6 ms·

> 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.

• dooglius 2y ago
  It sounds like you've just found one for >=2: 2, S(2), S(S(2)), ...
  • Someone 2y ago
    That, somewhat ironically, typically isn’t included in the set of elementary functions. ‘plus’ is, but ‘plus one’ isn’t.
    • tikhonj 2y ago
      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.
      • madcaptenor 2y ago
        For example it would not be weird to have ++ instead of +1.
        • StilesCrisis 2y ago
          That’s just S(n)
• thaumasiotes 2y ago
  > 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?
  • zeroonetwothree 2y ago
    Because of our standard notation for those
    • the__alchemist 2y ago
      You are inferring that as a rule of the game by making assumptions. There are other conclusions different people could reach.
      • karparov 2y ago
        And those other people are free to do whatever they like in their interpretation of the game.
• deleted 2y ago
  [deleted]
• omoikane 2y ago
  Maybe they meant symbolic operators feel alright but named functions feel like cheating, so 2+2+2+⌊√2⌋ is fine but 2+2+2+floor(sqrt(2)) is not.
• timerol 2y ago
  > 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
  • Karellen 2y ago
    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.
    • saulpw 2y ago
      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.
      • pyrale 2y ago
        APL programmers enter the room
      • Karellen 2y ago
        What if it turns out that the radical (square root) symbol is a letter from a language (if a little squished)? And we somehow figure out one day which letter it is, for sure? https://en.wikipedia.org/wiki/Radical_symbol#Origin https://en.wikipedia.org/wiki/Radical_symbol#Origin
    • jesterswilde 2y ago
      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.
• the__alchemist 2y ago
  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.
  • deleted 2y ago
    [deleted]
  • jrockway 2y ago
    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.
    • karparov 2y ago
      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!
      • jrockway 2y ago
        Perfect! Over time I will eventually win this game!
• jonahx 2y ago
  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.
• ajkjk 2y ago
  you shouldn't be able to use letters. You're supposed to use four 2s and symbols, not four 2s plus the letters "l", "o", "g".
  • necovek 2y ago
    But letters are symbols too. (Mostly goes to show that it's really hard to be precise and allow some mathematical language and disallow some)
    • RajBhai 2y ago
      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.
      • necovek 2y ago
        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.
        • RajBhai 2y ago
          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.
• quietbritishjim 2y ago
  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.
  • nwallin 2y ago
    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.
    • billyjmc 2y ago
      But repeated application of the unary minus basically results in a no-op. So it’s somewhat exceptional in that regard.