Sum to 100
- Task
Find solutions to the sum to one hundred puzzle.
Add (insert) the mathematical
operators + or ─ (plus
or minus) before any of the digits in the
decimal numeric string 123456789 such that the
resulting mathematical expression adds up to a
particular sum (in this iconic case, 100).
Example:
123 + 4 - 5 + 67 - 89 = 100
Show all output here.
- Show all solutions that sum to 100
- Show the sum that has the maximum number of solutions (from zero to infinity*)
- Show the lowest positive sum that can't be expressed (has no solutions), using the rules for this task
- Show the ten highest numbers that can be expressed using the rules for this task (extra credit)
An example of a sum that can't be expressed (within the rules of this task) is: 5074
which, of course, is not the lowest positive sum that can't be expressed.
* (where infinity would be a relatively small 123,456,789)
Haskell
<lang Haskell>import Data.Function (on) import Control.Arrow ((&&&)) import Data.Char (intToDigit) import Control.Monad (replicateM) import Data.List (nub, group, sort, sortBy, find, intercalate)
data Sign
= Unsigned | Plus | Minus deriving (Eq, Show)
universe :: (Int, Sign) universe =
zip [1 .. 9] <$> filter ((/= Plus) . head) (replicateM 9 [Unsigned, Plus, Minus])
allNonNegativeSums :: [Int] allNonNegativeSums = sort $ filter (>= 0) (asSum <$> universe)
uniqueNonNegativeSums :: [Int] uniqueNonNegativeSums = nub allNonNegativeSums
asSum :: [(Int, Sign)] -> Int asSum xs =
n +
(if null s
then 0
else read s :: Int)
where
(n, s) = foldr readSign (0, []) xs
readSign :: (Int, Sign) -> (Int, String) -> (Int, String)
readSign (i, x) (n, s)
| x == Unsigned = (n, intToDigit i : s)
| otherwise =
( (if x == Plus
then (+)
else (-))
n
(read (show i ++ s) :: Int)
, [])
asString :: [(Int, Sign)] -> String asString = foldr signedDigit []
where
signedDigit (i, x) s
| x == Unsigned = intToDigit i : s
| otherwise =
(if x == Plus
then " +"
else " -") ++
[intToDigit i] ++ s
main :: IO () main =
mapM_
putStrLn
[ "Sums to 100:\n"
, unlines $ asString <$> filter ((== 100) . asSum) universe
, "\n10 commonest sums [sum, number of routes to it]:\n"
, show
((head &&& length) <$>
take 10 (sortBy (on (flip compare) length) (group allNonNegativeSums)))
, "\nFirst positive integer not expressible as a sum of this kind:\n"
, maybeReport (find (uncurry (/=)) (zip [0 ..] uniqueNonNegativeSums))
, "\n10 largest sums:\n"
, show $ take 10 $ sortBy (flip compare) uniqueNonNegativeSums
, "\n"
]
where
maybeReport
:: Show a
=> Maybe (a, b) -> String
maybeReport (Just (x, _)) = show x
maybeReport _ = "No gaps found"</lang>
- Output:
(Run in Atom editor, through Script package)
Sums to 100: 123 +45 -67 +8 -9 123 +4 -5 +67 -89 123 -45 -67 +89 123 -4 -5 -6 -7 +8 -9 12 +3 +4 +5 -6 -7 +89 12 +3 -4 +5 +67 +8 +9 12 -3 -4 +5 -6 +7 +89 1 +23 -4 +56 +7 +8 +9 1 +23 -4 +5 +6 +78 -9 1 +2 +34 -5 +67 -8 +9 1 +2 +3 -4 +5 +6 +78 +9 -1 +2 -3 +4 +5 +6 +78 +9 10 commonest sums [sum, number of routes to it]: [(9,46),(27,44),(1,43),(15,43),(21,43),(45,42),(3,41),(5,40),(7,39),(17,39)] First positive integer not expressible as a sum of this kind: 211 10 largest sums: [123456789,23456790,23456788,12345687,12345669,3456801,3456792,3456790,3456788,3456786] [Finished in 1.237s]
JavaScript
ES5
<lang JavaScript>(function () {
'use strict';
// GENERIC FUNCTIONS ----------------------------------------------------
// permutationsWithRepetition :: Int -> [a] -> a var permutationsWithRepetition = function (n, as) { return as.length > 0 ? ( foldl1(curry(cartesianProduct)(as), replicate(n, as)) ) : []; };
// cartesianProduct :: [a] -> [b] -> a, b var cartesianProduct = function (xs, ys) { return [].concat.apply([], xs.map(function (x) { return [].concat.apply([], ys.map(function (y) { return [ [x].concat(y) ]; })); })); };
// curry :: ((a, b) -> c) -> a -> b -> c
var curry = function (f) {
return function (a) {
return function (b) {
return f(a, b);
};
};
};
// foldl1 :: (a -> a -> a) -> [a] -> a
var foldl1 = function (f, xs) {
return xs.length > 0 ? xs.slice(1)
.reduce(f, xs[0]) : [];
};
// replicate :: Int -> a -> [a]
var replicate = function (n, a) {
var v = [a],
o = [];
if (n < 1) return o;
while (n > 1) {
if (n & 1) o = o.concat(v);
n >>= 1;
v = v.concat(v);
}
return o.concat(v);
};
// group :: Eq a => [a] -> a var group = function (xs) { return groupBy(function (a, b) { return a === b; }, xs); };
// groupBy :: (a -> a -> Bool) -> [a] -> a var groupBy = function (f, xs) { var dct = xs.slice(1) .reduce(function (a, x) { var h = a.active.length > 0 ? a.active[0] : undefined, blnGroup = h !== undefined && f(h, x);
return {
active: blnGroup ? a.active.concat(x) : [x],
sofar: blnGroup ? a.sofar : a.sofar.concat([a.active])
};
}, {
active: xs.length > 0 ? [xs[0]] : [],
sofar: []
});
return dct.sofar.concat(dct.active.length > 0 ? [dct.active] : []);
};
// sort :: Ord a => [a] -> [a]
var sort = function (xs) {
return xs.sort();
};
// sortBy :: (a -> a -> Ordering) -> [a] -> [a]
var sortBy = function (f, xs) {
return xs.sort(f);
};
// nub :: [a] -> [a]
var nub = function (xs) {
return nubBy(function (a, b) {
return a === b;
}, xs);
};
// nubBy :: (a -> a -> Bool) -> [a] -> [a]
var nubBy = function (p, xs) {
var x = xs.length ? xs[0] : undefined;
return x !== undefined ? [x].concat(nubBy(p, xs.slice(1)
.filter(function (y) {
return !p(x, y);
}))) : [];
};
// find :: (a -> Bool) -> [a] -> Maybe a
var find = function (f, xs) {
for (var i = 0, lng = xs.length; i < lng; i++) {
if (f(xs[i], i)) return xs[i];
}
return undefined;
};
// unlines :: [String] -> String
var unlines = function (xs) {
return xs.join('\n');
};
// show :: a -> String
var show = function (x) {
return JSON.stringify(x);
}; //, null, 2);
// head :: [a] -> a
var head = function (xs) {
return xs.length ? xs[0] : undefined;
};
// SIGNED DIGIT SEQUENCES (mapped to sums and to strings)
// data Sign :: [ 0 | 1 | -1 ] = ( Unsigned | Plus | Minus )
// asSum :: [Sign] -> Int
var asSum = function (xs) {
var dct = xs.reduceRight(function (a, sign, i) {
var d = i + 1; // zero-based index to [1-9] positions
if (sign !== 0) {
// Sum increased, digits cleared
return {
digits: [],
n: a.n + sign * parseInt([d].concat(a.digits)
.join(), 10)
};
} else return { // Digits extended, sum unchanged
digits: [d].concat(a.digits),
n: a.n
};
}, {
digits: [],
n: 0
});
return dct.n + (dct.digits.length > 0 ? (
parseInt(dct.digits.join(), 10)
) : 0);
};
// data Sign :: [ 0 | 1 | -1 ] = ( Unsigned | Plus | Minus )
// asString :: [Sign] -> String
var asString = function (xs) {
var ns = xs.reduce(function (a, sign, i) {
var d = (i + 1)
.toString();
return sign === 0 ? a + d : a + (sign > 0 ? ' +' : ' -') + d;
}, );
return ns[0] === '+' ? ns.slice(1) : ns; };
// SUM T0 100 ------------------------------------------------------------
// universe :: Sign var universe = permutationsWithRepetition(9, [0, 1, -1]) .filter(function (x) { return x[0] !== 1; });
// allNonNegativeSums :: [Int]
var allNonNegativeSums = sort(universe.map(asSum)
.filter(function (x) {
return x >= 0;
}));
return [
"Sums to 100:\n", unlines(universe.filter(function (x) {
return asSum(x) === 100;
})
.map(asString)),
"\n\n10 commonest sums (sum, followed by number of routes to it):\n",
show(sortBy(function (a, b) {
var lngA = a.length,
lngB = b.length;
if (lngA > lngB) return -1;
return lngA < lngB ? 1 : 0;
}, group(allNonNegativeSums))
.slice(0, 10)
.map(function (xs) {
return [xs[0], xs.length];
})),
"\n\nFirst positive integer not expressible as a sum of this kind:\n",
show(find(function (x, i) {
return x !== i;
}, sortBy(function (a, b) {
if (a < b) return -1;
return a > b ? 1 : 0;
}, nub(allNonNegativeSums))) - 1) // i is the the zero-based Array index
].join('\n') + '\n';
})();</lang>
- Output:
(Run in Atom editor, through Script package)
Sums to 100: 123 +45 -67 +8 -9 123 +4 -5 +67 -89 123 -45 -67 +89 123 -4 -5 -6 -7 +8 -9 12 +3 +4 +5 -6 -7 +89 12 +3 -4 +5 +67 +8 +9 12 -3 -4 +5 -6 +7 +89 1 +23 -4 +56 +7 +8 +9 1 +23 -4 +5 +6 +78 -9 1 +2 +34 -5 +67 -8 +9 1 +2 +3 -4 +5 +6 +78 +9 -1 +2 -3 +4 +5 +6 +78 +9 10 commonest sums (sum, followed by number of routes to it): [[9,46],[27,44],[1,43],[15,43],[21,43],[45,42],[3,41],[5,40],[17,39],[7,39]] First positive integer not expressible as a sum of this kind: 211 [Finished in 0.355s]
ES6
<lang JavaScript>(() => {
'use strict';
// GENERIC FUNCTIONS ----------------------------------------------------
// permutationsWithRepetition :: Int -> [a] -> a const permutationsWithRepetition = (n, as) => as.length > 0 ? ( foldl1(curry(cartesianProduct)(as), replicate(n, as)) ) : [];
// cartesianProduct :: [a] -> [b] -> a, b const cartesianProduct = (xs, ys) => [].concat.apply([], xs.map(x => [].concat.apply([], ys.map(y => [[x].concat(y)]))));
// curry :: ((a, b) -> c) -> a -> b -> c const curry = f => a => b => f(a, b);
// foldl1 :: (a -> a -> a) -> [a] -> a
const foldl1 = (f, xs) =>
xs.length > 0 ? xs.slice(1)
.reduce(f, xs[0]) : [];
// replicate :: Int -> a -> [a]
const replicate = (n, a) => {
let v = [a],
o = [];
if (n < 1) return o;
while (n > 1) {
if (n & 1) o = o.concat(v);
n >>= 1;
v = v.concat(v);
}
return o.concat(v);
};
// group :: Eq a => [a] -> a const group = xs => groupBy((a, b) => a === b, xs);
// groupBy :: (a -> a -> Bool) -> [a] -> a const groupBy = (f, xs) => { const dct = xs.slice(1) .reduce((a, x) => { const h = a.active.length > 0 ? a.active[0] : undefined, blnGroup = h !== undefined && f(h, x);
return {
active: blnGroup ? a.active.concat(x) : [x],
sofar: blnGroup ? a.sofar : a.sofar.concat([a.active])
};
}, {
active: xs.length > 0 ? [xs[0]] : [],
sofar: []
});
return dct.sofar.concat(dct.active.length > 0 ? [dct.active] : []);
};
// sort :: Ord a => [a] -> [a] const sort = xs => xs.sort();
// sortBy :: (a -> a -> Ordering) -> [a] -> [a] const sortBy = (f, xs) => xs.sort(f);
// nub :: [a] -> [a] const nub = xs => nubBy((a, b) => a === b, xs);
// nubBy :: (a -> a -> Bool) -> [a] -> [a]
const nubBy = (p, xs) => {
const x = xs.length ? xs[0] : undefined;
return x !== undefined ? [x].concat(
nubBy(p, xs.slice(1)
.filter(y => !p(x, y)))
) : [];
};
// find :: (a -> Bool) -> [a] -> Maybe a
const find = (f, xs) => {
for (var i = 0, lng = xs.length; i < lng; i++) {
if (f(xs[i], i)) return xs[i];
}
return undefined;
}
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// show :: a -> String const show = x => JSON.stringify(x); //, null, 2);
// head :: [a] -> a const head = xs => xs.length ? xs[0] : undefined;
// SIGNED DIGIT SEQUENCES (mapped to sums and to strings)
// data Sign :: [ 0 | 1 | -1 ] = ( Unsigned | Plus | Minus )
// asSum :: [Sign] -> Int
const asSum = xs => {
const dct = xs.reduceRight((a, sign, i) => {
const d = i + 1; // zero-based index to [1-9] positions
if (sign !== 0) { // Sum increased, digits cleared
return {
digits: [],
n: a.n + (sign * parseInt([d].concat(a.digits)
.join(), 10))
};
} else return { // Digits extended, sum unchanged
digits: [d].concat(a.digits),
n: a.n
};
}, {
digits: [],
n: 0
});
return dct.n + (dct.digits.length > 0 ? (
parseInt(dct.digits.join(), 10)
) : 0);
};
// data Sign :: [ 0 | 1 | -1 ] = ( Unsigned | Plus | Minus )
// asString :: [Sign] -> String
const asString = xs => {
const ns = xs.reduce((a, sign, i) => {
const d = (i + 1)
.toString();
return (sign === 0 ? (
a + d
) : (a + (sign > 0 ? ' +' : ' -') + d));
}, );
return ns[0] === '+' ? ns.slice(1) : ns; };
// SUM T0 100 ------------------------------------------------------------
// universe :: Sign const universe = permutationsWithRepetition(9, [0, 1, -1]) .filter(x => x[0] !== 1);
// allNonNegativeSums :: [Int]
const allNonNegativeSums = sort(universe.map(asSum)
.filter(x => x >= 0));
// uniqueNonNegativeSums :: [Int] const uniqueNonNegativeSums = nub(allNonNegativeSums);
return [
"Sums to 100:\n",
unlines(universe.filter(x => asSum(x) === 100)
.map(asString)),
"\n\n10 commonest sums (sum, followed by number of routes to it):\n",
show(sortBy((a, b) => {
const
lngA = a.length,
lngB = b.length;
if (lngA > lngB) return -1;
return lngA < lngB ? 1 : 0;
}, group(allNonNegativeSums))
.slice(0, 10)
.map(xs => [xs[0], xs.length])),
"\n\nFirst positive integer not expressible as a sum of this kind:\n",
show(find(
(x, i) => x !== i,
sortBy((a, b) => {
if (a < b) return -1;
return a > b ? 1 : 0;
}, uniqueNonNegativeSums)
) - 1), // i is the the zero-based Array index.
"\n10 largest sums:\n",
show(uniqueNonNegativeSums.sort(
(a, b) => a < b ? 1 : (a > b ? -1 : 0)
)
.slice(0, 10))
].join('\n') + '\n';
})();</lang>
- Output:
(Run in Atom editor, through Script package)
Sums to 100: 123 +45 -67 +8 -9 123 +4 -5 +67 -89 123 -45 -67 +89 123 -4 -5 -6 -7 +8 -9 12 +3 +4 +5 -6 -7 +89 12 +3 -4 +5 +67 +8 +9 12 -3 -4 +5 -6 +7 +89 1 +23 -4 +56 +7 +8 +9 1 +23 -4 +5 +6 +78 -9 1 +2 +34 -5 +67 -8 +9 1 +2 +3 -4 +5 +6 +78 +9 -1 +2 -3 +4 +5 +6 +78 +9 10 commonest sums (sum, followed by number of routes to it): [[9,46],[27,44],[1,43],[15,43],[21,43],[45,42],[3,41],[5,40],[17,39],[7,39]] First positive integer not expressible as a sum of this kind: 211 10 largest sums: [123456789,23456790,23456788,12345687,12345669,3456801,3456792,3456790,3456788,3456786] [Finished in 0.397s]
Perl 6
<lang perl6>my @ops = ['-', ], |( [' + ', ' - ', ] xx 8 ); my @str = [X~] map { .Slip }, ( @ops Z 1..9 ); my %sol = @str.classify: *.subst( ' - ', ' -', :g )\
.subst( ' + ', ' ', :g ).words.sum;
my %count.push: %sol.map({ .value.elems => .key });
my $max_solutions = %count.max( + *.key ); my $first_unsolvable = first { %sol{$_} :!exists }, 1..*; my @two_largest_sums = %sol.keys.sort(-*)[^2];
given %sol{100}:p {
say "{.value.elems} solutions for sum {.key}:";
say " $_" for .value.list;
}
say .perl for :$max_solutions, :$first_unsolvable, :@two_largest_sums;</lang>
- Output:
12 solutions for sum 100:
-1 + 2 - 3 + 4 + 5 + 6 + 78 + 9
1 + 2 + 3 - 4 + 5 + 6 + 78 + 9
1 + 2 + 34 - 5 + 67 - 8 + 9
1 + 23 - 4 + 5 + 6 + 78 - 9
1 + 23 - 4 + 56 + 7 + 8 + 9
12 + 3 + 4 + 5 - 6 - 7 + 89
12 + 3 - 4 + 5 + 67 + 8 + 9
12 - 3 - 4 + 5 - 6 + 7 + 89
123 + 4 - 5 + 67 - 89
123 + 45 - 67 + 8 - 9
123 - 4 - 5 - 6 - 7 + 8 - 9
123 - 45 - 67 + 89
:max_solutions("46" => $["9", "-9"])
:first_unsolvable(211)
:two_largest_sums(["123456789", "23456790"])
REXX
<lang rexx>/*REXX pgm solves a puzzle: using the string 123456789, insert - or + to sum to 100*/ parse arg LO HI . /*obtain optional arguments from the CL*/ if LO== | LO=="," then LO=100 /*Not specified? Then use the default.*/ if HI== | HI=="," then HI=LO /* " " " " " " */ if LO==00 then HI=123456789 /*LOW specified as zero with leading 0s*/ ops= '+-'; L=length(ops) + 1 /*define operators (and their length). */ @.=; do i=1 to L-1; @.i=substr(ops,i,1) /* " some handy-dandy REXX literals*/
end /*i*/ /* " individual operators for speed*/
mx=0; mn=999999 /*initialize the minimums and maximums.*/ mxL=; mnL=; do j=LO to HI until LO==00 & mn==0 /*solve with a range of sums*/
z=solve(j) /*find # solutionson for J.*/
if z> mx then mxL= /*see if this is a new max. */
if z>=mx then do; mxL=mxL j; mx=z; end /*remember this new maximum.*/
if z< mn then mnL= /*see if this is a new min. */
if z<=mn then do; mnL=mnL j; mn=z; end /*remember this new minimum.*/
end /*j*/
if LO==HI then exit /*don't display max & min ? */ @@= 'number of solutions: '; say _=words(mxL); say 'sum's(_) "of" mxL ' 's(_,"have",'has') 'the maximum' @@ mx _=words(mnL); say 'sum's(_) "of" mnL ' 's(_,"have",'has') 'the minimum' @@ mn exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ s: if arg(1)==1 then return arg(3); return word(arg(2) "s",1) /*simple pluralizer*/ /*──────────────────────────────────────────────────────────────────────────────────────*/ solve: parse arg answer; #=0 /*obtain the answer (sum) to the puzzle*/
do a=L-1 to L; aa= @.a'1' /*choose one of ─ or nothing. */
do b=1 for L; bb=aa || @.b'2' /* " " " ─ +, or abutment.*/
do c=1 for L; cc=bb || @.c'3' /* " " " " " " " */
do d=1 for L; dd=cc || @.d'4' /* " " " " " " " */
do e=1 for L; ee=dd || @.e'5' /* " " " " " " " */
do f=1 for L; ff=ee || @.f'6' /* " " " " " " " */
do g=1 for L; gg=ff || @.g'7' /* " " " " " " " */
do h=1 for L; hh=gg || @.h'8' /* " " " " " " " */
do i=1 for L; ii=hh || @.i'9' /* " " " " " " " */
interpret '$=' ii /*calculate the sum of modified string.*/
if $\==answer then iterate /*Is sum not equal to answer? Then skip*/
#=#+1; if LO==HI then say 'solution: ' $ " ◄───► " ii
end /*i*/
end /*h*/
end /*g*/
end /*f*/
end /*e*/
end /*d*/
end /*c*/
end /*b*/
end /*a*/
y=#
if y==0 then y='no' /*maybe adjust the number of solutions.*/
if LO\==00 then say right(y, 9) 'solution's(y) 'found for' right(j, length(HI))
return # /*return the number of solutions found.*/</lang>
output when the default input is used:
solution: 100 ◄───► -1+2-3+4+5+6+78+9
solution: 100 ◄───► 1+2+3-4+5+6+78+9
solution: 100 ◄───► 1+2+34-5+67-8+9
solution: 100 ◄───► 1+23-4+5+6+78-9
solution: 100 ◄───► 1+23-4+56+7+8+9
solution: 100 ◄───► 12+3+4+5-6-7+89
solution: 100 ◄───► 12+3-4+5+67+8+9
solution: 100 ◄───► 12-3-4+5-6+7+89
solution: 100 ◄───► 123+4-5+67-89
solution: 100 ◄───► 123+45-67+8-9
solution: 100 ◄───► 123-4-5-6-7+8-9
solution: 100 ◄───► 123-45-67+89
12 solutions found for 100
output when the following input is used: 00
sum of 9 has the maximum number of solutions: 46 sum of 211 has the minimum number of solutions: 0
zkl
Taking a big clue from Haskell and just calculate the world. <lang zkl>var all = // ( (1,12,123...-1,-12,...), (2,23,...) ...)
(9).pump(List,fcn(n){ split("123456789"[n,*]) }) // 45
.apply(fcn(ns){ ns.extend(ns.copy().apply('*(-1))) }); // 90
fcn calcAllSums{ // calculate all 6572 sums (1715 unique)
fcn(n,sum,soFar,r){
if(n==9) return();
foreach b in (all[n]){
if(sum+b>=0 and b.abs()%10==9) r.appendV(sum+b,"%s%+d".fmt(soFar,b)); self.fcn(b.abs()%10,sum + b,"%s%+d".fmt(soFar,b),r);
} }(0,0,"",r:=Dictionary()); r
}
// "123" --> (1,12,123)
fcn split(nstr){ (1).pump(nstr.len(),List,nstr.get.fp(0),"toInt") }</lang> <lang zkl>fcn showSums(allSums,N=100,printSolutions=2){
slns:=allSums.find(N,T);
if(printSolutions) println("%d solutions for N=%d".fmt(slns.len(),N));
if(printSolutions==2) println(slns.concat("\n"));
println();
}
allSums:=calcAllSums(); showSums(allSums); showSums(allSums,0,1);
println("Smallest postive integer with no solution: ",
[1..].filter1('wrap(n){ Void==allSums.find(n) }));
println("5 commonest sums (sum, number of ways to calculate to it):"); ms:=allSums.values.apply("len").sort()[-5,*]; // 5 mostest sums allSums.pump(List, // get those pairs
'wrap([(k,v)]){ v=v.len(); ms.holds(v) and T(k.toInt(),v) or Void.Skip })
.sort(fcn(kv1,kv2){ kv1[1]>kv2[1] }) // and sort .println();</lang>
- Output:
12 solutions for N=100 +1+2+3-4+5+6+78+9 +1+2+34-5+67-8+9 +1+23-4+5+6+78-9 +1+23-4+56+7+8+9 +12+3+4+5-6-7+89 +12+3-4+5+67+8+9 +12-3-4+5-6+7+89 +123+4-5+67-89 +123+45-67+8-9 +123-4-5-6-7+8-9 +123-45-67+89 -1+2-3+4+5+6+78+9 22 solutions for N=0 Smallest postive integer with no solution: 211 5 commonest sums (sum, number of ways to calculate to it): L(L(9,46),L(27,44),L(15,43),L(1,43),L(21,43))