# Challenge

The challenge is to write a code that takes a positive integer 'n' as an input and displays all the possible ways in which the numbers from 1 - n can be written, with either positive or negative sign in between, such that their sum is equal to zero. Please remember that you may only use addition or subtraction.

For example, if the input is 3, then there are 2 ways to make the sum 0:

 1+2-3=0
-1-2+3=0


Note that, the numbers are in order, starting from 1 till n (which is 3 in this case). As it is evident from the example, the sign of the first number can also be negative, so be careful.

Now, 3 was pretty much simple. Let us list all the ways when we consider the number 7.

 1+2-3+4-5-6+7=0
1+2-3-4+5+6-7=0
1-2+3+4-5+6-7=0
1-2-3-4-5+6+7=0
-1+2+3+4+5-6-7=0
-1+2-3-4+5-6+7=0
-1-2+3+4-5-6+7=0
-1-2+3-4+5+6-7=0


So here, we have got a total of 8 possible ways.

# Input And Output

As stated before, the input would be a positive integer. Your output should contain all the possible ways in which the numbers give a sum of zero. In case there is no possible way to do the same, you can output anything you like.

Also, you can print the output in any format you like. But, it should be understandable. For example, you may print it as in the above example. Or, you may just print the signs of the numbers in order. Otherwise, you can also print '0's and '1's in order, where '0' would display negative sign and '1' would display positive sign (or vice versa).

For example, you can represent 1+2-3=0 using:

1+2-3=0
1+2-3
[1,2,-3]
++-
110
001


However, I would recommend using any of the first three formats for simplicity. You can assume all the inputs to be valid.

# Examples

7 ->

1+2-3+4-5-6+7=0
1+2-3-4+5+6-7=0
1-2+3+4-5+6-7=0
1-2-3-4-5+6+7=0
-1+2+3+4+5-6-7=0
-1+2-3-4+5-6+7=0
-1-2+3+4-5-6+7=0
-1-2+3-4+5+6-7=0

4 ->

1-2-3+4=0
-1+2+3-4=0

2 -> -

8 ->

1+2+3+4-5-6-7+8=0
1+2+3-4+5-6+7-8=0
1+2-3+4+5+6-7-8=0
1+2-3-4-5-6+7+8=0
1-2+3-4-5+6-7+8=0
1-2-3+4+5-6-7+8=0
1-2-3+4-5+6+7-8=0
-1+2+3-4+5-6-7+8=0
-1+2+3-4-5+6+7-8=0
-1+2-3+4+5-6+7-8=0
-1-2+3+4+5+6-7-8=0
-1-2+3-4-5-6+7+8=0
-1-2-3+4-5+6-7+8=0
-1-2-3-4+5+6+7-8=0


# Scoring

This is , so the shortest code wins!

• Please note that this is not a dupe of codegolf.stackexchange.com/questions/8655/… , because this challenge is meant to take only n as input and use all the numbers 1-n in order. – Manish Kundu Feb 3 '18 at 15:14
• May we represent + as N and - as -N, or is that taking it too far? (e.g. 3 -> [[-3,-3,3], [3,3,-3]]) – Jonathan Allan Feb 3 '18 at 16:05
• @JonathanAllan Isn't that mentioned in the list of output formats? Or did I wrongly interpret your question? – Manish Kundu Feb 3 '18 at 16:07
• I mean like the 0 and 1 option but using N and -N (see my edit above) – Jonathan Allan Feb 3 '18 at 16:09
• @JonathanAllan Yes thats certainly allowed. Make sure you mention that in the answer. – Manish Kundu Feb 3 '18 at 16:14

f n=[l|l<-mapM(\i->[i,-i])[1..n],0==sum l]


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• 42. – user202729 Feb 3 '18 at 15:40
• Shouldn't it be 0== ? – Laikoni Feb 3 '18 at 17:48

# Jelly, 9 bytes

1,-ṗ×RSÐḟ


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# Exp

1,-ṗ×RSÐḟ  Main link. Input = n. Assume n=2.
1,-        Literal list [1, -1].
ṗ       Cartesian power n. Get [[1, 1], [1, -1], [-1, 1], [-1, -1]]
×R     Multiply (each list) by Range 1..n.
Ðḟ  ḟilter out lists with truthy (nonzero)
S      Sum.


# Jelly, 9 bytes

Jonathan Allan's suggestion, output a list of signs.

1,-ṗæ.ÐḟR


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• How about (ab?)using the lax output format with ,Nṗæ.ÐḟR? – Jonathan Allan Feb 3 '18 at 15:53
• Or alternatively, this output the outputs multiplied by n. – user202729 Feb 3 '18 at 15:57
• The N and -N output I suggested has been allowed, so that saves one byte :) (just need to mention the format in the answer) – Jonathan Allan Feb 3 '18 at 16:34
• Not sure how recently it was added, but there's a builtin Ø+ to save a byte over 1,-. – Unrelated String May 25 at 19:08

# Python 2, 62 bytes

f=lambda n,*l:f(n-1,n,*l)+f(n-1,-n,*l)if n else[l]*(sum(l)==0)


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Mr. Xcoder saved 4 bytes with a nifty use of starred arguments.

• 62 bytes using *l instead of l=[] – Mr. Xcoder Feb 3 '18 at 22:16

# Perl, 37 36 bytes

perl -E 'map eval||say,glob join"{+,-}",0..<>' <<< 7

• Nicely done. You can drop -n and <<< if you replace $_ with pop. It doesn't actually improve your score, but it makes the overall expression shorter ;) – Chris Feb 4 '18 at 8:08 # 05AB1E, 11 bytes ®X‚¹ãʒ¹L*O_  Try it online! The output format for e.g. input 3 is: [[-1, -1, 1], [1, 1, -1]]  That is, -1-2+3, 1+2-3. • ¹L can be ā for -1 – Kevin Cruijssen May 25 at 15:08 # Wolfram Language (Mathematica), 36 bytes Pick[p={1,-1}~Tuples~#,p.Range@#,0]&  Try it online! # Husk, 10 bytes fo¬ΣΠmSe_ḣ  Try it online! ## Explanation Not too complicated. fo¬ΣΠmSe_ḣ Implicit input, say n=4 ḣ Range: [1,2,3,4] m Map over the range: Se pair element with _ its negation. Result: [[1,-1],[2,-2],[3,-3],[4,-4]] Π Cartesian product: [[1,2,3,4],[1,2,3,-4],..,[-1,-2,-3,-4]] f Keep those Σ whose sum o¬ is falsy (equals 0): [[-1,2,3,-4],[1,-2,-3,4]]  # Python 3, 105 bytes lambda n:[k for k in product(*[(1,-1)]*n)if sum(-~n*s for n,s in enumerate(k))==0] from itertools import*  Try it online! # JavaScript (V8), 69 61 58 bytes Saved 8 bytes thanks to @Neil Saved 3 bytes thanks to @l4m2 Prints all solutions. f=(n,o='')=>n?f(n-1,o+'+'+n)&f(n-1,o+-n):eval(o)||print(o)  Try it online! • Do you need k? Something like this: f=(n,o='')=>n?['+','-'].map(c=>f(n-1,c+n+o)):eval(o)||alert(o) – Neil Feb 3 '18 at 23:37 • @Neil I really don't... Thanks. – Arnauld Feb 4 '18 at 0:35 • o+'-'+n => o+-n – l4m2 May 25 at 13:09 # Swift, 116 bytes func f(n:Int){var r=[[Int]()] for i in 1...n{r=r.flatMap{[$0+[i],$0+[-i]]}} print(r.filter{$0.reduce(0){$0+$1}==0})}


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## Explanation

func f(n:Int){
var r=[[Int]()]                         // Initialize r with [[]]
// (list with one empty list)
for i in 1...n{                         // For i from 1 to n:
r=r.flatMap{[$0+[i],$0+[-i]]}         //   Replace every list in r with the list
}                                       //   prepended with i and prepended with -i
print(r.filter{$0.reduce(0){$0+$1}==0}) // Print all lists in r that sums to 0 }  # Python 2, 91 bytes lambda x:[s for s in[[~j*[1,-1][i>>j&1]for j in range(x)]for i in range(2**x)]if sum(s)==0]  Try it online! Returns a list of satisfying lists (e.g., f(3)=[[-1,-2,3], [1,2,-3]]) # APL (Dyalog), 38 bytes {k/⍨0=+/¨k←((,o∘.,⊢)⍣(⍵-1)⊢o←¯1 1)×⊂⍳⍵}  Try it online! # Pyth, 13 bytes f!sT.nM*F_BMS  Try it here! # Clean, 79 bytes import StdEnv$n=[k\\k<-foldr(\i l=[[p:s]\\s<-l,p<-[~i,i]])[[]][1..n]|sum k==0]


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# Retina, 73 bytes

.+
*
_
=_$ +0= -$%"+
(-(_)+|\+(_)+)+
$&=$#2=$#3= G(=.+)\1= =.* _+$.&


Try it online! Explanation:

.+
*


Convert the input to unary.

_
=_$  Convert the number to a list of =-prefixed numbers. +0= -$%"+


Replace each = in turn with both - and +, duplicating the number of lines each time.

(-(_)+|\+(_)+)+
$&=$#2=$#3=  Separately count the number of _s after -s and +s. This sums the negative and positive numbers. G(=.+)\1=  Keep only those lines where the -s and +s cancel out. =.*  Delete the counts. _+$.&


Convert to decimal.

# Python 3 + numpy, 104 103 bytes

import itertools as I,numpy as P
lambda N:[r for r in I.product(*[[-1,1]]*N)if sum(P.arange(N)*r+r)==0]


Output is [-1, 1] corresponding to the sign.

• You can remove the space before if for -1 byte – ovs Feb 3 '18 at 21:17

# C (gcc), 171 161 bytes

k,s;f(S,n,j)int*S;{if(s=j--)S[j]=~0,f(S,n,j),S[j]=1,f(S,n,j);else{for(k=n;k;)s+=k--*S[k];if(!s)for(puts("");k<n;)printf("%d",S[k++]+1);}}F(n){int S[n];f(S,n,n);}


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• @ceilingcat Thank you. – Jonathan Frech May 24 at 22:51

# C (gcc), 101100 83 bytes

Represents addition as 0 and subtraction as 1, but that ultimately that does not matter since for any valid sequence, the complement of the sequence will also result in zero. Returns the permutations by appending them as 32-bit ints to a caller-provided buffer.

i,j,s;f(n,p)int*p;{for(i=1<<n;--i;s||(*p++=i))for(j=s=0;j<n;)s+=i>>j++&1?j:-j;n=p;}


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# Explanation

i,j,s; // iterators i and j, sum s.
f(n,p)int*p; // declare function
{
for ( i=1<<n; --i; // iterate over all permutations of n bits (except all-0)
s||(*p++=i)) // if the permutation summed to 0, append it to the buffer
for(j=s=0;j<n;)
s+= i>>j++&1 // Determine the value of the jth bit
?j:-j; // 1=>+(j+1), 0=>-(j+1)
n=p; // return the pointer to the end of the appended values
}

• I do not think you need to use any prettifiers for your output, which could save some bytes. Note that printf("\n") can often be replaced by puts(""). – Jonathan Frech May 28 at 23:57
• Ah, sorry, I misinterpreted the escape sequences' function. In that case, I think standard I/O also forbids your method since one normally thinks of bytes as a program's output, not its terminal appearance. – Jonathan Frech May 29 at 19:03
• The general rule for this appears to be codegolf.meta.stackexchange.com/a/5515/46076, which I wasn't aware of until now. I've retracted my flag, since I honestly can't decide whether this answer is valid or not now. See also Default for Code Golf: Input/Output methods Also, fix ping to @JonathanFrech since the previous one didn't go through for me. – pppery Jun 21 at 20:01
• @pppery I also was not aware of this position, though as it appears to be consensus I guess that is the rule. Thank you for finding the meta post. – Jonathan Frech Jun 23 at 5:03
• I have updated the function to use a buffer instead of stdout since other submissions return lists, etc. as well. – CompilerPotato Jun 24 at 7:55

# Perl 6, 43 bytes

{grep *.sum==0,[X] (1..$_ X*1,-1).rotor(2)}  Try it Returns a sequence of lists ## Expanded: { # bare block lambda with implicit parameter ｢$_｣

grep              # only return the ones
*.sum == 0,     # that sum to zero

[X]             # reduce with cross meta operator

(
1 .. $_ # Range from 1 to the input X* # cross multiplied by 1, -1 ).rotor(2) # take 2 at a time (positive and negative) }  1..$_ X* 1,-1(1, -1, 2, -2)
(…).rotor(2)((1, -1), (2, -2))
[X] …((1, 2), (1, -2), (-1, 2), (-1, -2))

# J, 35 30 bytes

-5 bytes thanks to FrownyFrog!

>:@i.(]#~0=1#.*"1)_1^2#:@i.@^]


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# J, 35 bytes

[:(#~0=+/"1)>:@i.*"1(_1^[:#:@i.2^])


## How it works

I multiply the list 1..n with all possible lists of coefficients 1 / -1 and find the ones that add up to zero.

                    (             ) - the list of coefficients
i.     - list 0 to
2^]  - 2 to the power of the input
_1^[:          - -1 to the power of
#:@       - each binary digit of each number in 0..n-1 to
*"1                - each row multiplied by
>:@i.                   - list 1..n
(#~      )                        - copy those rows
0=+/"1                         - that add up to 0
[:                                  - compose


Try it online!

As an alternative I tried an explicit verb, using the approach of cartesian product of +/-:

# J, 37 bytes

3 :'(#~0=+/"1)(-y)]\;{(<"1@,.-)1+i.y'


{(<"1@,.-) finds the cartesian products for example:

{(<"1@,.-) 1 2 3
┌───────┬────────┐
│1 2 3  │1 2 _3  │
├───────┼────────┤
│1 _2 3 │1 _2 _3 │
└───────┴────────┘

┌───────┬────────┐
│_1 2 3 │_1 2 _3 │
├───────┼────────┤
│_1 _2 3│_1 _2 _3│
└───────┴────────┘


Too bad that it boxes the result, so I spent some bytes to unbox the values

Try it online!

• @FrownyFrog Thank you, I was not happy with the right side of my code. – Galen Ivanov Feb 5 '18 at 7:08

Not the best but I tried :)

# C (Linux GCC)

## Entire program: 282 characters

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
int n,x,i,s;int main(int c,char**z){n=atoi(z[1]);int a[n];for(x=0;x<n;x++)a[x]=x+1;for(i=0;i<pow(2, n)-1;i++){s=0;for(x=0;x<n;x++){printf("%c%d",i>>x&0x01?'-':'+',a[x]);s+=(i>>x&0x01?-1:1)*a[x];}printf(s!=0?"\r":"=%d \n",s);}}


## Just as function: 196 characters

void z(int n){int a[n],x,i,s;for(x=0;x<n;x++)a[x]=x+1;for(i=0;i<pow(2,n)-1;i++){s=0;for(x=0;x<n;x++){printf("%c%d",i>>x&0x01?'-':'+',a[x]);s+=(i>>x&0x01?-1:1)*a[x];}printf(s!=0?"\r":"=%d \n",s);}}


### Results

• n = 4:
+1-2-3+4=0
-1+2+3-4=0

• n = 7:
+1-2-3-4-5+6+7=0
-1+2-3-4+5-6+7=0
-1-2+3+4-5-6+7=0
+1+2-3+4-5-6+7=0
-1-2+3-4+5+6-7=0
+1+2-3-4+5+6-7=0
+1-2+3+4-5+6-7=0
-1+2+3+4+5-6-7=0


## Abstract Printing & Just Function: 172 characters

You could shrink the print statements down much more by simply outputting only the +/-'s which are represented by 1/-1 and abstractly representing the entire sum thing, but I still don't think it would beat the other C answers anyway.

That version is:

void z(int n){int a[n],x,i,j,s;for(x=0;x<n;x++)a[x]=x+1;for(i=0;i<pow(2,n)-1;i++){s=0;for(x=0;x<n;x++){j=i>>x&0x01?-1:1;printf("%d ",j);s+=j*a[x];}printf(s!=0?"\r":"\n");}}


## Results

• n = 4:
1 -1 -1 1
-1 1 1 -1

• n = 7:
1 -1 -1 -1 -1 1 1
-1 1 -1 -1 1 -1 1
-1 -1 1 1 -1 -1 1
1 1 -1 1 -1 -1 1
-1 -1 1 -1 1 1 -1 1
1 1 -1 -1 1 1 -1
1 -1 1 1 -1 1 -1 1
-1 1 1 1 1 -1 -1 -1


# T-SQL, 152 bytes

Input is a varchar(max)

Output format is +1+2-3

WITH C as(SELECT @*1d,@+null o,0f
UNION ALL
SELECT~-d,concat(char(44+s),d,o),f+s*d
FROM C,(values(-1),(1))x(s)WHERE d>0)SELECT
o FROM C WHERE f=0and d=0


Try it online

# R, 82 69 bytes

s=t(expand.grid(rep(list(c(-1,1)),n<-scan())))*1:n;t(s[,!colSums(s)])


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Edit: -13 bytes by using built-in expand.grid function to automatically create matrix of all combinations of +1 and -1.

Selects rows of matrix of 1:n in all combinations of positive & negative which sum to zero.

Commented version:

s=t(                    # transpose matrix of:
mapply(rep,             # apply the function 'rep'
list(c(-1,1)),      # to the values (-1, 1)
each=2^(0:(n-1)),   # repeating each by an increasing power-of-2
length=2^n)         # and making the output length always 2^n
)                       # so: this generates a matrix of all the
# combinations of -1 and 1

t(                      # now transpose the matrix s
s[,                     # taking only the columns with a
!colSums(s*1:n)]     # zero sum of multiplying the values
)                       # by the sequence 1:n


Previous (but sadly longer much longer) recursive strategy: 118 bytes

f=function(n,l=NULL,t=0)if(n>1,c(f(n-1,c(0,l),t-n),f(n-1,c(1,l),t+n)),if(t==n,list(c(0,l)),if(t==-n)list(c(1,l))))


Try it online!

Outputs 1 for + and 0 for - (or the other way around).

Commented version:

f=function(n,l=NULL,t=0){               # n = counts down to 1 recursively
# l = list of signs so far
# t = total so far
if(n==1){                           # if n is 1...
if(t-n==0){                     # ...then if t-n makes zero...
return(list(c("-",l))) }    # add a '-' to the list of signs & return it
else if(t+n==0){                # ...or, if t+n makes zero...
return(list(c("+",l))) }    # add a '+' to the list of signs & return it
else {                          # ...otherwise we can't make zero...
return(NULL) }              # so return NULL
} else {                            # if n is greater than 1
return(c(                       # return the result of calling this function
f(n-1,c("-",l),t-n),        # using n-1, after applying either a '-'
f(n-1,c("+",l),t+n)         # or a '+' and updating the total so far
))
}
}