I wasn't criticizing him for this, but rather fascinated that this is the variant that was chosen.

Sorry, the intention was just to show a cool use of complex numbers, not claim this is the simplest method to generate 12 which is pretty simple, as you demonstrate.

Because sqrt is the reverse of 2^2 and 2*2 (which is 2^2 unwrapped). Though there's no direct relationship between sqrt and 2+2 other than that it happens to be equal to 2*2.

Or put differently: N = sqrt(N^2) or sqrt(N * N) for every positive N, but x = sqrt(x + x) or sqrt(x + 2) is only true for x = 2 for both or x = 0 for the first representation.

Though as I said in the rest of my comment, the 2+2 unwrap only works for N=2. So it's not a general unwrap, but rather a specific example that happens to work for N=2.

I think my preference is more towards conciseness.

I made a stack machine with single character instructions and needed to solve a variation of this problem. I had just the digits 0 through 9. The characters '23' would be push 2 followed by push 3. To actually represent the number 23 you would use

    45*3+
 or something similar.

That left me with the problem of how to encode each integer in the fewest characters.

Tools at hand.

        The digits 0 through 9
        'P':  Pi
        '*':  (a * b),
        '/':  (a / b),
        '-':  (a - b),
        '+':  (a + b),
        's':  sin(a),
        'c':  cos(a),
        'q':  sqrt(a),
        'l':  log(a),
        '~':  abs(a),
        '#':  round(a),
        '$':  Math.floor(a),
        'C':  clamp(a),
      
        '<': min(a, b),
        '>': max(a, b),
        '^':  pow(a, b),
        'a': atan2(a, b),
        '%': positiveMod(a, b),
        '!': (1 - a),
        '?': (a <= 0 ? 0 : 1)
        'o':   a xor b scaled by c;   ((a*c) xor (b*c))/c

        'd':  duplicate the top stack entry
        ':':  swap the top two stack entries
        ';':  swap the top and third stack entries

I have wondered about revisiting the stack machine with a complex number stack to see what I can come up with.

(Next time I post something like this I am not going to use my phone)

Kolmogorov complexity is uncomputable because it admits Turing complete languages, and reduces to the halting problem. If the language you admit isn't Turing complete, then the Kolmogorov complexity of the thing is computable.

It looks like OP's language is not Turing complete. It always terminates. You can just do a breadth first search on the program space. The first program you get that outputs the number you want is the shortest program.

If it were Turing complete, you can't do this, because eventually you'll find a program that just keeps running for like a really long time. Is it running because the program never halts? Or is will it halt eventually and output the number you want? You can't know for sure.

I thought it would be possible to use the (undocumented?) T instruction and use the time variable as persistent memory, but it seems to reset each step. It would be nice if there were a way to store things across time steps!

What about making each digit be the instruction "times 10 plus digit", with a different instruction to push a 0, like a space? Then you can represent 23 with " 23".

> There's just one small wrinkle: it uses three instances of the digit 2, not four.

One small wrinkle, if you ignore the fact that the root notation conceals exponentiation by 1/2, by making that common value a default.

That's a lot of hidden 2's!

Root notation isn't really _concealing_ anything. The fact that it's mostly equivalent to exponentiation by half is a theorem. Do we need to admit that 2 is concealing 1+1, thus making the game impossible?

Given the prevalence of quadratic polynomials over higher-order ones, sqrt does feel somewhat more fundamental than arbitrary exponentiation.

Lots of people have pointed out the farcity of the game after allowing fancy functions, but IMHO, there is a lot of fun in just finding satisfying solutions, without the need for specific rule limitations.

The root notation is a syntax with a place for a number, to specify the degree. For instance ∛ -> cube root(s). When the degree is 2, it can be omitted.

So yes, the notation conceals something: its default value.

Related: there as a reverse engineering/CTF challenge (which shall remain nameless to prevent you from cheating) where my solution involved injecting shellcode that adds specific number to the stack pointer. But your shellcode -- including the number(s) you add -- can only involve bytes from the ascii alphanumeric set.

So I used a SAT solver to find a combination of numbers, not using prohibited bytes, that add up to the number I really want.

https://docs.google.com/presentation/d/19K7SK1L49reoFgjEPKCF...

I love this:

> Note that these are large files; both are over 1.6 Megabytes, so don’t load these if you have a slow Internet connection

Just shows how far internet speeds have come...

That's the one!

That's how I learned about false induction. I also liked the one about the men who were lined up and had something on their backs and they had to guess what it was.

Here are some values that are (understandably) not listed on the blog. They happen only due to the limited precision of floating point formats.

  128              = √(2 / √√(√2 - (2 / √2)))
  8192             = √√(2 / ((√2 * √2) - 2))
  16384            = (2 / √√(√2 - (2 / √2)))
  67108864         = √(2 / ((√2 * √2) - 2))
  134217728        = (2 / √(√2 - (2 / √2)))
  4503599627370496 = (2 / ((√2 * √2) - 2))
  9007199254740992 = (2 / (√2 - (2 / √2)))
  6369051672525773 = (√2 / (√2 - (2 / √2)))

I found these by accident a long time ago but kept them because they do "work". Try to input one expression in the lil box in https://www.wolframalpha.com/?source=nav and they will quickly evaluate to these values; the charade goes away after you press Enter and get the (mathematically) correct answer.

My old solvers from what feels like a previous life: https://madflame991.blogspot.com/2013/02/four-fours.html https://madflame991.blogspot.com/2013/02/return-of-four-four...

That was fun

Nice! Looking into them a bit deeper, they all rely on two facts involving quantities that are off by 1 ulp:

    √2 - 2/√2 as a float64 is exactly equal to 2^{-52}

    √2 * √2   as a float64 is exactly equal to 2 + 2^{-51}

and all other calculations after that are mathematically exact. For example:

    √(2 / √√(√2 - (2 / √2))) := √(2 / √√(2^{-52})) = √(2 / 2^{-13}) = 2^7 = 128

and

    √√(2 / ((√2 * √2) - 2)) := √√(2 / 2^{-51}) = √√(2^{52}) = 2^{13} = 8192

In your last one,

    6369051672525773 = 2^{52} * 1.4142135623730951454746218587388284504413604736328125

(https://float.exposed/0x4336a09e667f3bcd) where the second value is the exact value of the floating-point representation of √2 (i.e. the closest representable-in-float64 value to √2).

Very clever, but using an arbitrary number of square roots seems almost cheating since it’s practically another symbol for a “2” (exponent of 1/2)

> I've read about this story in Graham Farmelo's book The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius.

"The Strangest Man": https://en.wikipedia.org/wiki/The_Strangest_Man

Four Fours: https://en.wikipedia.org/wiki/Four_fours :

> Four fours is a mathematical puzzle, the goal of which is to find the simplest mathematical expression for every whole number from 0 to some maximum, using only common mathematical symbols and the digit four. No other digit is allowed. Most versions of the puzzle require that each expression have exactly four fours, but some variations require that each expression have some minimum number of fours.

I think I saw this one on an ancient arab math problems book. But it may be apocryphal, not sure how many tools they would have had, factorial symbols?

At any rate they invented algebra so maybe there's something to it

Edit: It was the man who counted, definitely apocryphal, as it was written in the 20th century

So, the formula is really about making any integer with three 2s, but historical precedent calls the game with four 2s more interesting, so the (stronger) result is monkey-patched by replacing a 2 with sqrt(2+2).

Why not use 3 2’s to make n + 2 or n - 2? That’s a lot easier than making a subscript so complicated.

This is the Curse of Knowledge. OP stared too long into the abyss.

The problem is allowing arbitrary numbers of unary operators. If you allowed ++ increment it would be trivialized even easier. Could even do all complex numbers with only 2 twos.

If you don't allow arbitrary numbers of unary operators and only a finite set of binary operators then you will only be able to represent a finite number of numbers. You need at least one "useful" unary operator for the problem to remain interesting.

You don't need the gamma function to get to 7. You can stay at an Algebra 1 level.

I solved the puzzle for 1-10 before looking at the answers, and this was my solution for 7:

⌊√222⌋/2

or more readably:

floor(sqrt(222)) / 2

Same here, our sixth grade teacher assigned it. I wrote a BASIC program on the TRS80 to try to find solutions. Printed each solution it found to the dot matrix printer.

Thanks—I think we'll merge the comments hither because this submission was the first, and because the other submitter currently has a second article on the frontpage right now.

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