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1 0 1 0 1 0 1 0 1 0 1


Making π from 1 2 3 4 5 6 7 8 9Number formation with digits givenA truly amazing way of making the number 2016A truly amazing way of making every possible positive integerRiddle with functionsExpressing numbers using 0, 1, 2, 3, and 4All digits with one operationMake numbers 1-100 with only four 8sMake 38, 44, 46 using 2,3,8,7?Create all numbers from 1-100 by using 1,3,3,7













1












$begingroup$


Start with the digits:



      $1 0 1 0 1 0 1 0 1 0 1$



You may then add (only!) these simple arithmetic operators:



      $+ - times div$



You may also remove the space between no more than 3 adjacent digits, to make larger numbers:



      $1 0 1$ can be concatenated to $101$, for example

      $1 0 1 0 1$ cannot be concatenated to $require{enclose}enclose{horizontalstrike}{10101} $ (too many digits)



You must then insert one equal sign $ = $ between two numbers to form a valid and correct equation.



What is the fewest operators you can add to form a valid equation?



[Added bonus question]
What is the fewest operators you can add to form a valid equation if operators only appear on one side of the equal sign?





Unary negation $-1$ is allowed.

Unary plus $+1$ is not allowed.

Binary addition, subtraction, multiplication, and division are allowed.

No other operators are allowed.

No decimal points, and no rounding.

Leading zeroes on numbers are not allowed.

Base 10 only.



Order of operations:

What Perl uses. Basically, BO(DM)(AS):

  • left to right

  • multiplication and division done together in one pass

  • addition and subtraction done together in a lower priority pass
You may add parentheses if required to reorder evaluation. Each pair counts as an operator.










share|improve this question











$endgroup$












  • $begingroup$
    Just to be clear: (1) you're allowed any number of arithmetic operators, not just one of each?, (2) you can do any number of space-removals, just not so as to make any single number longer than 3 digits?
    $endgroup$
    – Gareth McCaughan
    1 hour ago










  • $begingroup$
    Further clarification: (3) the number 0 is allowed despite the "no leading zeroes" rule?
    $endgroup$
    – Gareth McCaughan
    1 hour ago










  • $begingroup$
    Yes to all the above.
    $endgroup$
    – Rubio
    1 hour ago
















1












$begingroup$


Start with the digits:



      $1 0 1 0 1 0 1 0 1 0 1$



You may then add (only!) these simple arithmetic operators:



      $+ - times div$



You may also remove the space between no more than 3 adjacent digits, to make larger numbers:



      $1 0 1$ can be concatenated to $101$, for example

      $1 0 1 0 1$ cannot be concatenated to $require{enclose}enclose{horizontalstrike}{10101} $ (too many digits)



You must then insert one equal sign $ = $ between two numbers to form a valid and correct equation.



What is the fewest operators you can add to form a valid equation?



[Added bonus question]
What is the fewest operators you can add to form a valid equation if operators only appear on one side of the equal sign?





Unary negation $-1$ is allowed.

Unary plus $+1$ is not allowed.

Binary addition, subtraction, multiplication, and division are allowed.

No other operators are allowed.

No decimal points, and no rounding.

Leading zeroes on numbers are not allowed.

Base 10 only.



Order of operations:

What Perl uses. Basically, BO(DM)(AS):

  • left to right

  • multiplication and division done together in one pass

  • addition and subtraction done together in a lower priority pass
You may add parentheses if required to reorder evaluation. Each pair counts as an operator.










share|improve this question











$endgroup$












  • $begingroup$
    Just to be clear: (1) you're allowed any number of arithmetic operators, not just one of each?, (2) you can do any number of space-removals, just not so as to make any single number longer than 3 digits?
    $endgroup$
    – Gareth McCaughan
    1 hour ago










  • $begingroup$
    Further clarification: (3) the number 0 is allowed despite the "no leading zeroes" rule?
    $endgroup$
    – Gareth McCaughan
    1 hour ago










  • $begingroup$
    Yes to all the above.
    $endgroup$
    – Rubio
    1 hour ago














1












1








1





$begingroup$


Start with the digits:



      $1 0 1 0 1 0 1 0 1 0 1$



You may then add (only!) these simple arithmetic operators:



      $+ - times div$



You may also remove the space between no more than 3 adjacent digits, to make larger numbers:



      $1 0 1$ can be concatenated to $101$, for example

      $1 0 1 0 1$ cannot be concatenated to $require{enclose}enclose{horizontalstrike}{10101} $ (too many digits)



You must then insert one equal sign $ = $ between two numbers to form a valid and correct equation.



What is the fewest operators you can add to form a valid equation?



[Added bonus question]
What is the fewest operators you can add to form a valid equation if operators only appear on one side of the equal sign?





Unary negation $-1$ is allowed.

Unary plus $+1$ is not allowed.

Binary addition, subtraction, multiplication, and division are allowed.

No other operators are allowed.

No decimal points, and no rounding.

Leading zeroes on numbers are not allowed.

Base 10 only.



Order of operations:

What Perl uses. Basically, BO(DM)(AS):

  • left to right

  • multiplication and division done together in one pass

  • addition and subtraction done together in a lower priority pass
You may add parentheses if required to reorder evaluation. Each pair counts as an operator.










share|improve this question











$endgroup$




Start with the digits:



      $1 0 1 0 1 0 1 0 1 0 1$



You may then add (only!) these simple arithmetic operators:



      $+ - times div$



You may also remove the space between no more than 3 adjacent digits, to make larger numbers:



      $1 0 1$ can be concatenated to $101$, for example

      $1 0 1 0 1$ cannot be concatenated to $require{enclose}enclose{horizontalstrike}{10101} $ (too many digits)



You must then insert one equal sign $ = $ between two numbers to form a valid and correct equation.



What is the fewest operators you can add to form a valid equation?



[Added bonus question]
What is the fewest operators you can add to form a valid equation if operators only appear on one side of the equal sign?





Unary negation $-1$ is allowed.

Unary plus $+1$ is not allowed.

Binary addition, subtraction, multiplication, and division are allowed.

No other operators are allowed.

No decimal points, and no rounding.

Leading zeroes on numbers are not allowed.

Base 10 only.



Order of operations:

What Perl uses. Basically, BO(DM)(AS):

  • left to right

  • multiplication and division done together in one pass

  • addition and subtraction done together in a lower priority pass
You may add parentheses if required to reorder evaluation. Each pair counts as an operator.







no-computers formation-of-numbers arithmetic






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 1 hour ago







Rubio

















asked 1 hour ago









RubioRubio

29k565178




29k565178












  • $begingroup$
    Just to be clear: (1) you're allowed any number of arithmetic operators, not just one of each?, (2) you can do any number of space-removals, just not so as to make any single number longer than 3 digits?
    $endgroup$
    – Gareth McCaughan
    1 hour ago










  • $begingroup$
    Further clarification: (3) the number 0 is allowed despite the "no leading zeroes" rule?
    $endgroup$
    – Gareth McCaughan
    1 hour ago










  • $begingroup$
    Yes to all the above.
    $endgroup$
    – Rubio
    1 hour ago


















  • $begingroup$
    Just to be clear: (1) you're allowed any number of arithmetic operators, not just one of each?, (2) you can do any number of space-removals, just not so as to make any single number longer than 3 digits?
    $endgroup$
    – Gareth McCaughan
    1 hour ago










  • $begingroup$
    Further clarification: (3) the number 0 is allowed despite the "no leading zeroes" rule?
    $endgroup$
    – Gareth McCaughan
    1 hour ago










  • $begingroup$
    Yes to all the above.
    $endgroup$
    – Rubio
    1 hour ago
















$begingroup$
Just to be clear: (1) you're allowed any number of arithmetic operators, not just one of each?, (2) you can do any number of space-removals, just not so as to make any single number longer than 3 digits?
$endgroup$
– Gareth McCaughan
1 hour ago




$begingroup$
Just to be clear: (1) you're allowed any number of arithmetic operators, not just one of each?, (2) you can do any number of space-removals, just not so as to make any single number longer than 3 digits?
$endgroup$
– Gareth McCaughan
1 hour ago












$begingroup$
Further clarification: (3) the number 0 is allowed despite the "no leading zeroes" rule?
$endgroup$
– Gareth McCaughan
1 hour ago




$begingroup$
Further clarification: (3) the number 0 is allowed despite the "no leading zeroes" rule?
$endgroup$
– Gareth McCaughan
1 hour ago












$begingroup$
Yes to all the above.
$endgroup$
– Rubio
1 hour ago




$begingroup$
Yes to all the above.
$endgroup$
– Rubio
1 hour ago










2 Answers
2






active

oldest

votes


















3












$begingroup$

Here is a simple, but optimal, solution.




10+101+0=10+101




uses a total of




three additions, one =, and no other operators.




The best one could hope for is




two operators (and one =) -- since no number can be longer than 3 digits, we have to break up the string of digits in at least three places. So the above can't be improved much.




And in fact,




we have to break in at least four places, so the above is optimal. Why? Because if we ever break after a 1 then the next number starts with a 0 and hence must be exactly a 0; if there is a single-digit number then the remaining 10 digits must still form >3 groups (because no group is longer than 3), so we get at least 5 groups in all, hence at least 4 group-breakers, hence at least 3 operators. On the other hand, if we never break after a 1 then all our groups other than perhaps the last one are of even size, hence of size at most 2, which again means at least 5 groups in all.







share|improve this answer











$endgroup$













  • $begingroup$
    "The best one could hope for is" exactly what I'm thinking about right now. ;-) The solution you've provided (and its trivial variants) is the easy one.
    $endgroup$
    – EKons
    1 hour ago












  • $begingroup$
    I actually think it's impossible to do better than the easy one, but I haven't checked my reasoning yet or even written it down explicitly... [EDITED to add:] Yup, seems to be correct.
    $endgroup$
    – Gareth McCaughan
    1 hour ago





















0












$begingroup$

To answer the bonus question, you'll need:




$10-10+101+0=101$




This is optimal for the reason given in Gareth's answer by himself:




we have to break in at least four places, so the above is optimal. Why? Because if we ever break after a 1 then the next number starts with a 0 and hence must be exactly a 0; if there is a single-digit number then the remaining 10 digits must still form >3 groups (because no group is longer than 3), so we get at least 5 groups in all, hence at least 4 group-breakers, hence at least 3 operators. On the other hand, if we never break after a 1 then all our groups other than perhaps the last one are of even size, hence of size at most 2, which again means at least 5 groups in all.




Of course, the main solution has already been given.






share|improve this answer









$endgroup$













    Your Answer





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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    Here is a simple, but optimal, solution.




    10+101+0=10+101




    uses a total of




    three additions, one =, and no other operators.




    The best one could hope for is




    two operators (and one =) -- since no number can be longer than 3 digits, we have to break up the string of digits in at least three places. So the above can't be improved much.




    And in fact,




    we have to break in at least four places, so the above is optimal. Why? Because if we ever break after a 1 then the next number starts with a 0 and hence must be exactly a 0; if there is a single-digit number then the remaining 10 digits must still form >3 groups (because no group is longer than 3), so we get at least 5 groups in all, hence at least 4 group-breakers, hence at least 3 operators. On the other hand, if we never break after a 1 then all our groups other than perhaps the last one are of even size, hence of size at most 2, which again means at least 5 groups in all.







    share|improve this answer











    $endgroup$













    • $begingroup$
      "The best one could hope for is" exactly what I'm thinking about right now. ;-) The solution you've provided (and its trivial variants) is the easy one.
      $endgroup$
      – EKons
      1 hour ago












    • $begingroup$
      I actually think it's impossible to do better than the easy one, but I haven't checked my reasoning yet or even written it down explicitly... [EDITED to add:] Yup, seems to be correct.
      $endgroup$
      – Gareth McCaughan
      1 hour ago


















    3












    $begingroup$

    Here is a simple, but optimal, solution.




    10+101+0=10+101




    uses a total of




    three additions, one =, and no other operators.




    The best one could hope for is




    two operators (and one =) -- since no number can be longer than 3 digits, we have to break up the string of digits in at least three places. So the above can't be improved much.




    And in fact,




    we have to break in at least four places, so the above is optimal. Why? Because if we ever break after a 1 then the next number starts with a 0 and hence must be exactly a 0; if there is a single-digit number then the remaining 10 digits must still form >3 groups (because no group is longer than 3), so we get at least 5 groups in all, hence at least 4 group-breakers, hence at least 3 operators. On the other hand, if we never break after a 1 then all our groups other than perhaps the last one are of even size, hence of size at most 2, which again means at least 5 groups in all.







    share|improve this answer











    $endgroup$













    • $begingroup$
      "The best one could hope for is" exactly what I'm thinking about right now. ;-) The solution you've provided (and its trivial variants) is the easy one.
      $endgroup$
      – EKons
      1 hour ago












    • $begingroup$
      I actually think it's impossible to do better than the easy one, but I haven't checked my reasoning yet or even written it down explicitly... [EDITED to add:] Yup, seems to be correct.
      $endgroup$
      – Gareth McCaughan
      1 hour ago
















    3












    3








    3





    $begingroup$

    Here is a simple, but optimal, solution.




    10+101+0=10+101




    uses a total of




    three additions, one =, and no other operators.




    The best one could hope for is




    two operators (and one =) -- since no number can be longer than 3 digits, we have to break up the string of digits in at least three places. So the above can't be improved much.




    And in fact,




    we have to break in at least four places, so the above is optimal. Why? Because if we ever break after a 1 then the next number starts with a 0 and hence must be exactly a 0; if there is a single-digit number then the remaining 10 digits must still form >3 groups (because no group is longer than 3), so we get at least 5 groups in all, hence at least 4 group-breakers, hence at least 3 operators. On the other hand, if we never break after a 1 then all our groups other than perhaps the last one are of even size, hence of size at most 2, which again means at least 5 groups in all.







    share|improve this answer











    $endgroup$



    Here is a simple, but optimal, solution.




    10+101+0=10+101




    uses a total of




    three additions, one =, and no other operators.




    The best one could hope for is




    two operators (and one =) -- since no number can be longer than 3 digits, we have to break up the string of digits in at least three places. So the above can't be improved much.




    And in fact,




    we have to break in at least four places, so the above is optimal. Why? Because if we ever break after a 1 then the next number starts with a 0 and hence must be exactly a 0; if there is a single-digit number then the remaining 10 digits must still form >3 groups (because no group is longer than 3), so we get at least 5 groups in all, hence at least 4 group-breakers, hence at least 3 operators. On the other hand, if we never break after a 1 then all our groups other than perhaps the last one are of even size, hence of size at most 2, which again means at least 5 groups in all.








    share|improve this answer














    share|improve this answer



    share|improve this answer








    edited 1 hour ago

























    answered 1 hour ago









    Gareth McCaughanGareth McCaughan

    63.8k3163249




    63.8k3163249












    • $begingroup$
      "The best one could hope for is" exactly what I'm thinking about right now. ;-) The solution you've provided (and its trivial variants) is the easy one.
      $endgroup$
      – EKons
      1 hour ago












    • $begingroup$
      I actually think it's impossible to do better than the easy one, but I haven't checked my reasoning yet or even written it down explicitly... [EDITED to add:] Yup, seems to be correct.
      $endgroup$
      – Gareth McCaughan
      1 hour ago




















    • $begingroup$
      "The best one could hope for is" exactly what I'm thinking about right now. ;-) The solution you've provided (and its trivial variants) is the easy one.
      $endgroup$
      – EKons
      1 hour ago












    • $begingroup$
      I actually think it's impossible to do better than the easy one, but I haven't checked my reasoning yet or even written it down explicitly... [EDITED to add:] Yup, seems to be correct.
      $endgroup$
      – Gareth McCaughan
      1 hour ago


















    $begingroup$
    "The best one could hope for is" exactly what I'm thinking about right now. ;-) The solution you've provided (and its trivial variants) is the easy one.
    $endgroup$
    – EKons
    1 hour ago






    $begingroup$
    "The best one could hope for is" exactly what I'm thinking about right now. ;-) The solution you've provided (and its trivial variants) is the easy one.
    $endgroup$
    – EKons
    1 hour ago














    $begingroup$
    I actually think it's impossible to do better than the easy one, but I haven't checked my reasoning yet or even written it down explicitly... [EDITED to add:] Yup, seems to be correct.
    $endgroup$
    – Gareth McCaughan
    1 hour ago






    $begingroup$
    I actually think it's impossible to do better than the easy one, but I haven't checked my reasoning yet or even written it down explicitly... [EDITED to add:] Yup, seems to be correct.
    $endgroup$
    – Gareth McCaughan
    1 hour ago













    0












    $begingroup$

    To answer the bonus question, you'll need:




    $10-10+101+0=101$




    This is optimal for the reason given in Gareth's answer by himself:




    we have to break in at least four places, so the above is optimal. Why? Because if we ever break after a 1 then the next number starts with a 0 and hence must be exactly a 0; if there is a single-digit number then the remaining 10 digits must still form >3 groups (because no group is longer than 3), so we get at least 5 groups in all, hence at least 4 group-breakers, hence at least 3 operators. On the other hand, if we never break after a 1 then all our groups other than perhaps the last one are of even size, hence of size at most 2, which again means at least 5 groups in all.




    Of course, the main solution has already been given.






    share|improve this answer









    $endgroup$


















      0












      $begingroup$

      To answer the bonus question, you'll need:




      $10-10+101+0=101$




      This is optimal for the reason given in Gareth's answer by himself:




      we have to break in at least four places, so the above is optimal. Why? Because if we ever break after a 1 then the next number starts with a 0 and hence must be exactly a 0; if there is a single-digit number then the remaining 10 digits must still form >3 groups (because no group is longer than 3), so we get at least 5 groups in all, hence at least 4 group-breakers, hence at least 3 operators. On the other hand, if we never break after a 1 then all our groups other than perhaps the last one are of even size, hence of size at most 2, which again means at least 5 groups in all.




      Of course, the main solution has already been given.






      share|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        To answer the bonus question, you'll need:




        $10-10+101+0=101$




        This is optimal for the reason given in Gareth's answer by himself:




        we have to break in at least four places, so the above is optimal. Why? Because if we ever break after a 1 then the next number starts with a 0 and hence must be exactly a 0; if there is a single-digit number then the remaining 10 digits must still form >3 groups (because no group is longer than 3), so we get at least 5 groups in all, hence at least 4 group-breakers, hence at least 3 operators. On the other hand, if we never break after a 1 then all our groups other than perhaps the last one are of even size, hence of size at most 2, which again means at least 5 groups in all.




        Of course, the main solution has already been given.






        share|improve this answer









        $endgroup$



        To answer the bonus question, you'll need:




        $10-10+101+0=101$




        This is optimal for the reason given in Gareth's answer by himself:




        we have to break in at least four places, so the above is optimal. Why? Because if we ever break after a 1 then the next number starts with a 0 and hence must be exactly a 0; if there is a single-digit number then the remaining 10 digits must still form >3 groups (because no group is longer than 3), so we get at least 5 groups in all, hence at least 4 group-breakers, hence at least 3 operators. On the other hand, if we never break after a 1 then all our groups other than perhaps the last one are of even size, hence of size at most 2, which again means at least 5 groups in all.




        Of course, the main solution has already been given.







        share|improve this answer












        share|improve this answer



        share|improve this answer










        answered 42 mins ago









        EKonsEKons

        1,005826




        1,005826






























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