5 ms·

> use any mathematical operations Okay, then this is easy, just use the successor function. S(n) = n+1 6 = 222-2 7 = S(222-2) 8 = S(S(222-2))

> 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.

miningape 2y ago
lambda calculus has entered the chat
- Buttons840 2y ago
  https://wiki.haskell.org/Peano_numbers https://wiki.haskell.org/Peano_numbers
Hamuko 2y ago
I feel like having a 1 on the first line is cheating given the constraint "using no other digits".
- tessierashpool9 2y ago
  log is also using another digit.
- tantalor 2y ago
  By that argment you shouldn't be able to use factorial function either, because that has a "hidden 1" too: fac(n) = n * fac(n-1) n>0 = 1 n=0
  - Hamuko 2y ago
    The one here isn't hidden in any way – it's on the very first line. You might as well just define X = 7 and do 2-2+2-2+X.
    - tantalor 2y ago
      I'm not "defining" anything; I'm just using fairly bog standard math that has been around for hundreds of years.
      - empath75 2y ago
        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.
        account42 2y ago
        Unless you are talking about an alternative number system where 1+1 is not 2 then yes, S(0) is 1.
        empath75 2y ago
        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.
        account42 2y ago
        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.
Aardwolf 2y ago
Finding the shortest expression with 4 2's for a given integer would be a more interesting challenge
0xfffafaCrash 2y ago
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).
venusenvy47 2y ago
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.
- tantalor 2y ago
  It has a rich history going back to the formalization of arithmetic in the 1800s. https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms It's probably something that only folks who study the foundations of mathematics would know. You may have also heard about it if you learned about Gödel's incompleteness theorems, or read Gödel, Escher, Bach
- empath75 2y ago
  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.

> 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))

> 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.

miningape 2y ago

lambda calculus has entered the chat

Buttons840 2y ago
https://wiki.haskell.org/Peano_numbers https://wiki.haskell.org/Peano_numbers

Hamuko 2y ago

I feel like having a 1 on the first line is cheating given the constraint "using no other digits".

tessierashpool9 2y ago
log is also using another digit.
tantalor 2y ago
By that argment you shouldn't be able to use factorial function either, because that has a "hidden 1" too: fac(n) = n * fac(n-1) n>0 = 1 n=0
- Hamuko 2y ago
  The one here isn't hidden in any way – it's on the very first line. You might as well just define X = 7 and do 2-2+2-2+X.
  - tantalor 2y ago
    I'm not "defining" anything; I'm just using fairly bog standard math that has been around for hundreds of years.
    - empath75 2y ago
      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.
      - account42 2y ago
        Unless you are talking about an alternative number system where 1+1 is not 2 then yes, S(0) is 1.
        empath75 2y ago
        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.
        account42 2y ago
        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.

Aardwolf 2y ago

Finding the shortest expression with 4 2's for a given integer would be a more interesting challenge

0xfffafaCrash 2y ago

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).

venusenvy47 2y ago

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.

tantalor 2y ago
It has a rich history going back to the formalization of arithmetic in the 1800s. https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms It's probably something that only folks who study the foundations of mathematics would know. You may have also heard about it if you learned about Gödel's incompleteness theorems, or read Gödel, Escher, Bach
empath75 2y ago
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.