4 ms·

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".

by ajkjk 2y ago

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.
      - necovek 2y ago
        FWIW, "successor" is not really n + 1: you've got that the other way around. Successor simply "is" (it's a relation that satisfies a number of conditions), and summation is defined in terms of successor function. My point is that you can really define everything in terms of these primitive definitions, which means that there won't be any single use of a non-2 digit for any function, or you'll be going with a set of arbitrary allowances. But the whole point should be: what are those arbitrary constraints that make the game fun? And once you clear that bar, it's ok to open up the next one (this does not make them non-arbitrary though). Basically, I am saying your take at those arbitrary decisions is not a very fun one ;-)
- ajkjk 2y ago
  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.
  - necovek 2y ago
    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".
    - ajkjk 2y ago
      Those are clearly all letters in the sense of the spirit of the game. It's pretty unambiguous.

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".

by 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.
    - necovek 2y ago
      FWIW, "successor" is not really n + 1: you've got that the other way around. Successor simply "is" (it's a relation that satisfies a number of conditions), and summation is defined in terms of successor function. My point is that you can really define everything in terms of these primitive definitions, which means that there won't be any single use of a non-2 digit for any function, or you'll be going with a set of arbitrary allowances. But the whole point should be: what are those arbitrary constraints that make the game fun? And once you clear that bar, it's ok to open up the next one (this does not make them non-arbitrary though). Basically, I am saying your take at those arbitrary decisions is not a very fun one ;-)
ajkjk 2y ago
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.
- necovek 2y ago
  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".
  - ajkjk 2y ago
    Those are clearly all letters in the sense of the spirit of the game. It's pretty unambiguous.