# Subset sum problem

Subset sum problem is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Implement a function/procedure/method/subroutine that takes a set/array/list/stream/table/collection of words with integer weights, and identifies a non-empty subset of them whose weights sum to zero (cf. the Dropbox Diet candidate screening exercise and the Subset sum problem Wikipedia article).

For example

This set of weighted words, one solution would be the set of words:

•   {elysee,   efferent,   deploy,   departure,   centipede,   bonnet,   balm,   archbishop}

because their respective weights of:

•   -326,   54,   44,   952,   -658,   452,   397,   and   -915

sum to zero.

Table of weighted words
word weight
alliance -624
archbishop -915
balm 397
bonnet 452
brute 870
centipede -658
cobol 362
covariate 590
departure 952
deploy 44
diophantine 645
efferent 54
elysee -326
escritoire 856
exorcism -983
fiat 170
filmy -874
flatworm 503
gestapo 915
infra -847
isis -982
lindholm 999
markham 475
mincemeat -880
moresby 756
mycenae 183
plugging -266
smokescreen 423
speakeasy -745
vein 813

Another solution would be the set of words {flatworm, gestapo, infra, isis, lindholm, plugging, smokescreen, speakeasy}, because their respective weights of 503, 915, -847, -982, 999, -266, 423, and -745 also sum to zero.

You may assume the weights range from -1000 to 1000.

If there are multiple solutions, only one needs to be found.

Use any algorithm you want and demonstrate it on a set of at least 30 weighted words with the results shown in a human readable form.

Note that an implementation that depends on enumerating all possible subsets is likely to be infeasible.

`with Ada.Text_IO; use Ada.Text_IO;with Ada.Strings.Unbounded; use Ada.Strings.Unbounded;procedure SubsetSum is   function "+"(S:String) return Unbounded_String renames To_Unbounded_String;   type Point is record      str : Unbounded_String;      num : Integer;   end record;   type Points is array (Natural range <>) of Point;   type Indices is array (Natural range <>) of Natural;    procedure Print (data : Points; list : Indices; len : Positive) is begin      Put (len'Img & ":");      for i in 0..len-1 loop         Put (" "& To_String(data(list(i)).str));      end loop; New_Line;   end Print;    function Check (data : Points; list : Indices; len : Positive) return Boolean is      sum : Integer := 0;   begin      for i in 0..len-1 loop sum := sum + data(list(i)).num; end loop;      return sum = 0;   end Check;    procedure Next (list : in out Indices; n, r : Positive ) is begin      for i in reverse 0..r-1 loop         if list(i)/=i+n-r then list(i):=list(i)+1;            for j in i+1..r-1 loop list(j):=list(j-1)+1; end loop; exit;         end if;      end loop;   end Next;    data : constant Points := ((+"alliance", -624), (+"archbishop", -915),      (+"balm", 397), (+"bonnet", 452), (+"brute", 870),      (+"centipede", -658), (+"cobol", 362), (+"covariate", 590),      (+"departure", 952), (+"deploy", 44), (+"diophantine", 645),      (+"efferent", 54), (+"elysee", -326), (+"eradicate", 376),      (+"escritoire", 856), (+"exorcism", -983), (+"fiat", 170),      (+"filmy", -874), (+"flatworm", 503), (+"gestapo", 915),      (+"infra", -847), (+"isis", -982), (+"lindholm", 999),      (+"markham", 475), (+"mincemeat", -880), (+"moresby", 756),      (+"mycenae", 183), (+"plugging", -266), (+"smokescreen", 423),      (+"speakeasy", -745), (+"vein", 813));   list, last : Indices (data'Range);begin   for len in 2..data'Length loop      for i in 0..len-1 loop list(i):=i; end loop;      loop         if Check(data, list, len) then Print(data, list, len); exit; end if;         last := list;         Next(list, data'Length, len);         exit when last=list;      end loop;   end loop;end SubsetSum;`
Output:
```2: archbishop gestapo
3: centipede markham mycenae
4: alliance balm deploy mycenae
5: alliance brute covariate deploy mincemeat
6: alliance archbishop balm deploy gestapo mycenae
7: alliance archbishop bonnet cobol departure exorcism moresby
8: alliance archbishop balm bonnet fiat flatworm isis lindholm
9: alliance archbishop balm bonnet brute covariate eradicate mincemeat plugging
10: alliance archbishop balm bonnet brute centipede cobol departure deploy mincemeat
11: alliance archbishop balm bonnet brute centipede cobol departure infra moresby speakeasy
12: alliance archbishop balm bonnet brute centipede cobol covariate diophantine efferent elysee infra
13: alliance archbishop balm bonnet brute centipede cobol covariate departure efferent eradicate filmy isis
14: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee filmy markham speakeasy
15: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee exorcism flatworm infra mycenae
16: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine elysee exorcism filmy gestapo infra
17: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine exorcism isis mincemeat mycenae plugging vein
18: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee exorcism filmy isis mycenae vein
19: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism fiat infra isis smokescreen
20: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism gestapo infra isis smokescreen speakeasy
21: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism flatworm infra lindholm mincemeat plugging speakeasy
22: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy flatworm mincemeat plugging speakeasy
23: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism infra isis mincemeat moresby mycenae smokescreen speakeasy
24: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee exorcism filmy gestapo infra markham mincemeat moresby mycenae plugging smokescreen speakeasy
25: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine eradicate exorcism fiat filmy flatworm infra isis lindholm markham mincemeat moresby mycenae plugging speakeasy
26: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine elysee eradicate escritoire exorcism fiat filmy gestapo infra isis markham mincemeat mycenae plugging speakeasy vein
27: alliance archbishop balm bonnet brute centipede covariate departure deploy efferent elysee eradicate escritoire exorcism fiat filmy flatworm infra isis lindholm markham mincemeat moresby mycenae plugging smokescreen speakeasy```

## C

`#include <stdio.h>#include <stdlib.h> typedef struct {    char *word;    int weight;} item_t; item_t items[] = {    {"alliance",     -624},    {"archbishop",   -915},    {"balm",          397},    {"bonnet",        452},    {"brute",         870},    {"centipede",    -658},    {"cobol",         362},    {"covariate",     590},    {"departure",     952},    {"deploy",         44},    {"diophantine",   645},    {"efferent",       54},    {"elysee",       -326},    {"eradicate",     376},    {"escritoire",    856},    {"exorcism",     -983},    {"fiat",          170},    {"filmy",        -874},    {"flatworm",      503},    {"gestapo",       915},    {"infra",        -847},    {"isis",         -982},    {"lindholm",      999},    {"markham",       475},    {"mincemeat",    -880},    {"moresby",       756},    {"mycenae",       183},    {"plugging",     -266},    {"smokescreen",   423},    {"speakeasy",    -745},    {"vein",          813},}; int n = sizeof (items) / sizeof (item_t);int *set; void subsum (int i, int weight) {    int j;    if (i && !weight) {        for (j = 0; j < i; j++) {            item_t item = items[set[j]];            printf("%s%s", j ? " " : "", items[set[j]].word);        }        printf("\n");    }    for (j = i ? set[i - 1] + 1: 0; j < n; j++) {        set[i] = j;        subsum(i + 1, weight + items[j].weight);    }} int main () {    set = malloc(n * sizeof (int));    subsum(0, 0);    return 0;}`
Output:
```alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy flatworm mincemeat plugging speakeasy
...```

## C#

Translation of: Java
`using System;using System.Collections.Generic; namespace SubsetSum {    class Item {        public Item(string word, int weight) {            Word = word;            Weight = weight;        }         public string Word { get; set; }        public int Weight { get; set; }         public override string ToString() {            return string.Format("({0}, {1})", Word, Weight);        }    }     class Program {        private static readonly List<Item> items = new List<Item>() {            new Item("alliance", -624),            new Item("archbishop", -915),            new Item("balm", 397),            new Item("bonnet", 452),            new Item("brute", 870),            new Item("centipede", -658),            new Item("cobol", 362),            new Item("covariate", 590),            new Item("departure", 952),            new Item("deploy", 44),            new Item("diophantine", 645),            new Item("efferent", 54),            new Item("elysee", -326),            new Item("eradicate", 376),            new Item("escritoire", 856),            new Item("exorcism", -983),            new Item("fiat", 170),            new Item("filmy", -874),            new Item("flatworm", 503),            new Item("gestapo", 915),            new Item("infra", -847),            new Item("isis", -982),            new Item("lindholm", 999),            new Item("markham", 475),            new Item("mincemeat", -880),            new Item("moresby", 756),            new Item("mycenae", 183),            new Item("plugging", -266),            new Item("smokescreen", 423),            new Item("speakeasy", -745),            new Item("vein", 813),        };         private static readonly int n = items.Count;        private static readonly int LIMIT = 5;         private static int[] indices = new int[n];        private static int count = 0;         private static void ZeroSum(int i, int w) {            if (i != 0 && w == 0) {                for (int j = 0; j < i; j++) {                    Console.Write("{0} ", items[indices[j]]);                }                Console.WriteLine("\n");                if (count < LIMIT) count++;                else return;            }            int k = (i != 0) ? indices[i - 1] + 1 : 0;            for (int j = k; j < n; j++) {                indices[i] = j;                ZeroSum(i + 1, w + items[j].Weight);                if (count == LIMIT) return;            }        }         static void Main(string[] args) {            Console.WriteLine("The weights of the following {0} subsets add up to zero:\n", LIMIT);            ZeroSum(0, 0);        }    }}`
Output:
```The weights of the following 5 subsets add up to zero:

(alliance, -624) (archbishop, -915) (balm, 397) (bonnet, 452) (brute, 870) (centipede, -658) (cobol, 362) (covariate, 590) (departure, 952) (deploy, 44) (diophantine, 645) (efferent, 54) (elysee, -326) (eradicate, 376) (escritoire, 856) (exorcism, -983) (fiat, 170) (filmy, -874) (flatworm, 503) (mincemeat, -880) (plugging, -266) (speakeasy, -745)

(alliance, -624) (archbishop, -915) (balm, 397) (bonnet, 452) (brute, 870) (centipede, -658) (cobol, 362) (covariate, 590) (departure, 952) (deploy, 44) (diophantine, 645) (efferent, 54) (elysee, -326) (eradicate, 376) (escritoire, 856) (exorcism, -983) (infra, -847) (isis, -982) (mincemeat, -880) (moresby, 756) (mycenae, 183) (smokescreen, 423) (speakeasy, -745)

(alliance, -624) (archbishop, -915) (balm, 397) (bonnet, 452) (brute, 870) (centipede, -658) (cobol, 362) (covariate, 590) (departure, 952) (deploy, 44) (diophantine, 645) (efferent, 54) (elysee, -326) (eradicate, 376) (escritoire, 856) (fiat, 170) (infra, -847) (isis, -982) (markham, 475) (mincemeat, -880) (plugging, -266) (speakeasy, -745)

(alliance, -624) (archbishop, -915) (balm, 397) (bonnet, 452) (brute, 870) (centipede, -658) (cobol, 362) (covariate, 590) (departure, 952) (deploy, 44) (diophantine, 645) (efferent, 54) (elysee, -326) (eradicate, 376) (exorcism, -983) (fiat, 170) (infra, -847) (isis, -982) (mincemeat, -880) (moresby, 756) (plugging, -266) (vein, 813)

(alliance, -624) (archbishop, -915) (balm, 397) (bonnet, 452) (brute, 870) (centipede, -658) (cobol, 362) (covariate, 590) (departure, 952) (deploy, 44) (diophantine, 645) (efferent, 54) (elysee, -326) (eradicate, 376) (exorcism, -983) (fiat, 170) (infra, -847) (isis, -982) (smokescreen, 423)```

## D

A simple brute-force solution. This used the module of the third D solution of the Combinations Task.

Translation of: Ruby
`void main() {    import std.stdio, std.algorithm, std.typecons, combinations3;     alias P = tuple;    immutable items = [P("alliance",  -624),  P("archbishop",  -915),  P("balm",        397),P("bonnet",     452),  P("brute",        870),  P("centipede",  -658),P("cobol",      362),  P("covariate",    590),  P("departure",   952),P("deploy",      44),  P("diophantine",  645),  P("efferent",     54),P("elysee",    -326),  P("eradicate",    376),  P("escritoire",  856),P("exorcism",  -983),  P("fiat",         170),  P("filmy",      -874),P("flatworm",   503),  P("gestapo",      915),  P("infra",      -847),P("isis",      -982),  P("lindholm",     999),  P("markham",     475),P("mincemeat", -880),  P("moresby",      756),  P("mycenae",     183),P("plugging",  -266),  P("smokescreen",  423),  P("speakeasy",  -745),P("vein",       813)];     foreach (immutable n; 1 .. items.length)        foreach (const comb; items.combinations(n))            if (comb.map!q{ a[1] }.sum == 0)                return writefln("A subset of length %d: %-(%s, %)", n,                                comb.map!q{ a[0] });    "No solution found.".writeln;}`
Output:
`A subset of length 2: archbishop, gestapo`

### Alternative Version

This version prints all the 349_167 solutions in about 1.8 seconds and counts them in about 0.05 seconds.

Translation of: C
`import std.stdio, std.algorithm; enum showAllSolutions = true; struct Item { string data; int weight; }struct Sum { int sum; uint mask; } immutable Item[] em = [    {"alliance",  -624},  {"archbishop",  -915},  {"balm",        397},    {"bonnet",     452},  {"brute",        870},  {"centipede",  -658},    {"cobol",      362},  {"covariate",    590},  {"departure",   952},    {"deploy",      44},  {"diophantine",  645},  {"efferent",     54},    {"elysee",    -326},  {"eradicate",    376},  {"escritoire",  856},    {"exorcism",  -983},  {"fiat",         170},  {"filmy",      -874},    {"flatworm",   503},  {"gestapo",      915},  {"infra",      -847},    {"isis",      -982},  {"lindholm",     999},  {"markham",     475},    {"mincemeat", -880},  {"moresby",      756},  {"mycenae",     183},    {"plugging",  -266},  {"smokescreen",  423},  {"speakeasy",  -745},    {"vein",       813}]; Sum[] mkSums(in Item[] p, in size_t n, in size_t shift) {    auto r = new Sum[1 << n];    foreach (immutable i; 0 .. n)        r[1 << i].sum = p[i].weight;     foreach (immutable i, ref ri; r) {        immutable size_t b = i & -int(i);        ri = Sum(r[i & ~b].sum + r[b].sum, i << shift);    }     return r.sort!q{ a.sum < b.sum }.release;} void showMask(in uint mask) nothrow {    for (size_t m = 0; (1U << m) <= mask; m++)        if (mask & (1U << m))            // Much faster than writeln.            // The names are all zero-terminated.            printf("%s ", em[m].data.ptr);    if (mask)        putchar('\n');} int printList(in int i, in int j, in int i1, in int j1,              in Sum[] l, in Sum[] r) nothrow {    int s = (i1 - i) * (j - j1);    if (!l[i].sum)        s--;     static if (showAllSolutions)        foreach (immutable x; i .. i1)            foreach_reverse (immutable size_t y; j1 + 1 .. j + 1)                showMask(l[x].mask | r[y].mask);    return s;} void main() {    immutable N = em.length;    assert(N <= em[0].sizeof * 8, "Not enough bits in the mask");    immutable size_t n1 = N / 2;    immutable size_t n2 = N - n1;    immutable size_t n1p = 1 << n1;    immutable size_t n2p = 1 << n2;     auto l = mkSums(em[], n1, 0);    auto r = mkSums(em[n1 .. \$], n2, n1);     size_t sols = 0;    int i = 0;    int j = n2p - 1;    while (true) {        while (l[i].sum + r[j].sum) {            while (i < n1p && l[i].sum + r[j].sum < 0)                i++;            while (j >= 0 && l[i].sum + r[j].sum > 0)                j--;            if (i >= n1p || j < 0)                break;        }        if (i >= n1p || j < 0)            break;         int i1 = i + 1;        while (i1 < n1p && l[i1].sum == l[i].sum)            i1++;         int j1 = j - 1;        while (j1 >= 0 && r[j1].sum == r[j].sum)            j1--;         sols += printList(i, j, i1, j1, l, r);        i = i1;        j = j1;    }     writeln("Zero sums: ", sols);}`
Output:
`Zero sums: 349167`

## EchoLisp

### Dynamic programming

We use the Pseudo-polynomial time dynamic programming solution, found in the Subset sum problem Wikipedia article. If A and B are the min and max possible sums, the time and memory needed are O((B-A)*N). Q is an array such as Q(i,s) = true if there is a nonempty subset of x0, ..., xi which sums to s.

` ;; 0 <= i < N , A <= s <  B , -A = abs(A);; mapping two dims Q(i,s) to one-dim Q(qidx(i,s)) : (define-syntax-rule (qidx i s) (+ i (* (+ s -A) N))) ;; filling the Q array with true/false values;; Q(i, s) := Q(i − 1, s) or (xi == s) or Q(i − 1, s − xi),  for A ≤ s < B. (define (fillQ  xs (ds))    (define N (length xs))    (define A (apply + (filter negative? xs)))    (define B (1+ (apply + (filter positive? xs))))    (define -A (abs A))    (define Q (make-vector (* N (- B A))))    (set! xs (list->vector xs))     (printf "Q[%d] allocated." (vector-length Q))    (for ((s (in-range A B)))             (vector-set! Q (qidx 0 s ) (= [xs 0] s)))     (for*   ([i (in-range 1 N)]             [s (in-range A B)])         (set! ds (- s [xs i]))        (vector-set! Q (qidx i s)            (or                [Q (qidx (1- i) s)]                (= [xs i] s)                (and (>= ds A) (< ds B) [Q (qidx (1- i) ds )])))         ;; stop on first zero-sum found             #:break (and (zero? s) [Q (qidx i s)]) => (solQ Q xs i s -A N)        )) ;; backtracking to get the list of i's such as sum([xs i]) = 0;; start from q[i,0] === true  (define  (solQ Q xs i s -A N  (sol null))     (cond        (( = s [xs i]) (cons  i  sol))        ([Q (qidx (1- i ) s)] (solQ Q xs (1- i) s -A N sol))        (else  (solQ Q xs (1- i) (- s [xs i]) -A N (cons i sol))))) (define (task input)    (map  (lambda(i)  (first (list-ref input i))) (fillQ (map rest input))))  `
Output:
```
(define input
'({"alliance" . -624}
{"archbishop" . -915}
{"balm" . 397}
{"bonnet" . 452}
{"brute" . 870}
{"centipede" . -658}
{"cobol" . 362}
{"covariate" . 590}
{"departure" . 952}
{"deploy" . 44}
{"diophantine" . 645}
{"efferent" . 54}
{"elysee" . -326}
{"escritoire" . 856}
{"exorcism" . -983}
{"fiat" . 170}
{"filmy" . -874}
{"flatworm" . 503}
{"gestapo" . 915}
{"infra" . -847}
{"isis" . -982}
{"lindholm" . 999}
{"markham" . 475}
{"mincemeat" . -880}
{"moresby" . 756}
{"mycenae" . 183}
{"plugging" . -266}
{"smokescreen" . 423}
{"speakeasy" . -745}
{"vein" . 813}))

Q[587016] allocated.
→ ("archbishop" "balm" "bonnet" "centipede" "cobol" "covariate"
"deploy" "efferent" "elysee")

(define items
'[-61 1 32 373 311 249 311 32 -92 -185 -433
-402 -247 156 125 249 32 -464 -278 218 32 -123
-216 373 -185 -402 156 -402 -61 -31 902 ])

(map (lambda(i) (list-ref items i)) (fillQ items))

Q[221185] allocated.
→ (-61 32 373 311 249 311 32 -92 -185 -433 -402 -247 156 125 249 32 -
464 -278 218 32 -123 -216 373 -185 -402 156 -402 -61 902)
```

### Brute force

We use the powerset procrastinator which gives in sequence all subsets of the input list.

` (lib 'sequences) ;; for powerset (define (sum0? xs)	(zero? (apply + (map rest xs)))) ;; filter the powerset and ;; take first 5 solutions(for-each writeln (take (filter sum0? (powerset input)) 5)) ()  ;; empty    (("archbishop" . -915) ("balm" . 397) ("bonnet" . 452) ("centipede" . -658) ("cobol" . 362) ("covariate" . 590) ("deploy" . 44) ("efferent" . 54) ("elysee" . -326))    (("archbishop" . -915) ("balm" . 397) ("bonnet" . 452) ("centipede" . -658) ("departure" . 952) ("deploy" . 44) ("efferent" . 54) ("elysee" . -326))     (("alliance" . -624) ("brute" . 870) ("centipede" . -658) ("cobol" . 362) ("elysee" . -326) ("eradicate" . 376))    (("alliance" . -624) ("archbishop" . -915) ("bonnet" . 452) ("centipede" . -658) ("cobol" . 362) ("covariate" . 590) ("deploy" . 44) ("diophantine" . 645) ("efferent" . 54) ("elysee" . -326) ("eradicate" . 376))     `

## FunL

`def subsetSum( s, w, v ) =  def sumset( a ) = foldl1( (+), map(w, a) )   for i <- s.subsets() if i != {}    if sumset( i ) == v      return Some( i )   None s = {  ('alliance', -624),  ('archbishop', -915),  ('balm', 397),  ('bonnet', 452),  ('brute', 870),  ('centipede', -658),  ('cobol', 362),  ('covariate', 590),  ('departure', 952),  ('deploy', 44),  ('diophantine', 645),  ('efferent', 54),  ('elysee', -326),  ('eradicate', 376),  ('escritoire', 856),  ('exorcism', -983),  ('fiat', 170),  ('filmy', -874),  ('flatworm', 503),  ('gestapo', 915),  ('infra', -847),  ('isis', -982),  ('lindholm', 999),  ('markham', 475),  ('mincemeat', -880),  ('moresby', 756),  ('mycenae', 183),  ('plugging', -266),  ('smokescreen', 423),  ('speakeasy', -745),  ('vein', 813)  } for i <- 0..5  println( i, subsetSum(s, snd, i).get() )`
Output:
```0, {(archbishop, -915), (gestapo, 915)}
1, {(fiat, 170), (vein, 813), (isis, -982)}
2, {(alliance, -624), (departure, 952), (elysee, -326)}
3, {(alliance, -624), (archbishop, -915), (departure, 952), (covariate, 590)}
4, {(markham, 475), (infra, -847), (eradicate, 376)}
5, {(flatworm, 503), (eradicate, 376), (filmy, -874)}
```

## Go

`package main import "fmt" type ww struct {    word   string    weight int} var input = []*ww{    {"alliance", -624},    {"archbishop", -915},    {"balm", 397},    {"bonnet", 452},    {"brute", 870},    {"centipede", -658},    {"cobol", 362},    {"covariate", 590},    {"departure", 952},    {"deploy", 44},    {"diophantine", 645},    {"efferent", 54},    {"elysee", -326},    {"eradicate", 376},    {"escritoire", 856},    {"exorcism", -983},    {"fiat", 170},    {"filmy", -874},    {"flatworm", 503},    {"gestapo", 915},    {"infra", -847},    {"isis", -982},    {"lindholm", 999},    {"markham", 475},    {"mincemeat", -880},    {"moresby", 756},    {"mycenae", 183},    {"plugging", -266},    {"smokescreen", 423},    {"speakeasy", -745},    {"vein", 813},} type sss struct {    subset []*ww    sum    int} func main() {    ps := []sss{{nil, 0}}    for _, i := range input {        pl := len(ps)        for j := 0; j < pl; j++ {            subset := append([]*ww{i}, ps[j].subset...)            sum := i.weight + ps[j].sum            if sum == 0 {                fmt.Println("this subset sums to 0:")                for _, i := range subset {                    fmt.Println(*i)                }                return            }            ps = append(ps, sss{subset, sum})        }    }    fmt.Println("no subset sums to 0")}`
Output:
```this subset sums to 0:
{elysee -326}
{efferent 54}
{deploy 44}
{covariate 590}
{cobol 362}
{centipede -658}
{bonnet 452}
{balm 397}
{archbishop -915}
```

`combinations :: Int -> [a] -> [[a]]combinations 0 _      = [[]]combinations _ []     = []combinations k (x:xs) = map (x:) (combinations (k - 1) xs) ++                          combinations k xs data W = W { word   :: String,             weight :: Int } solver :: [W] -> [[W]]solver it = [comb | n <- [1 .. length it],                    comb <- combinations n it,                    sum (map weight comb) == 0] items =  [W "alliance"    (-624),  W "archbishop" (-915),          W "balm"          397,   W "bonnet"       452,          W "brute"         870,   W "centipede"  (-658),          W "cobol"         362,   W "covariate"    590,          W "departure"     952,   W "deploy"        44,          W "diophantine"   645,   W "efferent"      54,          W "elysee"      (-326),  W "eradicate"    376,          W "escritoire"    856,   W "exorcism"   (-983),          W "fiat"          170,   W "filmy"      (-874),          W "flatworm"      503,   W "gestapo"      915,          W "infra"       (-847),  W "isis"       (-982),          W "lindholm"      999,   W "markham"      475,          W "mincemeat"   (-880),  W "moresby"      756,          W "mycenae"       183,   W "plugging"   (-266),          W "smokescreen"   423,   W "speakeasy"  (-745),          W "vein"          813] main = print \$ map word \$ head \$ solver items`
Output:
`["archbishop","gestapo"]`

None bruteforce: the list of numbers used here are different, and difficult for a bruteforce method.

`subsum w = snd.head.filter ((==w).fst).(++[(w,[])]).foldl s [(0,[])]	where	s a x = merge a \$ map f a where f (a,l) = (a+x, l++[x]) 	-- keep list of sums sorted and unique	merge [] a = a	merge a [] = a	merge a@((av,al):as) b@((bv,bl):bs)		| av <  bv  = (av,al):merge as b		| av == bv  = (bv,bl):merge as bs		| otherwise = (bv,bl):merge a bs items = [-61, 1, 32, 373, 311, 249, 311, 32, -92, -185, -433,	-402, -247, 156, 125, 249, 32, -464, -278, 218, 32, -123,	-216, 373, -185, -402, 156, -402, -61, -31, 902	] main = print \$ subsum 0 items`
Output:
```[-61,32,373,311,249,311,32,-92,-185,-433,-402,-247,156,125,249,32,-464,-278,218,32,-123,-216,373,-185,-402,156,-402,-61,902]
```

## Icon and Unicon

Translation of: Ruby
`link printf,lists                        procedure main()   BruteZeroSubset(string2table(        "alliance/-624/archbishop/-915/balm/397/bonnet/452/brute/870/_         centipede/-658/cobol/362/covariate/590/departure/952/deploy/44/_         diophantine/645/efferent/54/elysee/-326/eradicate/376/escritoire/856/_         exorcism/-983/fiat/170/filmy/-874/flatworm/503/gestapo/915/infra/-847/_         isis/-982/lindholm/999/markham/475/mincemeat/-880/moresby/756/_         mycenae/183/plugging/-266/smokescreen/423/speakeasy/-745/vein/813/"))         end procedure BruteZeroSubset(words)                # brute force 1 of each length     every n := 1 to *words do {      every t := tcomb(words,n) do {            # generate combination            every (sum := 0) +:= words[!t]         # sum combination          if sum = 0 then {            printf("A zero-sum subset of length %d : %s\n",n,list2string(sort(t)))            break next                          # found one            }         }         printf("No zero-sum subsets of length %d\n",n)      }end   # helper procedures procedure tcomb(T, i)		    #: Table (key) combinations    local K   every put(K := [],key(T))        # list of keys   every suspend lcomb(K,i)         # return list combs end procedure list2string(L)            #: format list as a string   every (s := "[ ") ||:= !L || " " # reformat as string   return s || "]"end procedure string2table(s,d)         #: format string "k1/v1/.../kn/vn" as table    T := table()   /d := "/"   s ? until pos(0) do       T[1(tab(find(d)),=d)] := numeric(1(tab(find(d)),=d))    return Tend`
Output:
```No zero-sum subsets of length 1
A zero-sum subset of length 2 : [ archbishop gestapo ]
A zero-sum subset of length 3 : [ centipede markham mycenae ]
A zero-sum subset of length 4 : [ alliance balm deploy mycenae ]
A zero-sum subset of length 5 : [ balm eradicate isis markham plugging ]
A zero-sum subset of length 6 : [ archbishop balm escritoire exorcism fiat markham ]
A zero-sum subset of length 7 : [ balm bonnet cobol fiat filmy isis markham ]
A zero-sum subset of length 8 : [ balm bonnet cobol filmy markham mincemeat speakeasy vein ]
A zero-sum subset of length 9 : [ alliance archbishop balm bonnet cobol lindholm markham mincemeat plugging ]
A zero-sum subset of length 10 : [ archbishop balm bonnet cobol filmy gestapo markham mincemeat speakeasy vein ]
A zero-sum subset of length 11 : [ alliance archbishop balm bonnet cobol deploy gestapo isis markham mincemeat moresby ]
A zero-sum subset of length 12 : [ alliance archbishop balm bonnet cobol exorcism fiat lindholm markham mincemeat plugging vein ]
A zero-sum subset of length 13 : [ alliance archbishop balm bonnet brute cobol deploy diophantine exorcism markham mincemeat plugging smokescreen ]
A zero-sum subset of length 14 : [ alliance archbishop balm bonnet centipede cobol diophantine exorcism lindholm markham mincemeat mycenae plugging vein ]
A zero-sum subset of length 15 : [ alliance archbishop balm bonnet cobol diophantine fiat gestapo isis markham mincemeat mycenae plugging speakeasy vein ]
A zero-sum subset of length 16 : [ alliance archbishop balm bonnet brute cobol diophantine eradicate exorcism filmy infra lindholm markham mincemeat plugging vein ]
A zero-sum subset of length 17 : [ alliance archbishop balm bonnet centipede cobol covariate deploy diophantine exorcism filmy lindholm markham mincemeat plugging smokescreen vein ]
A zero-sum subset of length 18 : [ alliance archbishop balm bonnet centipede cobol diophantine eradicate escritoire exorcism filmy gestapo infra markham mincemeat moresby plugging vein ]
A zero-sum subset of length 19 : [ alliance archbishop balm bonnet cobol diophantine efferent exorcism filmy flatworm gestapo infra isis lindholm markham mincemeat moresby plugging vein ]
A zero-sum subset of length 20 : [ alliance archbishop balm bonnet centipede cobol deploy diophantine efferent escritoire exorcism fiat filmy gestapo isis lindholm markham mincemeat plugging vein ]
A zero-sum subset of length 21 : [ alliance archbishop balm bonnet brute centipede cobol covariate deploy diophantine efferent elysee exorcism filmy gestapo infra markham mincemeat moresby plugging vein ]
A zero-sum subset of length 22 : [ alliance archbishop balm bonnet centipede cobol deploy diophantine eradicate escritoire exorcism fiat filmy gestapo isis lindholm markham mincemeat plugging smokescreen speakeasy vein ]
A zero-sum subset of length 23 : [ alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine exorcism filmy flatworm gestapo infra isis markham mincemeat moresby plugging speakeasy vein ]
A zero-sum subset of length 24 : [ alliance archbishop balm bonnet brute centipede cobol departure deploy diophantine efferent escritoire exorcism filmy gestapo infra isis markham mincemeat moresby mycenae plugging speakeasy vein ]
A zero-sum subset of length 25 : [ alliance archbishop balm bonnet brute centipede cobol covariate deploy diophantine efferent elysee eradicate exorcism filmy gestapo infra isis markham mincemeat moresby mycenae plugging smokescreen vein ]
A zero-sum subset of length 26 : [ alliance archbishop balm bonnet centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy gestapo infra isis lindholm markham mincemeat plugging speakeasy vein ]
A zero-sum subset of length 27 : [ alliance archbishop balm bonnet brute centipede covariate departure deploy efferent elysee eradicate escritoire exorcism fiat filmy flatworm infra isis lindholm markham mincemeat moresby mycenae plugging smokescreen speakeasy ]
No zero-sum subsets of length 28
No zero-sum subsets of length 29
No zero-sum subsets of length 30

No zero-sum subsets of length 31```

## J

`text=:0 :0alliance     -624archbishop   -915balm          397bonnet        452brute         870centipede    -658cobol         362covariate     590departure     952deploy         44diophantine   645efferent       54elysee       -326eradicate     376escritoire    856exorcism     -983fiat          170filmy        -874flatworm      503gestapo       915infra        -847isis         -982lindholm      999markham       475mincemeat    -880moresby       756mycenae       183plugging     -266smokescreen   423speakeasy    -745vein          813) words=:{[email protected];:;._2 textnumbs=:+/|:0&".;._2 text`

Implementation:

`wsum0=:4 :0  p=:(#~ 0&<)y  n=:(#~ 0&>)y  poss=: +/@#~2#.inv 2 [email protected]^#  P=:poss p  N=:poss -n  choose=:(1{I.P e. N){P  keep=: [ #~ #&2@#@[ #: choose i.~ ]  ;:inv words #~y e. (p keep P),n keep N)`

`   words wsum0 numbscentipede markham mycenae`

Note also that there are over 300,000 valid solutions here. More than can be comfortably displayed:

`   Ps=: </.~ /:~ (I.P e. N){P   Ns=: </.~ /:~ (I.N e. P){N   +/#@,@{"1 Ps,.Ns349168`

(One of those is the empty solution, but the rest of them are valid.)

## Java

Translation of: Kotlin
`public class SubsetSum {    private static class Item {        private String word;        private int weight;         public Item(String word, int weight) {            this.word = word;            this.weight = weight;        }         @Override        public String toString() {            return String.format("(%s, %d)", word, weight);        }    }     private static Item[] items = new Item[]{        new Item("alliance", -624),        new Item("archbishop", -915),        new Item("balm", 397),        new Item("bonnet", 452),        new Item("brute", 870),        new Item("centipede", -658),        new Item("cobol", 362),        new Item("covariate", 590),        new Item("departure", 952),        new Item("deploy", 44),        new Item("diophantine", 645),        new Item("efferent", 54),        new Item("elysee", -326),        new Item("eradicate", 376),        new Item("escritoire", 856),        new Item("exorcism", -983),        new Item("fiat", 170),        new Item("filmy", -874),        new Item("flatworm", 503),        new Item("gestapo", 915),        new Item("infra", -847),        new Item("isis", -982),        new Item("lindholm", 999),        new Item("markham", 475),        new Item("mincemeat", -880),        new Item("moresby", 756),        new Item("mycenae", 183),        new Item("plugging", -266),        new Item("smokescreen", 423),        new Item("speakeasy", -745),        new Item("vein", 813),    };     private static final int n = items.length;    private static final int[] indices = new int[n];    private static int count = 0;     private static final int LIMIT = 5;     private static void zeroSum(int i, int w) {        if (i != 0 && w == 0) {            for (int j = 0; j < i; ++j) {                System.out.printf("%s ", items[indices[j]]);            }            System.out.println("\n");            if (count < LIMIT) count++;            else return;        }        int k = (i != 0) ? indices[i - 1] + 1 : 0;        for (int j = k; j < n; ++j) {            indices[i] = j;            zeroSum(i + 1, w + items[j].weight);            if (count == LIMIT) return;        }    }     public static void main(String[] args) {        System.out.printf("The weights of the following %d subsets add up to zero:\n\n", LIMIT);        zeroSum(0, 0);    }}`
Output:
```The weights of the following 5 subsets add up to zero:

(alliance, -624) (archbishop, -915) (balm, 397) (bonnet, 452) (brute, 870) (centipede, -658) (cobol, 362) (covariate, 590) (departure, 952) (deploy, 44) (diophantine, 645) (efferent, 54) (elysee, -326) (eradicate, 376) (escritoire, 856) (exorcism, -983) (fiat, 170) (filmy, -874) (flatworm, 503) (mincemeat, -880) (plugging, -266) (speakeasy, -745)

(alliance, -624) (archbishop, -915) (balm, 397) (bonnet, 452) (brute, 870) (centipede, -658) (cobol, 362) (covariate, 590) (departure, 952) (deploy, 44) (diophantine, 645) (efferent, 54) (elysee, -326) (eradicate, 376) (escritoire, 856) (exorcism, -983) (infra, -847) (isis, -982) (mincemeat, -880) (moresby, 756) (mycenae, 183) (smokescreen, 423) (speakeasy, -745)

(alliance, -624) (archbishop, -915) (balm, 397) (bonnet, 452) (brute, 870) (centipede, -658) (cobol, 362) (covariate, 590) (departure, 952) (deploy, 44) (diophantine, 645) (efferent, 54) (elysee, -326) (eradicate, 376) (escritoire, 856) (fiat, 170) (infra, -847) (isis, -982) (markham, 475) (mincemeat, -880) (plugging, -266) (speakeasy, -745)

(alliance, -624) (archbishop, -915) (balm, 397) (bonnet, 452) (brute, 870) (centipede, -658) (cobol, 362) (covariate, 590) (departure, 952) (deploy, 44) (diophantine, 645) (efferent, 54) (elysee, -326) (eradicate, 376) (exorcism, -983) (fiat, 170) (infra, -847) (isis, -982) (mincemeat, -880) (moresby, 756) (plugging, -266) (vein, 813)

(alliance, -624) (archbishop, -915) (balm, 397) (bonnet, 452) (brute, 870) (centipede, -658) (cobol, 362) (covariate, 590) (departure, 952) (deploy, 44) (diophantine, 645) (efferent, 54) (elysee, -326) (eradicate, 376) (exorcism, -983) (fiat, 170) (infra, -847) (isis, -982) (smokescreen, 423)```

## Kotlin

Translation of: C
`// version 1.1.2 class Item(val word: String, val weight: Int) {    override fun toString() = "(\$word \$weight)"} val items = arrayOf(     Item("alliance",   -624),    Item("archbishop", -915),    Item("balm",        397),    Item("bonnet",      452),    Item("brute",       870),    Item("centipede",  -658),    Item("cobol",       362),    Item("covariate",   590),    Item("departure",   952),    Item("deploy",       44),    Item("diophantine", 645),    Item("efferent",     54),    Item("elysee",     -326),    Item("eradicate",   376),    Item("escritoire",  856),    Item("exorcism",   -983),    Item("fiat",        170),    Item("filmy",      -874),    Item("flatworm",    503),    Item("gestapo",     915),    Item("infra",      -847),    Item("isis",       -982),    Item("lindholm",    999),    Item("markham",     475),    Item("mincemeat",  -880),    Item("moresby",     756),    Item("mycenae",     183),    Item("plugging",   -266),    Item("smokescreen", 423),    Item("speakeasy",  -745),    Item("vein",        813)) val n = items.sizeval indices = IntArray(n)var count = 0 const val LIMIT = 5 fun zeroSum(i: Int, w: Int) {    if (i != 0 && w == 0) {        for (j in 0 until i) print("\${items[indices[j]]} ")        println("\n")        if (count < LIMIT) count++ else return    }    val k = if (i != 0) indices[i - 1] + 1 else 0    for (j in k until n) {        indices[i] = j        zeroSum(i + 1, w + items[j].weight)        if (count == LIMIT) return     }} fun main(args: Array<String>) {    println("The weights of the following \$LIMIT subsets add up to zero:\n")    zeroSum(0, 0)}`
Output:
```The weights of the following 5 subsets add up to zero:

(alliance -624) (archbishop -915) (balm 397) (bonnet 452) (brute 870) (centipede -658) (cobol 362) (covariate 590) (departure 952) (deploy 44) (diophantine 645) (efferent 54) (elysee -326) (eradicate 376) (escritoire 856) (exorcism -983) (fiat 170) (filmy -874) (flatworm 503) (mincemeat -880) (plugging -266) (speakeasy -745)

(alliance -624) (archbishop -915) (balm 397) (bonnet 452) (brute 870) (centipede -658) (cobol 362) (covariate 590) (departure 952) (deploy 44) (diophantine 645) (efferent 54) (elysee -326) (eradicate 376) (escritoire 856) (exorcism -983) (infra -847) (isis -982) (mincemeat -880) (moresby 756) (mycenae 183) (smokescreen 423) (speakeasy -745)

(alliance -624) (archbishop -915) (balm 397) (bonnet 452) (brute 870) (centipede -658) (cobol 362) (covariate 590) (departure 952) (deploy 44) (diophantine 645) (efferent 54) (elysee -326) (eradicate 376) (escritoire 856) (fiat 170) (infra -847) (isis -982) (markham 475) (mincemeat -880) (plugging -266) (speakeasy -745)

(alliance -624) (archbishop -915) (balm 397) (bonnet 452) (brute 870) (centipede -658) (cobol 362) (covariate 590) (departure 952) (deploy 44) (diophantine 645) (efferent 54) (elysee -326) (eradicate 376) (exorcism -983) (fiat 170) (infra -847) (isis -982) (mincemeat -880) (moresby 756) (plugging -266) (vein 813)

(alliance -624) (archbishop -915) (balm 397) (bonnet 452) (brute 870) (centipede -658) (cobol 362) (covariate 590) (departure 952) (deploy 44) (diophantine 645) (efferent 54) (elysee -326) (eradicate 376) (exorcism -983) (fiat 170) (infra -847) (isis -982) (smokescreen 423)
```

## Mathematica

`a = {{"alliance", -624}, {"archbishop", -915}, {"balm", 397}, {"bonnet", 452}, {"brute", 870}, {"centipede", -658}, {"cobol", 362}, {"covariate",  590},{"departure", 952}, {"deploy", 44}, {"diophantine", 645}, {"efferent", 54}, {"elysee", -326}, {"eradicate", 376}, {"escritoire", 856}, {"exorcism", -983}, {"fiat", 170}, {"filmy", -874}, {"flatworm", 503}, {"gestapo", 915}, {"infra", -847}, {"isis", -982}, {"lindholm", 999}, {"markham", 475}, {"mincemeat", -880}, {"moresby", 756}, {"mycenae", 183}, {"plugging", -266}, {"smokescreen", 423}, {"speakeasy", -745}, {"vein", 813}}; result = [email protected][ Subsets[a, 7], (Total[#[[;; , 2]]] == 0) &]; Map[ (Print["A zero-sum subset of length ", Length[#],  " : ", #[[;; , 1]]])& , result ]`
```A zero-sum subset of length 2 : {archbishop,gestapo}
A zero-sum subset of length 3 : {centipede,markham,mycenae}
A zero-sum subset of length 3 : {exorcism,fiat,vein}
A zero-sum subset of length 4 : {alliance,balm,deploy,mycenae}
A zero-sum subset of length 4 : {balm,efferent,filmy,smokescreen}
A zero-sum subset of length 4 : {bonnet,elysee,escritoire,isis}
A zero-sum subset of length 4 : {brute,centipede,efferent,plugging}
....```

The above code uses a brute-force approach, but Mathematica includes several solution schemes that can be used to solve this problem. We can cast it as an integer linear programming problem, and thus find the largest or smallest subset sum, or even sums with specific constraints, such as a sum using three negative values and nine positive values.

`a = {{"alliance", -624}, {"archbishop", -915}, {"balm", 397}, {"bonnet", 452}, {"brute", 870}, {"centipede", -658}, {"cobol", 362}, {"covariate",  590},{"departure", 952}, {"deploy", 44}, {"diophantine", 645}, {"efferent", 54}, {"elysee", -326}, {"eradicate", 376}, {"escritoire", 856}, {"exorcism", -983}, {"fiat", 170}, {"filmy", -874}, {"flatworm", 503}, {"gestapo", 915}, {"infra", -847}, {"isis", -982}, {"lindholm", 999}, {"markham", 475}, {"mincemeat", -880}, {"moresby", 756}, {"mycenae", 183}, {"plugging", -266}, {"smokescreen", 423}, {"speakeasy", -745}, {"vein", 813}}; desiredValue = 0;aNames = #[[1]] & /@ a;aValues = #[[2]] & /@ a;aOnes = ConstantArray[1, Length[a]];aZeroOnes = ConstantArray[{0, 1}, Length[a]];Off[LinearProgramming::lpip]; maxSoln =  LinearProgramming[-aOnes, {aValues}, {{desiredValue, 0}}, aZeroOnes, Integers]; Print["Maximal solution: ", Select[Transpose[{maxSoln*aValues, aNames}], #[[1]] != 0 &]]; minSoln =  LinearProgramming[  aOnes, {aValues, aOnes}, {{desiredValue, 0}, {1, 1}}, aZeroOnes, Integers]; Print["Minimal solution: ", Select[Transpose[{minSoln*aValues, aNames}], #[[1]] != 0 &]]; threeNineSoln =  LinearProgramming[  aOnes, {aValues,          Boole[# < 0] & /@ aValues,           Boole[# > 0] & /@ aValues},  {{desiredValue, 0}, {3, 0}, {9, 0}}, aZeroOnes, Integers]; Print["3 -ves, 9 +ves: ", Select[Transpose[{threeNineSoln*aValues, aNames}], #[[1]] != 0 &]]; `
```Maximal solution: {{-624, alliance}, {-915, archbishop}, {397, balm},
{870, brute}, {-658, centipede}, {362, cobol}, {590, covariate},
{44, deploy}, {645, diophantine}, {54, efferent}, {-326, elysee},
{376, eradicate}, {-983, exorcism}, {170, fiat}, {-874, filmy},
{503, flatworm}, {915, gestapo}, {-847, infra}, {-982, isis},
{999, lindholm}, {-880, mincemeat}, {756, moresby}, {183, mycenae},
{-266, plugging}, {423, smokescreen}, {-745, speakeasy}, {813, vein}}

Minimal solution: {{-915, archbishop}, {915, gestapo}}

3 -ves, 9 +ves: {{-915, archbishop}, {397, balm}, {452, bonnet},
{362, cobol}, {44, deploy}, {54, efferent}, {-983, exorcism},
{170, fiat}, {503, flatworm}, {-982, isis}, {475, markham},
{423, smokescreen}}.```

## Modula-2

`MODULE SubsetSum;FROM FormatString IMPORT FormatString;FROM Terminal IMPORT WriteString,WriteLn,ReadChar; TYPE    String = ARRAY[0..63] OF CHAR;    Item = RECORD        word : String;        weight : INTEGER;    END; PROCEDURE WriteItem(self : Item);VAR buf : String;BEGIN    FormatString("(%s, %i)", buf, self.word, self.weight);    WriteString(buf);END WriteItem; CONST N = 31;VAR    items : ARRAY[0..N] OF Item;    indicies : ARRAY[0..N] OF INTEGER;    count : INTEGER;PROCEDURE Init;VAR i : INTEGER;BEGIN    items[0] := Item{"alliance", -624};    items[1] := Item{"archbishop", -915};    items[2] := Item{"balm", 397};    items[3] := Item{"bonnet", 452};    items[4] := Item{"brute", 870};    items[5] := Item{"centipede", -658};    items[6] := Item{"cobol", 362};    items[7] := Item{"covariate", 590};    items[8] := Item{"departure", 952};    items[9] := Item{"deploy", 44};    items[10] := Item{"diophantine", 645};    items[11] := Item{"efferent", 54};    items[12] := Item{"elysee", -326};    items[13] := Item{"eradicate", 376};    items[14] := Item{"escritoire", 856};    items[15] := Item{"exorcism", -983};    items[16] := Item{"fiat", 170};    items[17] := Item{"filmy", -874};    items[18] := Item{"flatworm", 503};    items[19] := Item{"gestapo", 915};    items[20] := Item{"infra", -847};    items[21] := Item{"isis", -982};    items[22] := Item{"lindholm", 999};    items[23] := Item{"markham", 475};    items[24] := Item{"mincemeat", -880};    items[25] := Item{"moresby", 756};    items[26] := Item{"mycenae", 183};    items[27] := Item{"plugging", -266};    items[28] := Item{"smokescreen", 423};    items[29] := Item{"speakeasy", -745};    items[30] := Item{"vein", 813};     count := 0;END Init; CONST LIMIT = 5;PROCEDURE ZeroSum(i,w : INTEGER);VAR j,k : INTEGER;BEGIN    IF (i#0) AND (w=0) THEN        FOR j:=0 TO i-1 DO            WriteItem(items[indicies[j]]);            WriteString(" ");        END;        WriteLn;        WriteString("---------------");        WriteLn;        IF count<LIMIT THEN            INC(count)        ELSE            RETURN;        END;    END;    IF i#0 THEN        k := indicies[i-1]+1;    ELSE        k := 0;    END;    FOR j:=k TO N-1 DO        indicies[i] := j;        ZeroSum(i+1,w+items[j].weight);        IF count=LIMIT THEN RETURN; END;    END;END ZeroSum; VAR buf : ARRAY[0..63] OF CHAR;VAR d : INTEGER;BEGIN    Init;    d := LIMIT;    FormatString("The weights of the following %i subsets add up to zero:\n\n", buf, d);    WriteString(buf);    ZeroSum(0,0);     ReadChar;END SubsetSum.`

## OCaml

Just search randomly until a result is found:

`let d =  [ "alliance", -624;  "archbishop", -915;  "balm", 397;  "bonnet", 452;    "brute", 870;  "centipede", -658;  "cobol", 362;  "covariate", 590;    "departure", 952;  "deploy", 44;  "diophantine", 645;  "efferent", 54;    "elysee", -326;  "eradicate", 376;  "escritoire", 856;  "exorcism", -983;    "fiat", 170;  "filmy", -874;  "flatworm", 503;  "gestapo", 915;    "infra", -847;  "isis", -982;  "lindholm", 999;  "markham", 475;    "mincemeat", -880;  "moresby", 756;  "mycenae", 183;  "plugging", -266;    "smokescreen", 423;  "speakeasy", -745;  "vein", 813; ] let sum = List.fold_left (fun sum (_,w) -> sum + w) 0let p = function [] -> false | lst -> (sum lst) = 0 let take lst set =  let x = List.nth set (Random.int (List.length set)) in  (x::lst, List.filter (fun y -> y <> x) set) let swap (a, b) = (b, a)let pop lst set = swap (take set lst) let () =  Random.self_init ();  let rec aux lst set =    let f =      match lst, set with      | [], _ -> take      | _, [] -> pop      | _ -> if Random.bool () then take else pop    in    let lst, set = f lst set in    if p lst then lst    else aux lst set  in  let res = aux [] d in  List.iter (fun (n,w) -> Printf.printf " %4d\t%s\n" w n) res`

## Perl

Library: ntheory
`use ntheory qw/:all/; my %pairs = (    alliance => -624, archbishop => -915, balm => 397, bonnet => 452,    brute => 870, centipede => -658, cobol => 362, covariate => 590,    departure => 952, deploy => 44, diophantine => 645, efferent => 54,    elysee => -326, eradicate => 376, escritoire => 856, exorcism => -983,    fiat => 170, filmy => -874, flatworm => 503, gestapo => 915,    infra => -847, isis => -982, lindholm => 999, markham => 475,    mincemeat => -880, moresby => 756, mycenae => 183, plugging => -266,    smokescreen => 423, speakeasy => -745, vein => 813 );# sort so we get the same order each timemy @names = sort keys(%pairs);my @weights = @pairs{@names};  # hash slice gives all values in same order foreach my \$n (1 .. @names) {    forcomb {        # Remove the "lastfor, " to get all combinations        lastfor, print "Length \$n: @names[@_]\n" if vecsum(@weights[@_]) == 0;    } @names, \$n;}`

Printing just the first one found for each number of elements:

Output:
```Length 2: archbishop gestapo
Length 3: centipede markham mycenae
Length 4: alliance balm deploy mycenae
Length 5: alliance brute covariate deploy mincemeat
Length 6: alliance archbishop balm deploy gestapo mycenae
Length 7: alliance archbishop bonnet cobol departure exorcism moresby
Length 8: alliance archbishop balm bonnet fiat flatworm isis lindholm
Length 9: alliance archbishop balm bonnet brute covariate eradicate mincemeat plugging
... to length 27 ...
```

We can also use different modules for this brute force method. Assuming the same pairs/names/weights variables:

`use List::Util qw/sum/;use Algorithm::Combinatorics qw/combinations/;foreach my \$n (1 .. @names) {  my \$iter = combinations([0..\$#weights], \$n);  while (my \$c = \$iter->next) {    next if sum(@weights[@\$c]);    print "Length \$n: @names[@\$c]\n";    last;  }}`

## Perl 6

`my @pairs =    alliance => -624, archbishop => -915, balm => 397, bonnet => 452,    brute => 870, centipede => -658, cobol => 362, covariate => 590,    departure => 952, deploy => 44, diophantine => 645, efferent => 54,    elysee => -326, eradicate => 376, escritoire => 856, exorcism => -983,    fiat => 170, filmy => -874, flatworm => 503, gestapo => 915,    infra => -847, isis => -982, lindholm => 999, markham => 475,    mincemeat => -880, moresby => 756, mycenae => 183, plugging => -266,    smokescreen => 423, speakeasy => -745, vein => 813;my @weights = @pairs».value;my %name = @pairs.hash.invert; .say for (1..27).hyper(:3batch).map: -> \$n {    given @weights.combinations(\$n).first({ 0 == [+] @^comb }) {        when .so { "Length \$n: ({.map: {%name{\$_}}})" }        default  { "Length \$n: (none)" }    }}`
Output:
```Length 1: (none)
Length 2: (archbishop gestapo)
Length 3: (centipede markham mycenae)
Length 4: (alliance balm deploy mycenae)
Length 5: (alliance brute covariate deploy mincemeat)
Length 6: (alliance archbishop balm deploy gestapo mycenae)
Length 7: (alliance archbishop bonnet cobol departure exorcism moresby)
Length 8: (alliance archbishop balm bonnet fiat flatworm isis lindholm)
Length 9: (alliance archbishop balm bonnet brute covariate eradicate mincemeat plugging)
Length 10: (alliance archbishop balm bonnet brute centipede cobol departure deploy mincemeat)
Length 11: (alliance archbishop balm bonnet brute centipede cobol departure infra moresby speakeasy)
Length 12: (alliance archbishop balm bonnet brute centipede cobol covariate diophantine efferent elysee infra)
Length 13: (alliance archbishop balm bonnet brute centipede cobol covariate departure efferent eradicate filmy isis)
Length 14: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee filmy markham speakeasy)
Length 15: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee exorcism flatworm infra mycenae)
Length 16: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine elysee exorcism filmy gestapo infra)
Length 17: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine exorcism isis mincemeat mycenae plugging vein)
Length 18: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee exorcism filmy isis mycenae vein)
Length 19: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism fiat infra isis smokescreen)
Length 20: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism gestapo infra isis smokescreen speakeasy)
Length 21: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism flatworm infra lindholm mincemeat plugging speakeasy)
Length 22: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy flatworm mincemeat plugging speakeasy)
Length 23: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism infra isis mincemeat moresby mycenae smokescreen speakeasy)
Length 24: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee exorcism filmy gestapo infra markham mincemeat moresby mycenae plugging smokescreen speakeasy)
Length 25: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine eradicate exorcism fiat filmy flatworm infra isis lindholm markham mincemeat moresby mycenae plugging speakeasy)
Length 26: (alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine elysee eradicate escritoire exorcism fiat filmy gestapo infra isis markham mincemeat mycenae plugging speakeasy vein)
Length 27: (alliance archbishop balm bonnet brute centipede covariate departure deploy efferent elysee eradicate escritoire exorcism fiat filmy flatworm infra isis lindholm markham mincemeat moresby mycenae plugging smokescreen speakeasy)```

## Phix

Simple Brute force

`sequence {words,weights} = columnize({{"alliance",   -624},                                      {"archbishop", -915},                                      {"balm",        397},                                      {"bonnet",      452},                                      {"brute",       870},                                      {"centipede",  -658},                                      {"cobol",       362},                                      {"covariate",   590},                                      {"departure",   952},                                      {"deploy",       44},                                      {"diophantine", 645},                                      {"efferent",     54},                                      {"elysee",     -326},                                      {"eradicate",   376},                                      {"escritoire",  856},                                      {"exorcism",   -983},                                      {"fiat",        170},                                      {"filmy",      -874},                                      {"flatworm",    503},                                      {"gestapo",     915},                                      {"infra",      -847},                                      {"isis",       -982},                                      {"lindholm",    999},                                      {"markham",     475},                                      {"mincemeat",  -880},                                      {"moresby",     756},                                      {"mycenae",     183},                                      {"plugging",   -266},                                      {"smokescreen", 423},                                      {"speakeasy",  -745},                                      {"vein",        813}}) function comb(sequence pool, integer needed, done=0, sequence chosen={})    if needed=0 then    -- got a full set        integer t = 0        for i=1 to length(chosen) do            t += weights[chosen[i]]        end for        if t=0 then            for i=1 to length(chosen) do                chosen[i] = words[chosen[i]]            end for            printf(1,"%d: %s\n",{length(chosen),sprint(chosen)})            return 1         end if    elsif done+needed<=length(pool) then        -- get all combinations with and without the next item:        done += 1        if comb(pool,needed-1,done,append(chosen,pool[done]))        or comb(pool,needed,done,chosen) then            return 1        end if    end if    return 0end function        integer n = length(weights)for i=1 to n do    if comb(tagset(n),i)=0 then        printf(1,"%d: No zero-sum subsets of that length\n",{i})    end ifend for`
Output:
```1: No zero-sum subsets of that length
2: {"archbishop","gestapo"}
3: {"centipede","markham","mycenae"}
4: {"alliance","balm","deploy","mycenae"}
5: {"alliance","brute","covariate","deploy","mincemeat"}
6: {"alliance","archbishop","balm","deploy","gestapo","mycenae"}
7: {"alliance","archbishop","bonnet","cobol","departure","exorcism","moresby"}
8: {"alliance","archbishop","balm","bonnet","fiat","flatworm","isis","lindholm"}
10: {"alliance","archbishop","balm","bonnet","brute","centipede","cobol","departure","deploy","mincemeat"}
11: {"alliance","archbishop","balm","bonnet","brute","centipede","cobol","departure","infra","moresby","speakeasy"}
12: {"alliance","archbishop","balm","bonnet","brute","centipede","cobol","covariate","diophantine","efferent","elysee","infra"}
14: {"alliance","archbishop","balm","bonnet","brute","centipede","cobol","covariate","departure","deploy","elysee","filmy","markham","speakeasy"}
15: {"alliance","archbishop","balm","bonnet","brute","centipede","cobol","covariate","departure","deploy","elysee","exorcism","flatworm","infra","mycenae"}
16: {"alliance","archbishop","balm","bonnet","brute","centipede","cobol","covariate","departure","deploy","diophantine","elysee","exorcism","filmy","gestapo","infra"}
17: {"alliance","archbishop","balm","bonnet","brute","centipede","cobol","covariate","departure","deploy","diophantine","exorcism","isis","mincemeat","mycenae","plugging","vein"}
18: {"alliance","archbishop","balm","bonnet","brute","centipede","cobol","covariate","departure","deploy","diophantine","efferent","elysee","exorcism","filmy","isis","mycenae","vein"}
24: {"alliance","archbishop","balm","bonnet","brute","centipede","cobol","covariate","departure","deploy","diophantine","efferent","elysee","exorcism","filmy","gestapo","infra","markham","mincemeat","moresby","mycenae","plugging","smokescreen","speakeasy"}
28: No zero-sum subsets of that length
29: No zero-sum subsets of that length
30: No zero-sum subsets of that length
31: No zero-sum subsets of that length
```

### Alternative

Using the harder set of weights from Go, and the version 1 approach of Python (modified to omit words and using a dictionary so that fractional weights can be accomodated).
This is significantly faster (near instant, in fact) than an "all possible combinations" approach.
Note that new_dict(tid) has been introduced for this task in 0.8.0, which has not yet been shipped.

Shows the first zero-sum subset found, only.

`constant weights = {-61, 1, 32, 373, 311, 249, 311, 32, -92, -185, -433,                    -402, -247, 156, 125, 249, 32, -464, -278, 218, 32,                     -123, -216, 373, -185, -402, 156, -402, -61, -31, 902 } integer sums = new_dict() for w=1 to length(weights) do    -- make a separate modifiable copy of sums, otherwise    -- it c/would mark sums[weights[w]*{2,3,etc}] as valid,    -- ie there cannot be any w in the getd_by_index(node).    integer s = new_dict(sums)    atom v = weights[w]    if getd_index(v,s)=NULL then setd(v,{w},s) end if     sequence sk = getd_all_keys(s)    for i=1 to length(sk) do        integer node = getd_index(sk[i],sums)        if node!=NULL and getd_index(sk[i]+v,s)=0 then            setd(sk[i]+v,getd_by_index(node,sums)&w,s)        end if    end for    destroy_dict(sums)    sums = s     integer node = getd_index(0,sums)    if node!=0 then        sequence s0 = getd_by_index(node,sums)        atom t = 0  -- (sanity check)        for i=1 to length(s0) do            integer si = s0[i]            t += weights[si]            s0[i] = weights[si]        end for        printf(1,"Total %d for %s\n",{t,sprint(s0)})        exit    end ifend for`
Output:
```Total 0 for {-61,32,373,311,249,311,32,-92,-185,-433,-402,-247,156,125,249,32,-464,-278,218,32,-123,-216,373,-185,-402,156,-402,-61,902}
```

## PicoLisp

`(de *Words   (alliance . -624) (archbishop . -915) (balm . 397) (bonnet . 452)   (brute . 870) (centipede . -658) (cobol . 362) (covariate . 590)   (departure . 952) (deploy . 44) (diophantine . 645) (efferent . 54)   (elysee . -326) (eradicate . 376) (escritoire . 856) (exorcism . -983)   (fiat . 170) (filmy . -874) (flatworm . 503) (gestapo . 915)   (infra . -847) (isis . -982) (lindholm . 999) (markham . 475)   (mincemeat . -880) (moresby . 756) (mycenae . 183) (plugging . -266)   (smokescreen . 423) (speakeasy . -745) (vein . 813) )`

Minimal brute force solution:

`(load "@lib/simul.l")  # For 'subsets' (pick   '((N)      (find '((L) (=0 (sum cdr L)))         (subsets N *Words) ) )   (range 1 (length *Words)) )`
Output:
`-> ((archbishop . -915) (gestapo . 915))`

## Python

### Version 1

`words = { # some values are different from example	"alliance": -624,	"archbishop": -925,	"balm":	397,	"bonnet": 452,		"brute": 870,		"centipede": -658,	"cobol": 362,		"covariate": 590,	"departure": 952,	"deploy": 44,		"diophantine": 645,	"efferent": 54,	"elysee": -326,		"eradicate": 376,	"escritoire": 856,	"exorcism": -983,	"fiat": 170,		"filmy": -874,	"flatworm": 503,	"gestapo": 915,		"infra": -847,	"isis": -982,		"lindholm": 999,	"markham": 475,	"mincemeat": -880,	"moresby": 756,		"mycenae": 183,	"plugging": -266,	"smokescreen": 423,	"speakeasy": -745,	"vein": 813} neg = 0pos = 0for (w,v) in words.iteritems():	if v > 0: pos += v	else:     neg += v sums = [0] * (pos - neg + 1) for (w,v) in words.iteritems():	s = sums[:]	if not s[v - neg]: s[v - neg] = (w,) 	for (i, w2) in enumerate(sums):		if w2 and not s[i + v]:			s[i + v] = w2 + (w,) 	sums = s	if s[-neg]:		for x in s[-neg]:			print(x, words[x])		break`
Output:
```
('mycenae', 183)
('speakeasy', -745)
('bonnet', 452)
('lindholm', 999)
('cobol', 362)
('archbishop', -925)
('elysee', -326)

```

### Brute force

`>>> from itertools import combinations>>> >>> word2weight = {"alliance": -624, "archbishop": -915, "balm": 397, "bonnet": 452,  "brute": 870, "centipede": -658, "cobol": 362, "covariate": 590,  "departure": 952, "deploy": 44, "diophantine": 645, "efferent": 54,  "elysee": -326, "eradicate": 376, "escritoire": 856, "exorcism": -983,  "fiat": 170, "filmy": -874, "flatworm": 503, "gestapo": 915,  "infra": -847, "isis": -982, "lindholm": 999, "markham": 475,  "mincemeat": -880, "moresby": 756, "mycenae": 183, "plugging": -266,  "smokescreen": 423, "speakeasy": -745, "vein": 813}>>> answer = None>>> for r in range(1, len(word2weight)+1):	if not answer:		for comb in combinations(word2weight, r):			if sum(word2weight[w] for w in comb) == 0:				answer = [(w, word2weight[w]) for w in comb]				break  >>> answer[('archbishop', -915), ('gestapo', 915)]`

## Racket

` #lang racket (define words  '([alliance -624] [archbishop -915] [balm 397] [bonnet 452] [brute 870]    [centipede -658] [cobol 362] [covariate 590] [departure 952] [deploy 44]    [diophantine 645] [efferent 54] [elysee -326] [eradicate 376]    [escritoire 856] [exorcism -983] [fiat 170] [filmy -874] [flatworm 503]    [gestapo 915] [infra -847] [isis -982] [lindholm 999] [markham 475]    [mincemeat -880] [moresby 756] [mycenae 183] [plugging -266]    [smokescreen 423] [speakeasy -745] [vein 813])) ;; Simple brute-force solution to find the smallest subset(define (nsubsets l n)  (cond [(zero? n) '(())] [(null? l) '()]        [else (append (for/list ([l2 (nsubsets (cdr l) (- n 1))])                        (cons (car l) l2))                      (nsubsets (cdr l) n))]))(for*/first ([i (sub1 (length words))] [s (nsubsets words (add1 i))]             #:when (zero? (apply + (map cadr s))))  (map car s));; => '(archbishop gestapo) ;; Alternative: customize the subsets to ones with zero sum, abort early;; if we're in a hopeless case (using the fact that weights are <1000)(define (zero-subsets l)  (define (loop l len n r sum)    (cond [(zero? n) (when (zero? sum) (displayln (reverse r)))]          [(and (pair? l) (<= sum (* 1000 n)))           (when (< n len) (loop (cdr l) (sub1 len) n r sum))           (loop (cdr l) (sub1 len) (sub1 n) (cons (caar l) r)                 (+ (cadar l) sum))]))  (define len (length l))  (for ([i (sub1 len)]) (loop l len (add1 i) '() 0)))(zero-subsets words) `
Output:
```'(archbishop gestapo) ; <- the first solution
(archbishop gestapo)
(exorcism fiat vein)
(centipede markham mycenae)
... 43M of printouts ...
(alliance archbishop balm bonnet brute centipede covariate departure deploy efferent elysee eradicate escritoire exorcism fiat filmy flatworm infra isis lindholm markham mincemeat moresby mycenae plugging smokescreen speakeasy)
```

## REXX

This REXX solution isn't limited to integers for the weights.     This isn't a brute force solution.

While optimizing the original program, it was found that sorting the names by weight could yield a vastly
improved algorithm (by an order of magnitude), so the extra code to sort the list was included, as well as
another sort to show the solutions in alphabetical order.   Support was also added to allow specification of
which "chunk" to search for solutions   (that is, out of the 31 names, take a "chunk" at a time).

Showing of the timing (elapsed time) was also added, as well as "que pasa" informational messages.   The
sum   (which is zero for this task)   can be any number, and can be specifiable on the command line.

`/*REXX program finds some  non─null subsets  of a  weighted list  whose  sum eqals zero.*/parse arg  target stopAt chunkette .             /*option optional arguments from the CL*/if target=='' | target==","  then target= 0      /*Not specified?  Then use the default.*/if stopAt=='' | stopAt==","  then stopAt= 1      /* "      "         "   "   "     "    */y=0zzz= 'alliance  -624              archbishop  -915                balm        397'  ,     'bonnet     452              brute        870                centipede  -658'  ,     'cobol      362              covariate    590                departure   952'  ,     'deploy      44              diophantine  645                efferent     54'  ,     'elysee    -326              eradicate    376                escritoire  856'  ,     'exorcism  -983              fiat         170                filmy      -874'  ,     'flatworm   503              gestapo      915                infra      -847'  ,     'isis      -982              lindholm     999                markham     475'  ,     'mincemeat -880              moresby      756                mycenae     183'  ,     'plugging  -266              smokescreen  423                speakeasy  -745'  ,     'vein       813'@.=0                do N=1  until zzz=''             /*construct an array from the ZZZ list.*/                parse var  zzz   @.N  #.N  zzz   /*pick from the list like a nose.      */                end   /*N*/call eSort N                                     /*sort the names with weights.         */call tellZ  'sorted'                             /*display the sorted list.             */chunkStart= 1                                    /*the default place to  start.         */chunkEnd  = N                                    /* "     "      "    "   end.          */if chunkette\==''  then do                       /*solutions just for a chunkette.      */                        chunkStart= chunkette                        chunkEnd  = chunkette                        endcall time 'Reset'                                /*reset the REXX elapsed time.         */??= 0                                            /*the number of solutions  (so far).   */      do chunk=chunkStart  to chunkEnd           /*traipse through the items.           */      call tello center(' doing chunk:'   chunk" ", 79, '─')      call combN N, chunk                        /*N  items,   a  CHUNK  at a time.     */      _= ??;            if _==0  then _= 'No'    /*Englishise for a zero count.         */      call tello _  'solution's(??)     "found so far."      end   /*chunk*/ if ??==0  then ??= 'no'                          /*Englishise the solutions number.     */call tello   'Found'    ??    "subset"s(??)    'whose summed weight's(??)    "="    targetexit                                             /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/combN: procedure expose @. #. ?? stopAt target;       parse arg x,y;        !.= @.0       base= x+1;        bbase= base - y         /*!.n   are the combination digits.    */         do n=1  for y;  !.n=n                   /*construct the first combination.     */         end   /*n*/       ym= y-1         do j=1;       _=!.1;         s= #._     /*obtain the first digit and the sum.  */         if s>target  then leave                 /*Is  1st dig>target?  Then we're done.*/           do k=2  for ym;   _= !.k;  s= s + #._ /*Σ the weights;  is sum > target ?    */           if s>target  then do;      if .combUp(k-1)  then return;    iterate j;    end           end   /*k*/         if s==target  then call telly           /*have we found a pot of gold?         */         !.y= !.y + 1;   if !.y==base  then  if .combUp(ym)  then leave    /*bump digit.*/         end      /*j*/;               return    /*done with this combination set.      *//*──────────────────────────────────────────────────────────────────────────────────────*/.combUp: procedure expose !. y bbase;  parse arg d;        if d==0  then return 1         p= !.d;   do u=d  to y;      !.u=p + 1  /*add one to digit we're pointing at.  */                   if !.u >= bbase+u  then return .combUp(u-1)                   p= !.u                        /*P   will be used for the next digt.  */                   end   /*u*/;       return 0   /*go back and sum this combination.    *//*──────────────────────────────────────────────────────────────────────────────────────*/eSort: procedure expose #. @. \$.;          parse arg N,\$;              h=N         do  while h>1;                    h= h%2           do i=1  for  N-h;     j=i;      k= h+i           if \$==. then do while \$.k<\$.j;  parse value \$.j \$.k         with \$.k \$.j                        if h>=j  then leave;     j= j-h;               k= k-h                        end   /*while \$.k<\$.j*/                   else do while #.k<#.j; parse value @.j @.k #.j #.k with @.k @.j #.k #.j                        if h>=j  then leave;     j= j-h;               k= k-h                        end   /*while #.k<#.j*/           end   /*i*/         end     /*while h>1*/;            return/*──────────────────────────────────────────────────────────────────────────────────────*/s:     if arg(1)==1  then return arg(3);  return word(arg(2) 's',1)  /*simple pluralizer*/tello: parse arg _,e;  if e==.  then say;  say _;  call lineout 'SUBSET.'y, _;      return/*──────────────────────────────────────────────────────────────────────────────────────*/telly: ??= ??+1;                     nameL=      /*start with a  "null"  name list.     */                 do gi=1  for y;     ggg= !.gi   /*build duplicate array (to be sorted).*/                 \$.gi= @.ggg                     /*transform from  index ──►  a name.   */                 end   /*gi*/                    /*build duplicate array (to be sorted).*/       call eSort y, .                           /*sort the names alphabetically.       */         do gs=1  for y;   nameL= nameL \$.gs     /*build a list of names whose  sum = 0 */         end   /*gs*/                            /*the list of names could be sorted.   */       call tello  '['y"   name"s(y)']'      space(nameL)       if ??<stopAt | stopAt==0  then return     /*see if we reached a  (or the)  limit.*/       call tello 'Stopped after finding '   ??   " subset"s(??)'.', .       exit                                      /*a short─timer,  we should quit then. *//*──────────────────────────────────────────────────────────────────────────────────────*/tellz:             do j=1  for N                 /*show a list of names and weights.    */                   call tello  right('['j']', 30)       right(@.j, 11)       right(#.j, 5)                   end   /*j*/       call tello       call tello    'There are  '     N     " entries in the (above)"   arg(1)   'table.'       call tello;                  return`

Output note:   this program also writes the displayed output to file(s):   SUBSET.nnn
──────── where   nnn   is the chunk number.

output  when using the input of:     0   12

(The above arguments set the target sum to zero, and limits the finding of a dozen solutions.)

```                           [1]    exorcism  -983
[2]        isis  -982
[3]  archbishop  -915
[4]   mincemeat  -880
[5]       filmy  -874
[6]       infra  -847
[7]   speakeasy  -745
[8]   centipede  -658
[9]    alliance  -624
[10]      elysee  -326
[11]    plugging  -266
[12]      deploy    44
[13]    efferent    54
[14]        fiat   170
[15]     mycenae   183
[16]       cobol   362
[18]        balm   397
[19] smokescreen   423
[20]      bonnet   452
[21]     markham   475
[22]    flatworm   503
[23]   covariate   590
[24] diophantine   645
[25]     moresby   756
[26]        vein   813
[27]  escritoire   856
[28]       brute   870
[29]     gestapo   915
[30]   departure   952
[31]    lindholm   999

There are   31  entries in the (above) sorted table.

─────────────────────────────── doing chunk: 1 ────────────────────────────────
No solutions found so far.
─────────────────────────────── doing chunk: 2 ────────────────────────────────
[2   names] archbishop gestapo
1 solution found so far.
─────────────────────────────── doing chunk: 3 ────────────────────────────────
[3   names] exorcism fiat vein
[3   names] centipede markham mycenae
3 solutions found so far.
─────────────────────────────── doing chunk: 4 ────────────────────────────────
[4   names] exorcism gestapo speakeasy vein
[4   names] deploy exorcism moresby mycenae
[4   names] bonnet elysee escritoire isis
[4   names] eradicate isis mycenae smokescreen
[4   names] balm efferent filmy smokescreen
[4   names] centipede covariate gestapo infra
[4   names] centipede covariate speakeasy vein
[4   names] brute centipede efferent plugging
[4   names] alliance balm deploy mycenae

Stopped after finding  12  subsets.
```

## Ring

` # Project : Subset sum problem knap = [["alliance", -624],            ["archbishop", -915],            ["balm", 397],            ["bonnet", 452],            ["brute", 870],            ["centipede", -658],            ["cobol", 362],            ["covariate", 590],            ["departure", 952],            ["deploy", 44],            ["diophantine", 645],            ["efferent", 54],            ["elysee", -326],            ["eradicate", 376],            ["escritoire", 856],            ["exorcism", -983],            ["fiat", 170],            ["filmy", -874],            ["flatworm", 503],            ["gestapo", 915],            ["infra", -847],            ["isis", -982],            ["lindholm", 999],            ["markham", 475],            ["mincemeat", -880],            ["moresby", 756],            ["mycenae", 183],            ["plugging", -266],            ["smokescreen", 423],            ["speakeasy", -745],            ["vein", 813]] knapsack = createDimList([pow(2, len(knap)),len(knap)+2])knapweight = createDimList([pow(2, len(knap)),len(knap)+2])lenknap = list(pow(2, len(knap))) powerset(knap)  func powerset(list)        n1 = 0        num = 0        for i = 2 to (2 << len(list)) - 1 step 2             n2 = 0             n1 = n1 + 1             weight = 0             for j = 1 to len(list)                   if i & (1 << j)                     n2 = n2 + 1                     knapsack[n1][n2] = list[j][1]                     weight = weight + list[j][2]                      knapweight[n1][n2] = list[j][2]                  ok             next             lenknap[n1] = n2+1             if weight = 0               see "" + num + ": "                               for p = 1 to lenknap[n1]-1                      see "{" + knapsack[n1][p] + " " + knapweight[n1][p]+ "}"                next                see nl                num = num + 1             ok         next func createDimList(dimArray)        sizeList = len(dimArray)        newParms = []        for i = 2 to sizeList            Add(newParms, dimArray[i])         next              alist = list(dimArray[1])        if sizeList = 1           return aList        ok        for t in alist              t = createDimList(newParms)        next               return alist `

Output:

```0: {(archbishop, -915), (gestapo, 915)}
1: {(fiat, 170), (vein, 813), (isis, -982)}
2: {(alliance, -624), (departure, 952), (elysee, -326)}
3: {(alliance, -624), (archbishop, -915), (departure, 952), (covariate, 590)}
4: {(markham, 475), (infra, -847), (eradicate, 376)}
5: {(flatworm, 503), (eradicate, 376), (filmy, -874)}
```

## Ruby

a brute force solution:

`weights = {  'alliance'   =>-624, 'archbishop'=>-915, 'balm'       => 397, 'bonnet'   => 452,  'brute'      => 870, 'centipede' =>-658, 'cobol'      => 362, 'covariate'=> 590,  'departure'  => 952, 'deploy'    =>  44, 'diophantine'=> 645, 'efferent' =>  54,  'elysee'     =>-326, 'eradicate' => 376, 'escritoire' => 856, 'exorcism' =>-983,  'fiat'       => 170, 'filmy'     =>-874, 'flatworm'   => 503, 'gestapo'  => 915,  'infra'      =>-847, 'isis'      =>-982, 'lindholm'   => 999, 'markham'  => 475,  'mincemeat'  =>-880, 'moresby'   => 756, 'mycenae'    => 183, 'plugging' =>-266,  'smokescreen'=> 423, 'speakeasy' =>-745, 'vein'       => 813,} words = weights.keys1.upto(words.length) do |n|  zerosum = words.combination(n).find do |subset|    subset.reduce(0) {|sum, word| sum + weights[word]} == 0  end   if zerosum    puts "a subset of length #{n} that sums to zero: #{zerosum}"  else    puts "no subsets of length #{n} sum to zero"  endend`
Output:
```no subsets of length 1 sum to zero
a subset of length 2 that sums to zero: ["archbishop", "gestapo"]
a subset of length 3 that sums to zero: ["centipede", "markham", "mycenae"]
a subset of length 4 that sums to zero: ["alliance", "balm", "deploy", "mycenae"]
a subset of length 5 that sums to zero: ["alliance", "brute", "covariate", "deploy", "mincemeat"]
a subset of length 6 that sums to zero: ["alliance", "archbishop", "balm", "deploy", "gestapo", "mycenae"]
a subset of length 7 that sums to zero: ["alliance", "archbishop", "bonnet", "cobol", "departure", "exorcism", "moresby"]
a subset of length 8 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "fiat", "flatworm", "isis", "lindholm"]
a subset of length 9 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "covariate", "eradicate", "mincemeat", "plugging"]
a subset of length 10 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "departure", "deploy", "mincemeat"]
a subset of length 11 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "departure", "infra", "moresby", "speakeasy"]
a subset of length 12 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "diophantine", "efferent", "elysee", "infra"]
a subset of length 13 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "efferent", "eradicate", "filmy", "isis"]
a subset of length 14 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "elysee", "filmy", "markham", "speakeasy"]
a subset of length 15 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "elysee", "exorcism", "flatworm", "infra", "mycenae"]
a subset of length 16 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "elysee", "exorcism", "filmy", "gestapo", "infra"]
a subset of length 17 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "exorcism", "isis", "mincemeat", "mycenae", "plugging", "vein"]
a subset of length 18 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "exorcism", "filmy", "isis", "mycenae", "vein"]
a subset of length 19 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "exorcism", "fiat", "infra", "isis", "smokescreen"]
a subset of length 20 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "exorcism", "gestapo", "infra", "isis", "smokescreen", "speakeasy"]
a subset of length 21 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "exorcism", "flatworm", "infra", "lindholm", "mincemeat", "plugging", "speakeasy"]
a subset of length 22 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "escritoire", "exorcism", "fiat", "filmy", "flatworm", "mincemeat", "plugging", "speakeasy"]
a subset of length 23 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "escritoire", "exorcism", "infra", "isis", "mincemeat", "moresby", "mycenae", "smokescreen", "speakeasy"]
a subset of length 24 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "exorcism", "filmy", "gestapo", "infra", "markham", "mincemeat", "moresby", "mycenae", "plugging", "smokescreen", "speakeasy"]
a subset of length 25 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "eradicate", "exorcism", "fiat", "filmy", "flatworm", "infra", "isis", "lindholm", "markham", "mincemeat", "moresby", "mycenae", "plugging", "speakeasy"]
a subset of length 26 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "elysee", "eradicate", "escritoire", "exorcism", "fiat", "filmy", "gestapo", "infra", "isis", "markham", "mincemeat", "mycenae", "plugging", "speakeasy", "vein"]
a subset of length 27 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "covariate", "departure", "deploy", "efferent", "elysee", "eradicate", "escritoire", "exorcism", "fiat", "filmy", "flatworm", "infra", "isis", "lindholm", "markham", "mincemeat", "moresby", "mycenae", "plugging", "smokescreen", "speakeasy"]
no subsets of length 28 sum to zero
no subsets of length 29 sum to zero
no subsets of length 30 sum to zero
no subsets of length 31 sum to zero```

## Scala

Output:
Best seen running in your browser by Scastie (remote JVM).
`object SubsetSum extends App {  private val LIMIT = 5  private val n = items.length  private val indices = new Array[Int](n)  private var count = 0   private def items = Seq(    Item("alliance", -624),    Item("archbishop", -915),    Item("balm", 397),    Item("bonnet", 452),    Item("brute", 870),    Item("centipede", -658),    Item("cobol", 362),    Item("covariate", 590),    Item("departure", 952),    Item("deploy", 44),    Item("diophantine", 645),    Item("efferent", 54),    Item("elysee", -326),    Item("eradicate", 376),    Item("escritoire", 856),    Item("exorcism", -983),    Item("fiat", 170),    Item("filmy", -874),    Item("flatworm", 503),    Item("gestapo", 915),    Item("infra", -847),    Item("isis", -982),    Item("lindholm", 999),    Item("markham", 475),    Item("mincemeat", -880),    Item("moresby", 756),    Item("mycenae", 183),    Item("plugging", -266),    Item("smokescreen", 423),    Item("speakeasy", -745),    Item("vein", 813)  )   private def zeroSum(i: Int, w: Int): Unit = {    if (count < LIMIT) {      if (i != 0 && w == 0) {        for (j <- 0 until i) print(f"\${items(indices(j))}%s ")        println        count += 1      } else        for (j <- (if (i != 0) indices(i - 1) + 1 else 0) until n) {          indices(i) = j          zeroSum(i + 1, w + items(j).weight)        }    }  }   // Not optimized  private case class Item(word: String, weight: Int) {    override def toString: String = f"(\$word%s, \$weight%d)"  }   println(f"The weights of the following \$LIMIT%d subsets add up to zero:")  zeroSum(0, 0) }`

## Sidef

`var pairs = Hash(    alliance    => -624, archbishop => -915,    brute       =>  870, centipede  => -658,    departure   =>  952, deploy     =>   44,    elysee      => -326, eradicate  =>  376,    fiat        =>  170, filmy      => -874,    infra       => -847, isis       => -982,    mincemeat   => -880, moresby    =>  756,    smokescreen =>  423, speakeasy  => -745,    balm        =>  397, bonnet     =>  452,    cobol       =>  362, covariate  =>  590,    diophantine =>  645, efferent   =>   54,    escritoire  =>  856, exorcism   => -983,    flatworm    =>  503, gestapo    =>  915,    lindholm    =>  999, markham    =>  475,    mycenae     =>  183, plugging   => -266,    vein        =>  813,) var weights = pairs.keys.sort.map{|k| pairs{k} }var inverse = pairs.flip for n in (1 .. weights.end) {    var found = false    weights.combinations(n, {|*comb|        if (comb.sum == 0) {            say "Length #{n}: "+" ".join(inverse{comb...})            found = true            break        }    })    found || say "Length #{n}: (none)"}`
Output:
```Length 1: (none)
Length 2: archbishop gestapo
Length 3: centipede markham mycenae
Length 4: alliance balm deploy mycenae
Length 5: alliance brute covariate deploy mincemeat
Length 6: alliance archbishop balm deploy gestapo mycenae
Length 7: alliance archbishop bonnet cobol departure exorcism moresby
Length 8: alliance archbishop balm bonnet fiat flatworm isis lindholm
Length 9: alliance archbishop balm bonnet brute covariate eradicate mincemeat plugging
Length 10: alliance archbishop balm bonnet brute centipede cobol departure deploy mincemeat
Length 11: alliance archbishop balm bonnet brute centipede cobol departure infra moresby speakeasy
Length 12: alliance archbishop balm bonnet brute centipede cobol covariate diophantine efferent elysee infra
Length 13: alliance archbishop balm bonnet brute centipede cobol covariate departure efferent eradicate filmy isis
Length 14: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee filmy markham speakeasy
Length 15: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee exorcism flatworm infra mycenae
Length 16: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine elysee exorcism filmy gestapo infra
Length 17: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine exorcism isis mincemeat mycenae plugging vein
Length 18: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee exorcism filmy isis mycenae vein
Length 19: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism fiat infra isis smokescreen
Length 20: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism gestapo infra isis smokescreen speakeasy
Length 21: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism flatworm infra lindholm mincemeat plugging speakeasy
Length 22: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy flatworm mincemeat plugging speakeasy
Length 23: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism infra isis mincemeat moresby mycenae smokescreen speakeasy
Length 24: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee exorcism filmy gestapo infra markham mincemeat moresby mycenae plugging smokescreen speakeasy
Length 25: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine eradicate exorcism fiat filmy flatworm infra isis lindholm markham mincemeat moresby mycenae plugging speakeasy
Length 26: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine elysee eradicate escritoire exorcism fiat filmy gestapo infra isis markham mincemeat mycenae plugging speakeasy vein
Length 27: alliance archbishop balm bonnet brute centipede covariate departure deploy efferent elysee eradicate escritoire exorcism fiat filmy flatworm infra isis lindholm markham mincemeat moresby mycenae plugging smokescreen speakeasy
Length 28: (none)
Length 29: (none)
Length 30: (none)```

## Tcl

As it turns out that the problem space has small subsets that sum to zero, it is more efficient to enumerate subsets in order of their size rather than doing a simple combination search. This is not true of all possible input data sets though; the problem is known to be NP-complete after all.

`proc subsetsOfSize {set size} {    if {\$size <= 0} {	return    } elseif {\$size == 1} {	foreach elem \$set {lappend result [list \$elem]}    } else {	incr size [set i -1]	foreach elem \$set {	    foreach sub [subsetsOfSize [lreplace \$set [incr i] \$i] \$size] {		lappend result [lappend sub \$elem]	    }	}    }    return \$result}proc searchForSubset {wordweights {minsize 1}} {    set words [dict keys \$wordweights]    for {set i \$minsize} {\$i < [llength \$words]} {incr i} {	foreach subset [subsetsOfSize \$words \$i] {	    set w 0	    foreach elem \$subset {incr w [dict get \$wordweights \$elem]}	    if {!\$w} {return \$subset}	}    }    # Nothing was found    return -code error "no subset sums to zero"}`

Demonstrating:

`set wordweights {    alliance	 -624    archbishop	 -915    balm	 397    bonnet	 452    brute	 870    centipede	 -658    cobol	 362    covariate	 590    departure	 952    deploy	 44    diophantine	 645    efferent	 54    elysee	 -326    eradicate	 376    escritoire	 856    exorcism	 -983    fiat	 170    filmy	 -874    flatworm	 503    gestapo	 915    infra	 -847    isis	 -982    lindholm	 999    markham	 475    mincemeat	 -880    moresby	 756    mycenae	 183    plugging	 -266    smokescreen	 423    speakeasy	 -745    vein	 813}set zsss [searchForSubset \$wordweights]puts "Found zero-summing subset: [join [lsort \$zsss] {, }]"`
Output:
```Found zero-summing subset: archbishop, gestapo
```

## Ursala

This solution scans the set sequentially while maintaining a record of all distinct sums obtainable by words encountered thus far, and stops when a zero sum is found.

`#import std#import int weights = {   'alliance': -624,   'archbishop': -915,   'balm': 397,   'bonnet': 452,   'brute': 870,   'centipede': -658,   'cobol': 362,   'covariate': 590,   'departure': 952,   'deploy': 44,   'diophantine': 645,   'efferent': 54,   'elysee': -326,   'eradicate': 376,   'escritoire': 856,   'exorcism': -983,   'fiat': 170,   'filmy': -874,   'flatworm': 503,   'gestapo': 915,   'infra': -847,   'isis': -982,   'lindholm': 999,   'markham': 475,   'mincemeat': -880,   'moresby': 756,   'mycenae': 183,   'plugging': -266,   'smokescreen': 423,   'speakeasy': -745,   'vein': 813} nullset = ~&nZFihmPB+ =><> ~&ng?r\~&r ^TnK2hS\~&r ^C/~&lmPlNCX *D ^A/[email protected] ~&lrmPC #cast %zm main = nullset weights`

The name of the function that takes the weighted set is `nullset`. It manipulates a partial result represented as a list of pairs, each containing a subset of weighted words and the sum of their weights. Here is a rough translation:

• `=><>` fold right combinator with the empty list as the vacuuous case
• `~&ng?r\~&r` If the partial result contains a zero sum, return it.
• `^TnK2hS\~&r` Concatenate the partial result with the new list of subsets (computed as follows) and delete duplicate sums.
• `^C/~&lmPlNCX` Cons a singleton subset containing the next word to the partial results.
• `*D` Distribute the next word in the set to the partial results and do the following to each.
• `[email protected]` Add the weight of the new word to the existing sum.
• `~&lrmPC` Cons the new word to the list of existing ones.
• `~&nZFihmPB+` To conclude, search for a result with a zero sum, if any, and return its associated subset of weighted words.
Output:
```<
'flatworm': 503,
'gestapo': 915,
'infra': -847,
'isis': -982,
'lindholm': 999,
'plugging': -266,
'smokescreen': 423,
'speakeasy': -745>
```

## zkl

Translation of: C
`var items=T(T("alliance",  -624),  T("archbishop",  -915),  T("balm",        397),T("bonnet",     452),  T("brute",        870),  T("centipede",  -658),T("cobol",      362),  T("covariate",    590),  T("departure",   952),T("deploy",      44),  T("diophantine",  645),  T("efferent",     54),T("elysee",    -326),  T("eradicate",    376),  T("escritoire",  856),T("exorcism",  -983),  T("fiat",         170),  T("filmy",      -874),T("flatworm",   503),  T("gestapo",      915),  T("infra",      -847),T("isis",      -982),  T("lindholm",     999),  T("markham",     475),T("mincemeat", -880),  T("moresby",      756),  T("mycenae",     183),T("plugging",  -266),  T("smokescreen",  423),  T("speakeasy",  -745),T("vein",       813)); fcn subSum(set,i,weight){   if(i and not weight){      itms:=i.pump(List,'wrap(n){ items[set[n]][0] });      println(itms.len(),": ",itms.concat(","));      throw(Exception.TheEnd);   }   foreach j in ([i and set[i-1] + 1 or 0 .. items.len()-1]){      set[i]=j;      self.fcn(set, i+1, weight + items[j][1]);   }} set:=List.createLong(items.len(),0);try{ subSum(set,0,0); }catch(TheEnd){}`
Output:
```22: alliance,archbishop,balm,bonnet,brute,centipede,cobol,covariate,departure,deploy,diophantine,efferent,elysee,eradicate,escritoire,exorcism,fiat,filmy,flatworm,mincemeat,plugging,speakeasy
```