Why does Python start at index -1 (as opposed to 0) when indexing a list from the end? [duplicate]Understanding slice notationAngular ContentChildren of extended classFinding the index of an item given a list containing it in PythonWhat does the “yield” keyword do?What is the difference between Python's list methods append and extend?Why is it string.join(list) instead of list.join(string)?Accessing the index in 'for' loops?How to remove an element from a list by index?How to make a flat list out of list of listsHow do I get the number of elements in a list?How to clone or copy a list?How do I list all files of a directory?

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Why does Python start at index -1 (as opposed to 0) when indexing a list from the end? [duplicate]


Understanding slice notationAngular ContentChildren of extended classFinding the index of an item given a list containing it in PythonWhat does the “yield” keyword do?What is the difference between Python's list methods append and extend?Why is it string.join(list) instead of list.join(string)?Accessing the index in 'for' loops?How to remove an element from a list by index?How to make a flat list out of list of listsHow do I get the number of elements in a list?How to clone or copy a list?How do I list all files of a directory?






.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty
margin-bottom:0;









78
















This question already has an answer here:



  • Understanding slice notation

    32 answers



list = ["a", "b", "c", "d"]
print(list[3]) # Number 3 is "d"

print(list[-4]) # Number -4 is "a"









share|improve this question
















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  • 64





    Don't use list as a variable name, it's the name of a standard class.

    – Barmar
    Apr 15 at 16:22






  • 10





    It isn't starting at 1, it's starting at -1. ?!?

    – Thomas Weller
    Apr 15 at 20:21







  • 7





    mod arithmetic should really be mentioned on this page somewhere...

    – Nacht
    Apr 16 at 0:14






  • 6





    Should that say as opposed to -0? Since it starts at 0 when indexing from the start, it is trivial that it can't be 0 from the end, so I think -0 is what is meant.

    – Raimund Krämer
    Apr 16 at 9:30






  • 2





    Did you try accessing index 0?

    – jpmc26
    Apr 16 at 14:30

















78
















This question already has an answer here:



  • Understanding slice notation

    32 answers



list = ["a", "b", "c", "d"]
print(list[3]) # Number 3 is "d"

print(list[-4]) # Number -4 is "a"









share|improve this question
















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  • 64





    Don't use list as a variable name, it's the name of a standard class.

    – Barmar
    Apr 15 at 16:22






  • 10





    It isn't starting at 1, it's starting at -1. ?!?

    – Thomas Weller
    Apr 15 at 20:21







  • 7





    mod arithmetic should really be mentioned on this page somewhere...

    – Nacht
    Apr 16 at 0:14






  • 6





    Should that say as opposed to -0? Since it starts at 0 when indexing from the start, it is trivial that it can't be 0 from the end, so I think -0 is what is meant.

    – Raimund Krämer
    Apr 16 at 9:30






  • 2





    Did you try accessing index 0?

    – jpmc26
    Apr 16 at 14:30













78












78








78


20







This question already has an answer here:



  • Understanding slice notation

    32 answers



list = ["a", "b", "c", "d"]
print(list[3]) # Number 3 is "d"

print(list[-4]) # Number -4 is "a"









share|improve this question

















This question already has an answer here:



  • Understanding slice notation

    32 answers



list = ["a", "b", "c", "d"]
print(list[3]) # Number 3 is "d"

print(list[-4]) # Number -4 is "a"




This question already has an answer here:



  • Understanding slice notation

    32 answers







python list






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited Apr 16 at 7:03









glglgl

71.4k8 gold badges106 silver badges174 bronze badges




71.4k8 gold badges106 silver badges174 bronze badges










asked Apr 15 at 8:12









abrahamabraham

4251 gold badge2 silver badges6 bronze badges




4251 gold badge2 silver badges6 bronze badges





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This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









  • 64





    Don't use list as a variable name, it's the name of a standard class.

    – Barmar
    Apr 15 at 16:22






  • 10





    It isn't starting at 1, it's starting at -1. ?!?

    – Thomas Weller
    Apr 15 at 20:21







  • 7





    mod arithmetic should really be mentioned on this page somewhere...

    – Nacht
    Apr 16 at 0:14






  • 6





    Should that say as opposed to -0? Since it starts at 0 when indexing from the start, it is trivial that it can't be 0 from the end, so I think -0 is what is meant.

    – Raimund Krämer
    Apr 16 at 9:30






  • 2





    Did you try accessing index 0?

    – jpmc26
    Apr 16 at 14:30












  • 64





    Don't use list as a variable name, it's the name of a standard class.

    – Barmar
    Apr 15 at 16:22






  • 10





    It isn't starting at 1, it's starting at -1. ?!?

    – Thomas Weller
    Apr 15 at 20:21







  • 7





    mod arithmetic should really be mentioned on this page somewhere...

    – Nacht
    Apr 16 at 0:14






  • 6





    Should that say as opposed to -0? Since it starts at 0 when indexing from the start, it is trivial that it can't be 0 from the end, so I think -0 is what is meant.

    – Raimund Krämer
    Apr 16 at 9:30






  • 2





    Did you try accessing index 0?

    – jpmc26
    Apr 16 at 14:30







64




64





Don't use list as a variable name, it's the name of a standard class.

– Barmar
Apr 15 at 16:22





Don't use list as a variable name, it's the name of a standard class.

– Barmar
Apr 15 at 16:22




10




10





It isn't starting at 1, it's starting at -1. ?!?

– Thomas Weller
Apr 15 at 20:21






It isn't starting at 1, it's starting at -1. ?!?

– Thomas Weller
Apr 15 at 20:21





7




7





mod arithmetic should really be mentioned on this page somewhere...

– Nacht
Apr 16 at 0:14





mod arithmetic should really be mentioned on this page somewhere...

– Nacht
Apr 16 at 0:14




6




6





Should that say as opposed to -0? Since it starts at 0 when indexing from the start, it is trivial that it can't be 0 from the end, so I think -0 is what is meant.

– Raimund Krämer
Apr 16 at 9:30





Should that say as opposed to -0? Since it starts at 0 when indexing from the start, it is trivial that it can't be 0 from the end, so I think -0 is what is meant.

– Raimund Krämer
Apr 16 at 9:30




2




2





Did you try accessing index 0?

– jpmc26
Apr 16 at 14:30





Did you try accessing index 0?

– jpmc26
Apr 16 at 14:30












7 Answers
7






active

oldest

votes


















168
















To explain it in another way, because -0 is equal to 0, if backward starts from 0, it is ambiguous to the interpreter.




If you are confused about -, and looking for another way to index backwards more understandably, you can try ~, it is a mirror of forward:



arr = ["a", "b", "c", "d"]
print(arr[~0]) # d
print(arr[~1]) # c


The typical usages for ~ are like "swap mirror node" or "find median in a sort list":



"""swap mirror node"""
def reverse(arr: List[int]) -> None:
for i in range(len(arr) // 2):
arr[i], arr[~i] = arr[~i], arr[i]

"""find median in a sort list"""
def median(arr: List[float]) -> float:
mid = len(arr) // 2
return (arr[mid] + arr[~mid]) / 2

"""deal with mirror pairs"""
# verify the number is strobogrammatic, strobogrammatic number looks the same when rotated 180 degrees
def is_strobogrammatic(num: str) -> bool:
return all(num[i] + num[~i] in '696 00 11 88' for i in range(len(num) // 2 + 1))


~ actually is a math trick of inverse code and complement code, and it is more easy to understand in some situations.




Discussion about whether should use python tricks like ~:



In my opinion, if it is a code maintained by yourself, you can use any trick to avoid potential bug or achieve goal easier, because of maybe a high readability and usability. But in team work, avoid using 'too clever' code, may bring troubles to your co-workers.



For example, here is one concise code from Stefan Pochmann to solve this problem. I learned a lot from his code. But some are just for fun, too hackish to use.



# a strobogrammatic number is a number that looks the same when rotated 180 degrees (looked at upside down)
# find all strobogrammatic numbers that are of length = n
def findStrobogrammatic(self, n):
nums = n % 2 * list('018') or ['']
while n > 1:
n -= 2
# n < 2 is so genius here
nums = [a + num + b for a, b in '00 11 88 69 96'.split()[n < 2:] for num in nums]
return nums


I have summarized python tricks like this, in case you are interested.






share|improve this answer



























  • Comments are not for extended discussion; this conversation has been moved to chat.

    – Samuel Liew
    Apr 16 at 0:27






  • 1





    Could you provide description of the problem that Stefan's code solves? I don't want to register at leetcode only to access that information.It's also good when the answers contain all relevant details.

    – Konrad
    Apr 16 at 12:01






  • 2





    Problem Description: A strobogrammatic number is a number that looks the same when rotated 180 degrees (looked at upside down). Find all strobogrammatic numbers that are of length = n. And the key points of Stefan's code: First line, if nums is odd, the middle will in '018', we will add pair to leftmost and rightmost in while loop, but we should consider number can not start with '0', so n<2 is used here. just 5 line solve a complex problem. @Konrad

    – recnac
    Apr 16 at 12:38







  • 1





    Donor use this method. You still need to understand the underlying use of negative indexing for indexing from the end. This is just further obfuscation.

    – Paddy3118
    Apr 16 at 16:22


















165
















list[-1]


Is short hand for:



list[len(list)-1]


The len(list) part is implicit. That's why the -1 is the last element. That goes for any negative index - the subtraction from len(list) is always implicit






share|improve this answer




















  • 11





    In my opinion this answer is better than the accepted one.

    – NicolasB
    Apr 15 at 12:17






  • 9





    Be aware that list[-n] and list[len(list)-n] are only equivilent for values of n between 1 and len(list). This becomes especially important when slicing rather than indexing.

    – plugwash
    Apr 15 at 16:56


















24
















This is the mnemonic method I use. It is just an approach of what is happening, but it works.




Don't think of those as indexes. Think of them as offsets on a circular list.



Let's use the list x = [a,b,c,d,e,f,g,h] as an example. Think about x[2] and x[-2]:



enter image description here



You start at offset zero. If you move two steps forward, you're going from a to b (0 to 1), and them from b to c (1 to 2).



If you move two steps backward, you're going from a to h (0 to -1), and then from h to g (-1 to -2)






share|improve this answer



























  • "Think about a[2] and a[-2]" is a the list, or an element of the list?

    – detly
    Apr 15 at 22:54






  • 1





    "a" is an hypothetical list which contains the values a-h! I'll clarify!

    – T. Sar
    Apr 16 at 2:27












  • Note the similarity with modular arithmetic.

    – Code-Apprentice
    Apr 16 at 17:54


















22
















Because -0 in Python is 0.

With 0 you get first element of list and
with -1 you get the last element of the list



list = ["a", "b", "c", "d"]
print(list[0]) # "a"
print(list[-1]) # d



You can also think it as shorthand for list[len(list) - x] where x is the element position from the back.
This is valid only if 0 < -(-x) < len(list)




print(list[-1]) # d
print(list[len(list) - 1]) # d
print(list[-5]) # list index out of range
print(list[len(list) - 5]) # a





share|improve this answer






















  • 10





    I think -0 is 0 pretty much everywhere.

    – Koray Tugay
    Apr 15 at 15:43






  • 9





    @KorayTugay Except in floating point.

    – Barmar
    Apr 15 at 16:24











  • I guess that is not 0 anymore. @Barmar

    – Koray Tugay
    Apr 15 at 16:37






  • 7





    @KorayTugay It is. The binary representation is even all 0s. Floating point also just has another 0, where the sign bit is 1.

    – curiousdannii
    Apr 15 at 23:41






  • 3





    @Barmar There were architectures where 0 and -0 were distinct values even for integral values (sign and magnitude). I don't believe any processors now on the market use this representation though.

    – Martin Bonner
    Apr 16 at 8:07


















13
















This idiom can be justified using modular arithmetic. We can think of indices as referring to a cell in a list obtained by walking forward i elements. -1 referring to the last element of the list is a natural generalization of this, since we arrive at the last element in the list if we walk backwards one step from the start of the list.



For any list xs and index i positive or negative, the expression



xs[i]


will either have the same value as the expression below or produce an IndexError:



xs[i % len(xs)]


The index of the last element is -1 + len(xs) which is congruent to -1 mod len(xs). For example, in an array of length 12, the canonical index of the last element is 11. 11 is congruent to -1 mod 12.



In Python, though, arrays are more often used as linear data structures than circular ones, so indices larger than -1 + len(xs) or smaller than -len(xs) are out of bounds since there's seldom a need for them and the effects would be really counterintuitive if the size of the array ever changed.






share|improve this answer


































    8
















    Another explanation:



    Your finger points to the first element. The index decides how many places you shift your finger to the right. If the number is negative, you shift your finger to the left.



    Of course, you can't step to the left from the first element, so the first step to the left wraps around to the last element.






    share|improve this answer


































      6
















      You could intuitively understand it this way



      steps= ["a", "b", "c", "d"]


      Suppose you start from a to d, a is your staring point where you stand (or your home), so mark it as 0(because you did not move yet),



      Move one step to b, second step to c and arrive at third d.



      Then how about you return from d to a (or return from your office to your home). Your home is 0 because your family live there, so your office cannot be a 0.It's your last stop.



      So when you return back to home. d is the last first stop to where you start off for home, c is the last second ....






      share|improve this answer



























      • d is the last first stop to start, c is the last second This is kind of difficult to understand, could you dissect that sentence a bit?

        – Raimund Krämer
        Apr 16 at 9:35











      • last here is ending or final, not recent @RaimundKrämer

        – Algebra
        Apr 16 at 10:02


















      7 Answers
      7






      active

      oldest

      votes








      7 Answers
      7






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      168
















      To explain it in another way, because -0 is equal to 0, if backward starts from 0, it is ambiguous to the interpreter.




      If you are confused about -, and looking for another way to index backwards more understandably, you can try ~, it is a mirror of forward:



      arr = ["a", "b", "c", "d"]
      print(arr[~0]) # d
      print(arr[~1]) # c


      The typical usages for ~ are like "swap mirror node" or "find median in a sort list":



      """swap mirror node"""
      def reverse(arr: List[int]) -> None:
      for i in range(len(arr) // 2):
      arr[i], arr[~i] = arr[~i], arr[i]

      """find median in a sort list"""
      def median(arr: List[float]) -> float:
      mid = len(arr) // 2
      return (arr[mid] + arr[~mid]) / 2

      """deal with mirror pairs"""
      # verify the number is strobogrammatic, strobogrammatic number looks the same when rotated 180 degrees
      def is_strobogrammatic(num: str) -> bool:
      return all(num[i] + num[~i] in '696 00 11 88' for i in range(len(num) // 2 + 1))


      ~ actually is a math trick of inverse code and complement code, and it is more easy to understand in some situations.




      Discussion about whether should use python tricks like ~:



      In my opinion, if it is a code maintained by yourself, you can use any trick to avoid potential bug or achieve goal easier, because of maybe a high readability and usability. But in team work, avoid using 'too clever' code, may bring troubles to your co-workers.



      For example, here is one concise code from Stefan Pochmann to solve this problem. I learned a lot from his code. But some are just for fun, too hackish to use.



      # a strobogrammatic number is a number that looks the same when rotated 180 degrees (looked at upside down)
      # find all strobogrammatic numbers that are of length = n
      def findStrobogrammatic(self, n):
      nums = n % 2 * list('018') or ['']
      while n > 1:
      n -= 2
      # n < 2 is so genius here
      nums = [a + num + b for a, b in '00 11 88 69 96'.split()[n < 2:] for num in nums]
      return nums


      I have summarized python tricks like this, in case you are interested.






      share|improve this answer



























      • Comments are not for extended discussion; this conversation has been moved to chat.

        – Samuel Liew
        Apr 16 at 0:27






      • 1





        Could you provide description of the problem that Stefan's code solves? I don't want to register at leetcode only to access that information.It's also good when the answers contain all relevant details.

        – Konrad
        Apr 16 at 12:01






      • 2





        Problem Description: A strobogrammatic number is a number that looks the same when rotated 180 degrees (looked at upside down). Find all strobogrammatic numbers that are of length = n. And the key points of Stefan's code: First line, if nums is odd, the middle will in '018', we will add pair to leftmost and rightmost in while loop, but we should consider number can not start with '0', so n<2 is used here. just 5 line solve a complex problem. @Konrad

        – recnac
        Apr 16 at 12:38







      • 1





        Donor use this method. You still need to understand the underlying use of negative indexing for indexing from the end. This is just further obfuscation.

        – Paddy3118
        Apr 16 at 16:22















      168
















      To explain it in another way, because -0 is equal to 0, if backward starts from 0, it is ambiguous to the interpreter.




      If you are confused about -, and looking for another way to index backwards more understandably, you can try ~, it is a mirror of forward:



      arr = ["a", "b", "c", "d"]
      print(arr[~0]) # d
      print(arr[~1]) # c


      The typical usages for ~ are like "swap mirror node" or "find median in a sort list":



      """swap mirror node"""
      def reverse(arr: List[int]) -> None:
      for i in range(len(arr) // 2):
      arr[i], arr[~i] = arr[~i], arr[i]

      """find median in a sort list"""
      def median(arr: List[float]) -> float:
      mid = len(arr) // 2
      return (arr[mid] + arr[~mid]) / 2

      """deal with mirror pairs"""
      # verify the number is strobogrammatic, strobogrammatic number looks the same when rotated 180 degrees
      def is_strobogrammatic(num: str) -> bool:
      return all(num[i] + num[~i] in '696 00 11 88' for i in range(len(num) // 2 + 1))


      ~ actually is a math trick of inverse code and complement code, and it is more easy to understand in some situations.




      Discussion about whether should use python tricks like ~:



      In my opinion, if it is a code maintained by yourself, you can use any trick to avoid potential bug or achieve goal easier, because of maybe a high readability and usability. But in team work, avoid using 'too clever' code, may bring troubles to your co-workers.



      For example, here is one concise code from Stefan Pochmann to solve this problem. I learned a lot from his code. But some are just for fun, too hackish to use.



      # a strobogrammatic number is a number that looks the same when rotated 180 degrees (looked at upside down)
      # find all strobogrammatic numbers that are of length = n
      def findStrobogrammatic(self, n):
      nums = n % 2 * list('018') or ['']
      while n > 1:
      n -= 2
      # n < 2 is so genius here
      nums = [a + num + b for a, b in '00 11 88 69 96'.split()[n < 2:] for num in nums]
      return nums


      I have summarized python tricks like this, in case you are interested.






      share|improve this answer



























      • Comments are not for extended discussion; this conversation has been moved to chat.

        – Samuel Liew
        Apr 16 at 0:27






      • 1





        Could you provide description of the problem that Stefan's code solves? I don't want to register at leetcode only to access that information.It's also good when the answers contain all relevant details.

        – Konrad
        Apr 16 at 12:01






      • 2





        Problem Description: A strobogrammatic number is a number that looks the same when rotated 180 degrees (looked at upside down). Find all strobogrammatic numbers that are of length = n. And the key points of Stefan's code: First line, if nums is odd, the middle will in '018', we will add pair to leftmost and rightmost in while loop, but we should consider number can not start with '0', so n<2 is used here. just 5 line solve a complex problem. @Konrad

        – recnac
        Apr 16 at 12:38







      • 1





        Donor use this method. You still need to understand the underlying use of negative indexing for indexing from the end. This is just further obfuscation.

        – Paddy3118
        Apr 16 at 16:22













      168














      168










      168









      To explain it in another way, because -0 is equal to 0, if backward starts from 0, it is ambiguous to the interpreter.




      If you are confused about -, and looking for another way to index backwards more understandably, you can try ~, it is a mirror of forward:



      arr = ["a", "b", "c", "d"]
      print(arr[~0]) # d
      print(arr[~1]) # c


      The typical usages for ~ are like "swap mirror node" or "find median in a sort list":



      """swap mirror node"""
      def reverse(arr: List[int]) -> None:
      for i in range(len(arr) // 2):
      arr[i], arr[~i] = arr[~i], arr[i]

      """find median in a sort list"""
      def median(arr: List[float]) -> float:
      mid = len(arr) // 2
      return (arr[mid] + arr[~mid]) / 2

      """deal with mirror pairs"""
      # verify the number is strobogrammatic, strobogrammatic number looks the same when rotated 180 degrees
      def is_strobogrammatic(num: str) -> bool:
      return all(num[i] + num[~i] in '696 00 11 88' for i in range(len(num) // 2 + 1))


      ~ actually is a math trick of inverse code and complement code, and it is more easy to understand in some situations.




      Discussion about whether should use python tricks like ~:



      In my opinion, if it is a code maintained by yourself, you can use any trick to avoid potential bug or achieve goal easier, because of maybe a high readability and usability. But in team work, avoid using 'too clever' code, may bring troubles to your co-workers.



      For example, here is one concise code from Stefan Pochmann to solve this problem. I learned a lot from his code. But some are just for fun, too hackish to use.



      # a strobogrammatic number is a number that looks the same when rotated 180 degrees (looked at upside down)
      # find all strobogrammatic numbers that are of length = n
      def findStrobogrammatic(self, n):
      nums = n % 2 * list('018') or ['']
      while n > 1:
      n -= 2
      # n < 2 is so genius here
      nums = [a + num + b for a, b in '00 11 88 69 96'.split()[n < 2:] for num in nums]
      return nums


      I have summarized python tricks like this, in case you are interested.






      share|improve this answer















      To explain it in another way, because -0 is equal to 0, if backward starts from 0, it is ambiguous to the interpreter.




      If you are confused about -, and looking for another way to index backwards more understandably, you can try ~, it is a mirror of forward:



      arr = ["a", "b", "c", "d"]
      print(arr[~0]) # d
      print(arr[~1]) # c


      The typical usages for ~ are like "swap mirror node" or "find median in a sort list":



      """swap mirror node"""
      def reverse(arr: List[int]) -> None:
      for i in range(len(arr) // 2):
      arr[i], arr[~i] = arr[~i], arr[i]

      """find median in a sort list"""
      def median(arr: List[float]) -> float:
      mid = len(arr) // 2
      return (arr[mid] + arr[~mid]) / 2

      """deal with mirror pairs"""
      # verify the number is strobogrammatic, strobogrammatic number looks the same when rotated 180 degrees
      def is_strobogrammatic(num: str) -> bool:
      return all(num[i] + num[~i] in '696 00 11 88' for i in range(len(num) // 2 + 1))


      ~ actually is a math trick of inverse code and complement code, and it is more easy to understand in some situations.




      Discussion about whether should use python tricks like ~:



      In my opinion, if it is a code maintained by yourself, you can use any trick to avoid potential bug or achieve goal easier, because of maybe a high readability and usability. But in team work, avoid using 'too clever' code, may bring troubles to your co-workers.



      For example, here is one concise code from Stefan Pochmann to solve this problem. I learned a lot from his code. But some are just for fun, too hackish to use.



      # a strobogrammatic number is a number that looks the same when rotated 180 degrees (looked at upside down)
      # find all strobogrammatic numbers that are of length = n
      def findStrobogrammatic(self, n):
      nums = n % 2 * list('018') or ['']
      while n > 1:
      n -= 2
      # n < 2 is so genius here
      nums = [a + num + b for a, b in '00 11 88 69 96'.split()[n < 2:] for num in nums]
      return nums


      I have summarized python tricks like this, in case you are interested.







      share|improve this answer














      share|improve this answer



      share|improve this answer








      edited Apr 28 at 2:32

























      answered Apr 15 at 8:16









      recnacrecnac

      3,1976 gold badges9 silver badges38 bronze badges




      3,1976 gold badges9 silver badges38 bronze badges















      • Comments are not for extended discussion; this conversation has been moved to chat.

        – Samuel Liew
        Apr 16 at 0:27






      • 1





        Could you provide description of the problem that Stefan's code solves? I don't want to register at leetcode only to access that information.It's also good when the answers contain all relevant details.

        – Konrad
        Apr 16 at 12:01






      • 2





        Problem Description: A strobogrammatic number is a number that looks the same when rotated 180 degrees (looked at upside down). Find all strobogrammatic numbers that are of length = n. And the key points of Stefan's code: First line, if nums is odd, the middle will in '018', we will add pair to leftmost and rightmost in while loop, but we should consider number can not start with '0', so n<2 is used here. just 5 line solve a complex problem. @Konrad

        – recnac
        Apr 16 at 12:38







      • 1





        Donor use this method. You still need to understand the underlying use of negative indexing for indexing from the end. This is just further obfuscation.

        – Paddy3118
        Apr 16 at 16:22

















      • Comments are not for extended discussion; this conversation has been moved to chat.

        – Samuel Liew
        Apr 16 at 0:27






      • 1





        Could you provide description of the problem that Stefan's code solves? I don't want to register at leetcode only to access that information.It's also good when the answers contain all relevant details.

        – Konrad
        Apr 16 at 12:01






      • 2





        Problem Description: A strobogrammatic number is a number that looks the same when rotated 180 degrees (looked at upside down). Find all strobogrammatic numbers that are of length = n. And the key points of Stefan's code: First line, if nums is odd, the middle will in '018', we will add pair to leftmost and rightmost in while loop, but we should consider number can not start with '0', so n<2 is used here. just 5 line solve a complex problem. @Konrad

        – recnac
        Apr 16 at 12:38







      • 1





        Donor use this method. You still need to understand the underlying use of negative indexing for indexing from the end. This is just further obfuscation.

        – Paddy3118
        Apr 16 at 16:22
















      Comments are not for extended discussion; this conversation has been moved to chat.

      – Samuel Liew
      Apr 16 at 0:27





      Comments are not for extended discussion; this conversation has been moved to chat.

      – Samuel Liew
      Apr 16 at 0:27




      1




      1





      Could you provide description of the problem that Stefan's code solves? I don't want to register at leetcode only to access that information.It's also good when the answers contain all relevant details.

      – Konrad
      Apr 16 at 12:01





      Could you provide description of the problem that Stefan's code solves? I don't want to register at leetcode only to access that information.It's also good when the answers contain all relevant details.

      – Konrad
      Apr 16 at 12:01




      2




      2





      Problem Description: A strobogrammatic number is a number that looks the same when rotated 180 degrees (looked at upside down). Find all strobogrammatic numbers that are of length = n. And the key points of Stefan's code: First line, if nums is odd, the middle will in '018', we will add pair to leftmost and rightmost in while loop, but we should consider number can not start with '0', so n<2 is used here. just 5 line solve a complex problem. @Konrad

      – recnac
      Apr 16 at 12:38






      Problem Description: A strobogrammatic number is a number that looks the same when rotated 180 degrees (looked at upside down). Find all strobogrammatic numbers that are of length = n. And the key points of Stefan's code: First line, if nums is odd, the middle will in '018', we will add pair to leftmost and rightmost in while loop, but we should consider number can not start with '0', so n<2 is used here. just 5 line solve a complex problem. @Konrad

      – recnac
      Apr 16 at 12:38





      1




      1





      Donor use this method. You still need to understand the underlying use of negative indexing for indexing from the end. This is just further obfuscation.

      – Paddy3118
      Apr 16 at 16:22





      Donor use this method. You still need to understand the underlying use of negative indexing for indexing from the end. This is just further obfuscation.

      – Paddy3118
      Apr 16 at 16:22













      165
















      list[-1]


      Is short hand for:



      list[len(list)-1]


      The len(list) part is implicit. That's why the -1 is the last element. That goes for any negative index - the subtraction from len(list) is always implicit






      share|improve this answer




















      • 11





        In my opinion this answer is better than the accepted one.

        – NicolasB
        Apr 15 at 12:17






      • 9





        Be aware that list[-n] and list[len(list)-n] are only equivilent for values of n between 1 and len(list). This becomes especially important when slicing rather than indexing.

        – plugwash
        Apr 15 at 16:56















      165
















      list[-1]


      Is short hand for:



      list[len(list)-1]


      The len(list) part is implicit. That's why the -1 is the last element. That goes for any negative index - the subtraction from len(list) is always implicit






      share|improve this answer




















      • 11





        In my opinion this answer is better than the accepted one.

        – NicolasB
        Apr 15 at 12:17






      • 9





        Be aware that list[-n] and list[len(list)-n] are only equivilent for values of n between 1 and len(list). This becomes especially important when slicing rather than indexing.

        – plugwash
        Apr 15 at 16:56













      165














      165










      165









      list[-1]


      Is short hand for:



      list[len(list)-1]


      The len(list) part is implicit. That's why the -1 is the last element. That goes for any negative index - the subtraction from len(list) is always implicit






      share|improve this answer













      list[-1]


      Is short hand for:



      list[len(list)-1]


      The len(list) part is implicit. That's why the -1 is the last element. That goes for any negative index - the subtraction from len(list) is always implicit







      share|improve this answer












      share|improve this answer



      share|improve this answer










      answered Apr 15 at 8:14









      rdasrdas

      7,6223 gold badges14 silver badges28 bronze badges




      7,6223 gold badges14 silver badges28 bronze badges










      • 11





        In my opinion this answer is better than the accepted one.

        – NicolasB
        Apr 15 at 12:17






      • 9





        Be aware that list[-n] and list[len(list)-n] are only equivilent for values of n between 1 and len(list). This becomes especially important when slicing rather than indexing.

        – plugwash
        Apr 15 at 16:56












      • 11





        In my opinion this answer is better than the accepted one.

        – NicolasB
        Apr 15 at 12:17






      • 9





        Be aware that list[-n] and list[len(list)-n] are only equivilent for values of n between 1 and len(list). This becomes especially important when slicing rather than indexing.

        – plugwash
        Apr 15 at 16:56







      11




      11





      In my opinion this answer is better than the accepted one.

      – NicolasB
      Apr 15 at 12:17





      In my opinion this answer is better than the accepted one.

      – NicolasB
      Apr 15 at 12:17




      9




      9





      Be aware that list[-n] and list[len(list)-n] are only equivilent for values of n between 1 and len(list). This becomes especially important when slicing rather than indexing.

      – plugwash
      Apr 15 at 16:56





      Be aware that list[-n] and list[len(list)-n] are only equivilent for values of n between 1 and len(list). This becomes especially important when slicing rather than indexing.

      – plugwash
      Apr 15 at 16:56











      24
















      This is the mnemonic method I use. It is just an approach of what is happening, but it works.




      Don't think of those as indexes. Think of them as offsets on a circular list.



      Let's use the list x = [a,b,c,d,e,f,g,h] as an example. Think about x[2] and x[-2]:



      enter image description here



      You start at offset zero. If you move two steps forward, you're going from a to b (0 to 1), and them from b to c (1 to 2).



      If you move two steps backward, you're going from a to h (0 to -1), and then from h to g (-1 to -2)






      share|improve this answer



























      • "Think about a[2] and a[-2]" is a the list, or an element of the list?

        – detly
        Apr 15 at 22:54






      • 1





        "a" is an hypothetical list which contains the values a-h! I'll clarify!

        – T. Sar
        Apr 16 at 2:27












      • Note the similarity with modular arithmetic.

        – Code-Apprentice
        Apr 16 at 17:54















      24
















      This is the mnemonic method I use. It is just an approach of what is happening, but it works.




      Don't think of those as indexes. Think of them as offsets on a circular list.



      Let's use the list x = [a,b,c,d,e,f,g,h] as an example. Think about x[2] and x[-2]:



      enter image description here



      You start at offset zero. If you move two steps forward, you're going from a to b (0 to 1), and them from b to c (1 to 2).



      If you move two steps backward, you're going from a to h (0 to -1), and then from h to g (-1 to -2)






      share|improve this answer



























      • "Think about a[2] and a[-2]" is a the list, or an element of the list?

        – detly
        Apr 15 at 22:54






      • 1





        "a" is an hypothetical list which contains the values a-h! I'll clarify!

        – T. Sar
        Apr 16 at 2:27












      • Note the similarity with modular arithmetic.

        – Code-Apprentice
        Apr 16 at 17:54













      24














      24










      24









      This is the mnemonic method I use. It is just an approach of what is happening, but it works.




      Don't think of those as indexes. Think of them as offsets on a circular list.



      Let's use the list x = [a,b,c,d,e,f,g,h] as an example. Think about x[2] and x[-2]:



      enter image description here



      You start at offset zero. If you move two steps forward, you're going from a to b (0 to 1), and them from b to c (1 to 2).



      If you move two steps backward, you're going from a to h (0 to -1), and then from h to g (-1 to -2)






      share|improve this answer















      This is the mnemonic method I use. It is just an approach of what is happening, but it works.




      Don't think of those as indexes. Think of them as offsets on a circular list.



      Let's use the list x = [a,b,c,d,e,f,g,h] as an example. Think about x[2] and x[-2]:



      enter image description here



      You start at offset zero. If you move two steps forward, you're going from a to b (0 to 1), and them from b to c (1 to 2).



      If you move two steps backward, you're going from a to h (0 to -1), and then from h to g (-1 to -2)







      share|improve this answer














      share|improve this answer



      share|improve this answer








      edited Apr 16 at 2:27

























      answered Apr 15 at 18:15









      T. SarT. Sar

      3712 silver badges11 bronze badges




      3712 silver badges11 bronze badges















      • "Think about a[2] and a[-2]" is a the list, or an element of the list?

        – detly
        Apr 15 at 22:54






      • 1





        "a" is an hypothetical list which contains the values a-h! I'll clarify!

        – T. Sar
        Apr 16 at 2:27












      • Note the similarity with modular arithmetic.

        – Code-Apprentice
        Apr 16 at 17:54

















      • "Think about a[2] and a[-2]" is a the list, or an element of the list?

        – detly
        Apr 15 at 22:54






      • 1





        "a" is an hypothetical list which contains the values a-h! I'll clarify!

        – T. Sar
        Apr 16 at 2:27












      • Note the similarity with modular arithmetic.

        – Code-Apprentice
        Apr 16 at 17:54
















      "Think about a[2] and a[-2]" is a the list, or an element of the list?

      – detly
      Apr 15 at 22:54





      "Think about a[2] and a[-2]" is a the list, or an element of the list?

      – detly
      Apr 15 at 22:54




      1




      1





      "a" is an hypothetical list which contains the values a-h! I'll clarify!

      – T. Sar
      Apr 16 at 2:27






      "a" is an hypothetical list which contains the values a-h! I'll clarify!

      – T. Sar
      Apr 16 at 2:27














      Note the similarity with modular arithmetic.

      – Code-Apprentice
      Apr 16 at 17:54





      Note the similarity with modular arithmetic.

      – Code-Apprentice
      Apr 16 at 17:54











      22
















      Because -0 in Python is 0.

      With 0 you get first element of list and
      with -1 you get the last element of the list



      list = ["a", "b", "c", "d"]
      print(list[0]) # "a"
      print(list[-1]) # d



      You can also think it as shorthand for list[len(list) - x] where x is the element position from the back.
      This is valid only if 0 < -(-x) < len(list)




      print(list[-1]) # d
      print(list[len(list) - 1]) # d
      print(list[-5]) # list index out of range
      print(list[len(list) - 5]) # a





      share|improve this answer






















      • 10





        I think -0 is 0 pretty much everywhere.

        – Koray Tugay
        Apr 15 at 15:43






      • 9





        @KorayTugay Except in floating point.

        – Barmar
        Apr 15 at 16:24











      • I guess that is not 0 anymore. @Barmar

        – Koray Tugay
        Apr 15 at 16:37






      • 7





        @KorayTugay It is. The binary representation is even all 0s. Floating point also just has another 0, where the sign bit is 1.

        – curiousdannii
        Apr 15 at 23:41






      • 3





        @Barmar There were architectures where 0 and -0 were distinct values even for integral values (sign and magnitude). I don't believe any processors now on the market use this representation though.

        – Martin Bonner
        Apr 16 at 8:07















      22
















      Because -0 in Python is 0.

      With 0 you get first element of list and
      with -1 you get the last element of the list



      list = ["a", "b", "c", "d"]
      print(list[0]) # "a"
      print(list[-1]) # d



      You can also think it as shorthand for list[len(list) - x] where x is the element position from the back.
      This is valid only if 0 < -(-x) < len(list)




      print(list[-1]) # d
      print(list[len(list) - 1]) # d
      print(list[-5]) # list index out of range
      print(list[len(list) - 5]) # a





      share|improve this answer






















      • 10





        I think -0 is 0 pretty much everywhere.

        – Koray Tugay
        Apr 15 at 15:43






      • 9





        @KorayTugay Except in floating point.

        – Barmar
        Apr 15 at 16:24











      • I guess that is not 0 anymore. @Barmar

        – Koray Tugay
        Apr 15 at 16:37






      • 7





        @KorayTugay It is. The binary representation is even all 0s. Floating point also just has another 0, where the sign bit is 1.

        – curiousdannii
        Apr 15 at 23:41






      • 3





        @Barmar There were architectures where 0 and -0 were distinct values even for integral values (sign and magnitude). I don't believe any processors now on the market use this representation though.

        – Martin Bonner
        Apr 16 at 8:07













      22














      22










      22









      Because -0 in Python is 0.

      With 0 you get first element of list and
      with -1 you get the last element of the list



      list = ["a", "b", "c", "d"]
      print(list[0]) # "a"
      print(list[-1]) # d



      You can also think it as shorthand for list[len(list) - x] where x is the element position from the back.
      This is valid only if 0 < -(-x) < len(list)




      print(list[-1]) # d
      print(list[len(list) - 1]) # d
      print(list[-5]) # list index out of range
      print(list[len(list) - 5]) # a





      share|improve this answer















      Because -0 in Python is 0.

      With 0 you get first element of list and
      with -1 you get the last element of the list



      list = ["a", "b", "c", "d"]
      print(list[0]) # "a"
      print(list[-1]) # d



      You can also think it as shorthand for list[len(list) - x] where x is the element position from the back.
      This is valid only if 0 < -(-x) < len(list)




      print(list[-1]) # d
      print(list[len(list) - 1]) # d
      print(list[-5]) # list index out of range
      print(list[len(list) - 5]) # a






      share|improve this answer














      share|improve this answer



      share|improve this answer








      edited Apr 18 at 11:31

























      answered Apr 15 at 8:15









      AshishAshish

      2,0886 silver badges30 bronze badges




      2,0886 silver badges30 bronze badges










      • 10





        I think -0 is 0 pretty much everywhere.

        – Koray Tugay
        Apr 15 at 15:43






      • 9





        @KorayTugay Except in floating point.

        – Barmar
        Apr 15 at 16:24











      • I guess that is not 0 anymore. @Barmar

        – Koray Tugay
        Apr 15 at 16:37






      • 7





        @KorayTugay It is. The binary representation is even all 0s. Floating point also just has another 0, where the sign bit is 1.

        – curiousdannii
        Apr 15 at 23:41






      • 3





        @Barmar There were architectures where 0 and -0 were distinct values even for integral values (sign and magnitude). I don't believe any processors now on the market use this representation though.

        – Martin Bonner
        Apr 16 at 8:07












      • 10





        I think -0 is 0 pretty much everywhere.

        – Koray Tugay
        Apr 15 at 15:43






      • 9





        @KorayTugay Except in floating point.

        – Barmar
        Apr 15 at 16:24











      • I guess that is not 0 anymore. @Barmar

        – Koray Tugay
        Apr 15 at 16:37






      • 7





        @KorayTugay It is. The binary representation is even all 0s. Floating point also just has another 0, where the sign bit is 1.

        – curiousdannii
        Apr 15 at 23:41






      • 3





        @Barmar There were architectures where 0 and -0 were distinct values even for integral values (sign and magnitude). I don't believe any processors now on the market use this representation though.

        – Martin Bonner
        Apr 16 at 8:07







      10




      10





      I think -0 is 0 pretty much everywhere.

      – Koray Tugay
      Apr 15 at 15:43





      I think -0 is 0 pretty much everywhere.

      – Koray Tugay
      Apr 15 at 15:43




      9




      9





      @KorayTugay Except in floating point.

      – Barmar
      Apr 15 at 16:24





      @KorayTugay Except in floating point.

      – Barmar
      Apr 15 at 16:24













      I guess that is not 0 anymore. @Barmar

      – Koray Tugay
      Apr 15 at 16:37





      I guess that is not 0 anymore. @Barmar

      – Koray Tugay
      Apr 15 at 16:37




      7




      7





      @KorayTugay It is. The binary representation is even all 0s. Floating point also just has another 0, where the sign bit is 1.

      – curiousdannii
      Apr 15 at 23:41





      @KorayTugay It is. The binary representation is even all 0s. Floating point also just has another 0, where the sign bit is 1.

      – curiousdannii
      Apr 15 at 23:41




      3




      3





      @Barmar There were architectures where 0 and -0 were distinct values even for integral values (sign and magnitude). I don't believe any processors now on the market use this representation though.

      – Martin Bonner
      Apr 16 at 8:07





      @Barmar There were architectures where 0 and -0 were distinct values even for integral values (sign and magnitude). I don't believe any processors now on the market use this representation though.

      – Martin Bonner
      Apr 16 at 8:07











      13
















      This idiom can be justified using modular arithmetic. We can think of indices as referring to a cell in a list obtained by walking forward i elements. -1 referring to the last element of the list is a natural generalization of this, since we arrive at the last element in the list if we walk backwards one step from the start of the list.



      For any list xs and index i positive or negative, the expression



      xs[i]


      will either have the same value as the expression below or produce an IndexError:



      xs[i % len(xs)]


      The index of the last element is -1 + len(xs) which is congruent to -1 mod len(xs). For example, in an array of length 12, the canonical index of the last element is 11. 11 is congruent to -1 mod 12.



      In Python, though, arrays are more often used as linear data structures than circular ones, so indices larger than -1 + len(xs) or smaller than -len(xs) are out of bounds since there's seldom a need for them and the effects would be really counterintuitive if the size of the array ever changed.






      share|improve this answer































        13
















        This idiom can be justified using modular arithmetic. We can think of indices as referring to a cell in a list obtained by walking forward i elements. -1 referring to the last element of the list is a natural generalization of this, since we arrive at the last element in the list if we walk backwards one step from the start of the list.



        For any list xs and index i positive or negative, the expression



        xs[i]


        will either have the same value as the expression below or produce an IndexError:



        xs[i % len(xs)]


        The index of the last element is -1 + len(xs) which is congruent to -1 mod len(xs). For example, in an array of length 12, the canonical index of the last element is 11. 11 is congruent to -1 mod 12.



        In Python, though, arrays are more often used as linear data structures than circular ones, so indices larger than -1 + len(xs) or smaller than -len(xs) are out of bounds since there's seldom a need for them and the effects would be really counterintuitive if the size of the array ever changed.






        share|improve this answer





























          13














          13










          13









          This idiom can be justified using modular arithmetic. We can think of indices as referring to a cell in a list obtained by walking forward i elements. -1 referring to the last element of the list is a natural generalization of this, since we arrive at the last element in the list if we walk backwards one step from the start of the list.



          For any list xs and index i positive or negative, the expression



          xs[i]


          will either have the same value as the expression below or produce an IndexError:



          xs[i % len(xs)]


          The index of the last element is -1 + len(xs) which is congruent to -1 mod len(xs). For example, in an array of length 12, the canonical index of the last element is 11. 11 is congruent to -1 mod 12.



          In Python, though, arrays are more often used as linear data structures than circular ones, so indices larger than -1 + len(xs) or smaller than -len(xs) are out of bounds since there's seldom a need for them and the effects would be really counterintuitive if the size of the array ever changed.






          share|improve this answer















          This idiom can be justified using modular arithmetic. We can think of indices as referring to a cell in a list obtained by walking forward i elements. -1 referring to the last element of the list is a natural generalization of this, since we arrive at the last element in the list if we walk backwards one step from the start of the list.



          For any list xs and index i positive or negative, the expression



          xs[i]


          will either have the same value as the expression below or produce an IndexError:



          xs[i % len(xs)]


          The index of the last element is -1 + len(xs) which is congruent to -1 mod len(xs). For example, in an array of length 12, the canonical index of the last element is 11. 11 is congruent to -1 mod 12.



          In Python, though, arrays are more often used as linear data structures than circular ones, so indices larger than -1 + len(xs) or smaller than -len(xs) are out of bounds since there's seldom a need for them and the effects would be really counterintuitive if the size of the array ever changed.







          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited Apr 23 at 21:13

























          answered Apr 16 at 0:32









          Gregory NisbetGregory Nisbet

          3,4622 gold badges15 silver badges36 bronze badges




          3,4622 gold badges15 silver badges36 bronze badges
























              8
















              Another explanation:



              Your finger points to the first element. The index decides how many places you shift your finger to the right. If the number is negative, you shift your finger to the left.



              Of course, you can't step to the left from the first element, so the first step to the left wraps around to the last element.






              share|improve this answer































                8
















                Another explanation:



                Your finger points to the first element. The index decides how many places you shift your finger to the right. If the number is negative, you shift your finger to the left.



                Of course, you can't step to the left from the first element, so the first step to the left wraps around to the last element.






                share|improve this answer





























                  8














                  8










                  8









                  Another explanation:



                  Your finger points to the first element. The index decides how many places you shift your finger to the right. If the number is negative, you shift your finger to the left.



                  Of course, you can't step to the left from the first element, so the first step to the left wraps around to the last element.






                  share|improve this answer















                  Another explanation:



                  Your finger points to the first element. The index decides how many places you shift your finger to the right. If the number is negative, you shift your finger to the left.



                  Of course, you can't step to the left from the first element, so the first step to the left wraps around to the last element.







                  share|improve this answer














                  share|improve this answer



                  share|improve this answer








                  edited Apr 24 at 12:46

























                  answered Apr 15 at 13:39









                  OscarOscar

                  814 bronze badges




                  814 bronze badges
























                      6
















                      You could intuitively understand it this way



                      steps= ["a", "b", "c", "d"]


                      Suppose you start from a to d, a is your staring point where you stand (or your home), so mark it as 0(because you did not move yet),



                      Move one step to b, second step to c and arrive at third d.



                      Then how about you return from d to a (or return from your office to your home). Your home is 0 because your family live there, so your office cannot be a 0.It's your last stop.



                      So when you return back to home. d is the last first stop to where you start off for home, c is the last second ....






                      share|improve this answer



























                      • d is the last first stop to start, c is the last second This is kind of difficult to understand, could you dissect that sentence a bit?

                        – Raimund Krämer
                        Apr 16 at 9:35











                      • last here is ending or final, not recent @RaimundKrämer

                        – Algebra
                        Apr 16 at 10:02















                      6
















                      You could intuitively understand it this way



                      steps= ["a", "b", "c", "d"]


                      Suppose you start from a to d, a is your staring point where you stand (or your home), so mark it as 0(because you did not move yet),



                      Move one step to b, second step to c and arrive at third d.



                      Then how about you return from d to a (or return from your office to your home). Your home is 0 because your family live there, so your office cannot be a 0.It's your last stop.



                      So when you return back to home. d is the last first stop to where you start off for home, c is the last second ....






                      share|improve this answer



























                      • d is the last first stop to start, c is the last second This is kind of difficult to understand, could you dissect that sentence a bit?

                        – Raimund Krämer
                        Apr 16 at 9:35











                      • last here is ending or final, not recent @RaimundKrämer

                        – Algebra
                        Apr 16 at 10:02













                      6














                      6










                      6









                      You could intuitively understand it this way



                      steps= ["a", "b", "c", "d"]


                      Suppose you start from a to d, a is your staring point where you stand (or your home), so mark it as 0(because you did not move yet),



                      Move one step to b, second step to c and arrive at third d.



                      Then how about you return from d to a (or return from your office to your home). Your home is 0 because your family live there, so your office cannot be a 0.It's your last stop.



                      So when you return back to home. d is the last first stop to where you start off for home, c is the last second ....






                      share|improve this answer















                      You could intuitively understand it this way



                      steps= ["a", "b", "c", "d"]


                      Suppose you start from a to d, a is your staring point where you stand (or your home), so mark it as 0(because you did not move yet),



                      Move one step to b, second step to c and arrive at third d.



                      Then how about you return from d to a (or return from your office to your home). Your home is 0 because your family live there, so your office cannot be a 0.It's your last stop.



                      So when you return back to home. d is the last first stop to where you start off for home, c is the last second ....







                      share|improve this answer














                      share|improve this answer



                      share|improve this answer








                      edited Apr 23 at 11:02

























                      answered Apr 16 at 8:54









                      AlgebraAlgebra

                      5,9382 gold badges27 silver badges51 bronze badges




                      5,9382 gold badges27 silver badges51 bronze badges















                      • d is the last first stop to start, c is the last second This is kind of difficult to understand, could you dissect that sentence a bit?

                        – Raimund Krämer
                        Apr 16 at 9:35











                      • last here is ending or final, not recent @RaimundKrämer

                        – Algebra
                        Apr 16 at 10:02

















                      • d is the last first stop to start, c is the last second This is kind of difficult to understand, could you dissect that sentence a bit?

                        – Raimund Krämer
                        Apr 16 at 9:35











                      • last here is ending or final, not recent @RaimundKrämer

                        – Algebra
                        Apr 16 at 10:02
















                      d is the last first stop to start, c is the last second This is kind of difficult to understand, could you dissect that sentence a bit?

                      – Raimund Krämer
                      Apr 16 at 9:35





                      d is the last first stop to start, c is the last second This is kind of difficult to understand, could you dissect that sentence a bit?

                      – Raimund Krämer
                      Apr 16 at 9:35













                      last here is ending or final, not recent @RaimundKrämer

                      – Algebra
                      Apr 16 at 10:02





                      last here is ending or final, not recent @RaimundKrämer

                      – Algebra
                      Apr 16 at 10:02



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