Sum to 100

From Rosetta Code
Task
Sum to 100
You are encouraged to solve this task according to the task description, using any language you may know.
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)

ALGOL 68[edit]

BEGIN
# find the numbers the string 123456789 ( with "+/-" optionally inserted #
# before each digit ) can generate #
 
# experimentation shows that the largest hundred numbers that can be #
# generated are are greater than or equal to 56795 #
# as we can't declare an array with bounds -123456789 : 123456789 in #
# Algol 68G, we use -60000 : 60000 and keep counts for the top hundred #
 
INT max number = 60 000;
[ - max number : max number ]STRING solutions;
[ - max number : max number ]INT count;
FOR i FROM LWB solutions TO UPB solutions DO solutions[ i ] := ""; count[ i ] := 0 OD;
 
# calculate the numbers ( up to max number ) we can generate and the strings leading to them #
# also determine the largest numbers we can generate #
[ 100 ]INT largest;
[ 100 ]INT largest count;
INT impossible number = - 999 999 999;
FOR i FROM LWB largest TO UPB largest DO
largest [ i ] := impossible number;
largest count[ i ] := 0
OD;
[ 1 : 18 ]CHAR sum string := ".1.2.3.4.5.6.7.8.9";
[]CHAR sign char = []CHAR( "-", " ", "+" )[ AT -1 ];
# we don't distinguish between strings starting "+1" and starting " 1" #
FOR s1 FROM -1 TO 0 DO
sum string[ 1 ] := sign char[ s1 ];
FOR s2 FROM -1 TO 1 DO
sum string[ 3 ] := sign char[ s2 ];
FOR s3 FROM -1 TO 1 DO
sum string[ 5 ] := sign char[ s3 ];
FOR s4 FROM -1 TO 1 DO
sum string[ 7 ] := sign char[ s4 ];
FOR s5 FROM -1 TO 1 DO
sum string[ 9 ] := sign char[ s5 ];
FOR s6 FROM -1 TO 1 DO
sum string[ 11 ] := sign char[ s6 ];
FOR s7 FROM -1 TO 1 DO
sum string[ 13 ] := sign char[ s7 ];
FOR s8 FROM -1 TO 1 DO
sum string[ 15 ] := sign char[ s8 ];
FOR s9 FROM -1 TO 1 DO
sum string[ 17 ] := sign char[ s9 ];
INT number := 0;
INT part := IF s1 < 0 THEN -1 ELSE 1 FI;
IF s2 = 0 THEN part *:= 10 +:= 2 * SIGN part ELSE number +:= part; part := 2 * s2 FI;
IF s3 = 0 THEN part *:= 10 +:= 3 * SIGN part ELSE number +:= part; part := 3 * s3 FI;
IF s4 = 0 THEN part *:= 10 +:= 4 * SIGN part ELSE number +:= part; part := 4 * s4 FI;
IF s5 = 0 THEN part *:= 10 +:= 5 * SIGN part ELSE number +:= part; part := 5 * s5 FI;
IF s6 = 0 THEN part *:= 10 +:= 6 * SIGN part ELSE number +:= part; part := 6 * s6 FI;
IF s7 = 0 THEN part *:= 10 +:= 7 * SIGN part ELSE number +:= part; part := 7 * s7 FI;
IF s8 = 0 THEN part *:= 10 +:= 8 * SIGN part ELSE number +:= part; part := 8 * s8 FI;
IF s9 = 0 THEN part *:= 10 +:= 9 * SIGN part ELSE number +:= part; part := 9 * s9 FI;
number +:= part;
IF number >= LWB solutions
AND number <= UPB solutions
THEN
solutions[ number ] +:= ";" + sum string;
count [ number ] +:= 1
FI;
BOOL inserted := FALSE;
FOR l pos FROM LWB largest TO UPB largest WHILE NOT inserted DO
IF number > largest[ l pos ] THEN
# found a new larger number #
FOR m pos FROM UPB largest BY -1 TO l pos + 1 DO
largest [ m pos ] := largest [ m pos - 1 ];
largest count[ m pos ] := largest count[ m pos - 1 ]
OD;
largest [ l pos ] := number;
largest count[ l pos ] := 1;
inserted := TRUE
ELIF number = largest[ l pos ] THEN
# have another way of generating this number #
largest count[ l pos ] +:= 1;
inserted := TRUE
FI
OD
OD
OD
OD
OD
OD
OD
OD
OD
OD;
 
# show the solutions for 100 #
print( ( "100 has ", whole( count[ 100 ], 0 ), " solutions:" ) );
STRING s := solutions[ 100 ];
FOR s pos FROM LWB s TO UPB s DO
IF s[ s pos ] = ";" THEN print( ( newline, " " ) )
ELIF s[ s pos ] /= " " THEN print( ( s[ s pos ] ) )
FI
OD;
print( ( newline ) );
# find the number with the most solutions #
INT max solutions := 0;
INT number with max := LWB count - 1;
FOR n FROM 0 TO max number DO
IF count[ n ] > max solutions THEN
max solutions := count[ n ];
number with max := n
FI
OD;
FOR n FROM LWB largest count TO UPB largest count DO
IF largest count[ n ] > max solutions THEN
max solutions := largest count[ n ];
number with max := largest[ n ]
FI
OD;
print( ( whole( number with max, 0 ), " has the maximum number of solutions: ", whole( max solutions, 0 ), newline ) );
# find the smallest positive number that has no solutions #
BOOL have solutions := TRUE;
FOR n FROM 0 TO max number
WHILE IF NOT ( have solutions := count[ n ] > 0 )
THEN print( ( whole( n, 0 ), " is the lowest positive number with no solutions", newline ) )
FI;
have solutions
DO SKIP OD;
IF have solutions
THEN print( ( "All positive numbers up to ", whole( max number, 0 ), " have solutions", newline ) )
FI;
print( ( "The 10 largest numbers that can be generated are:", newline ) );
FOR t pos FROM 1 TO 10 DO
print( ( " ", whole( largest[ t pos ], 0 ) ) )
OD;
print( ( newline ) )
 
END
Output:
100 has 12 solutions:
        -1+2-3+4+5+6+78+9
        12-3-4+5-6+7+89
        123-4-5-6-7+8-9
        123-45-67+89
        123+4-5+67-89
        123+45-67+8-9
        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+3-4+5+6+78+9
        1+2+34-5+67-8+9
9 has the maximum number of solutions: 46
211 is the lowest positive number with no solutions
The 10 largest numbers that can be generated are:
 123456789 23456790 23456788 12345687 12345669 3456801 3456792 3456790 3456788 3456786

AppleScript[edit]

Translation of: JavaScript

AppleScript is essentially out of its depth at this scale. The first task (number of distinct paths to 100) is accessible within a few seconds. Subsequent tasks, however, terminate only (if at all) after impractical amounts of time. Note the contrast with the lighter and more optimised JavaScript interpreter, which takes less than half a second to return full results for all the listed tasks.

use framework "Foundation" -- for basic NSArray sort
 
property pSigns : {1, 0, -1} --> ( + | unsigned | - )
property plst100 : {"Sums to 100:", ""}
property plstSums : {}
property plstSumsSorted : missing value
property plstSumGroups : missing value
 
-- data Sign :: [ 1 | 0 | -1 ] = ( Plus | Unsigned | Minus )
-- asSum :: [Sign] -> Int
on asSum(xs)
script
on lambda(a, sign, i)
if sign ≠ 0 then
{digits:{}, n:(n of a) + (sign * ((i & digits of a) as string as integer))}
else
{digits:{i} & (digits of a), n:n of a}
end if
end lambda
end script
 
set rec to foldr(result, {digits:{}, n:0}, xs)
set ds to digits of rec
if length of ds > 0 then
(n of rec) + (ds as string as integer)
else
n of rec
end if
end asSum
 
-- data Sign :: [ 1 | 0 | -1 ] = ( Plus | Unisigned | Minus )
-- asString :: [Sign] -> String
on asString(xs)
script
on lambda(a, sign, i)
set d to i as string
if sign ≠ 0 then
if sign > 0 then
a & " +" & d
else
a & " -" & d
end if
else
a & d
end if
end lambda
end script
 
foldl(result, "", xs)
end asString
 
-- sumsTo100 :: () -> String
on sumsTo100()
-- From first permutation without leading '+' (3 ^ 8) to end of universe (3 ^ 9)
repeat with i from 6561 to 19683
set xs to nthPermutationWithRepn(pSigns, 9, i)
if asSum(xs) = 100 then set end of plst100 to asString(xs)
end repeat
intercalate(linefeed, plst100)
end sumsTo100
 
 
-- mostCommonSum :: () -> String
on mostCommonSum()
-- From first permutation without leading '+' (3 ^ 8) to end of universe (3 ^ 9)
repeat with i from 6561 to 19683
set intSum to asSum(nthPermutationWithRepn(pSigns, 9, i))
if intSum ≥ 0 then set end of plstSums to intSum
end repeat
 
set plstSumsSorted to sort(plstSums)
set plstSumGroups to group(plstSumsSorted)
 
script groupLength
on lambda(a, b)
set intA to length of a
set intB to length of b
if intA < intB then
-1
else if intA > intB then
1
else
0
end if
end lambda
end script
 
set lstMaxSum to maximumBy(groupLength, plstSumGroups)
intercalate(linefeed, {"Most common sum: " & item 1 of lstMaxSum, "Number of instances: " & length of lstMaxSum})
end mostCommonSum
 
 
-- TEST ----------------------------------------------------------------------
on run
return sumsTo100()
 
-- Also returns a value, but slow:
-- mostCommonSum()
end run
 
 
-- GENERIC FUNCTIONS ---------------------------------------------------------
 
-- nthPermutationWithRepn :: [a] -> Int -> Int -> [a]
on nthPermutationWithRepn(xs, groupSize, iIndex)
set intBase to length of xs
set intSetSize to intBase ^ groupSize
 
if intBase < 1 or iIndex > intSetSize then
{}
else
set baseElems to inBaseElements(xs, iIndex)
set intZeros to groupSize - (length of baseElems)
 
if intZeros > 0 then
replicate(intZeros, item 1 of xs) & baseElems
else
baseElems
end if
end if
end nthPermutationWithRepn
 
-- inBaseElements :: [a] -> Int -> [String]
on inBaseElements(xs, n)
set intBase to length of xs
 
script nextDigit
on lambda(residue)
set {divided, remainder} to quotRem(residue, intBase)
 
{valid:divided > 0, value:(item (remainder + 1) of xs), new:divided}
end lambda
end script
 
reverse of unfoldr(nextDigit, n)
end inBaseElements
 
-- sort :: [a] -> [a]
on sort(lst)
((current application's NSArray's arrayWithArray:lst)'s ¬
sortedArrayUsingSelector:"compare:") as list
end sort
 
-- maximumBy :: (a -> a -> Ordering) -> [a] -> a
on maximumBy(f, xs)
set cmp to mReturn(f)
script max
on lambda(a, b)
if a is missing value or cmp's lambda(a, b) < 0 then
b
else
a
end if
end lambda
end script
 
foldl(max, missing value, xs)
end maximumBy
 
-- group :: Eq a => [a] -> [[a]]
on group(xs)
script eq
on lambda(a, b)
a = b
end lambda
end script
 
groupBy(eq, xs)
end group
 
-- groupBy :: (a -> a -> Bool) -> [a] -> [[a]]
on groupBy(f, xs)
set mf to mReturn(f)
 
script enGroup
on lambda(a, x)
if length of (active of a) > 0 then
set h to item 1 of active of a
else
set h to missing value
end if
 
if h is not missing value and mf's lambda(h, x) then
{active:(active of a) & x, sofar:sofar of a}
else
{active:{x}, sofar:(sofar of a) & {active of a}}
end if
end lambda
end script
 
if length of xs > 0 then
set dct to foldl(enGroup, {active:{item 1 of xs}, sofar:{}}, tail(xs))
if length of (active of dct) > 0 then
sofar of dct & {active of dct}
else
sofar of dct
end if
else
{}
end if
end groupBy
 
-- tail :: [a] -> [a]
on tail(xs)
if length of xs > 1 then
items 2 thru -1 of xs
else
{}
end if
end tail
 
 
-- intercalate :: Text -> [Text] -> Text
on intercalate(strText, lstText)
set {dlm, my text item delimiters} to {my text item delimiters, strText}
set strJoined to lstText as text
set my text item delimiters to dlm
return strJoined
end intercalate
 
-- quotRem :: Integral a => a -> a -> (a, a)
on quotRem(m, n)
{m div n, m mod n}
end quotRem
 
-- replicate :: Int -> a -> [a]
on replicate(n, a)
set out to {}
if n < 1 then return out
set dbl to {a}
 
repeat while (n > 1)
if (n mod 2) > 0 then set out to out & dbl
set n to (n div 2)
set dbl to (dbl & dbl)
end repeat
return out & dbl
end replicate
 
-- foldr :: (a -> b -> a) -> a -> [b] -> a
on foldr(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from lng to 1 by -1
set v to lambda(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldr
 
-- foldl :: (a -> b -> a) -> a -> [b] -> a
on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to lambda(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
 
-- unfoldr :: (b -> Maybe (a, b)) -> b -> [a]
on unfoldr(f, v)
set mf to mReturn(f)
set lst to {}
set recM to mf's lambda(v)
repeat while (valid of recM) is true
set end of lst to value of recM
set recM to mf's lambda(new of recM)
end repeat
lst & value of recM
end unfoldr
 
-- until :: (a -> Bool) -> (a -> a) -> a -> a
on |until|(p, f, x)
set mp to mReturn(p)
set v to x
 
tell mReturn(f)
repeat until mp's lambda(v)
set v to lambda(v)
end repeat
end tell
return v
end |until|
 
-- range :: Int -> Int -> [Int]
on range(m, n)
if n < m then
set d to -1
else
set d to 1
end if
set lst to {}
repeat with i from m to n by d
set end of lst to i
end repeat
return lst
end range
 
-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to lambda(item i of xs, i, xs)
end repeat
return lst
end tell
end map
 
 
-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: Handler -> Script
on mReturn(f)
if class of f is script then
f
else
script
property lambda : f
end script
end if
end mReturn
Output:
Sums to 100:

1 +2 +34 -5 +67 -8 +9
1 +2 +3 -4 +5 +6 +78 +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
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
 -1 +2 -3 +4 +5 +6 +78 +9

AutoHotkey[edit]

This example is incomplete.

The output is incomplete, please address the 2nd and 3rd task requirements.

Please ensure that it meets all task requirements and remove this message.

Inspired by https://autohotkey.com/board/topic/149914-five-challenges-to-do-in-an-hour/

global Matches:=[]
AllPossibilities100()
for eq, val in matches
res .= eq "`n"
MsgBox % res
return
 
AllPossibilities100(n:=0, S:="") {
if (n = 0) ; First call
AllPossibilities100(n+1, n) ; Recurse
else if (n < 10){
AllPossibilities100(n+1, S ",-" n) ; Recurse. Concatenate S, ",-" and n
AllPossibilities100(n+1, S ",+" n) ; Recurse. Concatenate S, ",+" and n
AllPossibilities100(n+1, S n) ; Recurse. Concatenate S and n
} else { ; 10th level recursion
Loop, Parse, S, CSV ; Total the values of S and check if equal to 100
{
SubS := SubStr(A_LoopField, 2) ; The number portion of A_LoopField
if (A_Index = 1)
Total := A_LoopField
else if (SubStr(A_LoopField, 1, 1) = "+") ; If the first character is + add
Total += SubS
else ; else subtract
Total -= SubS
}
if (Total = 100)
matches[LTrim(LTrim(StrReplace(S, ","), "0"),"+")] := true ; remove leading 0's, +'s and all commas
}
}
Outputs:
-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

C#[edit]

using System;
using System.Collections.Generic;
using System.Linq;
 
class Program
{
static void Main(string[] args)
{
// All unique expressions that have a plus sign in front of the 1; calculated in parallel
var expressionsPlus = Enumerable.Range(0, (int)Math.Pow(3, 8)).AsParallel().Select(i => new Expression(i, 1));
// All unique expressions that have a minus sign in front of the 1; calculated in parallel
var expressionsMinus = Enumerable.Range(0, (int)Math.Pow(3, 8)).AsParallel().Select(i => new Expression(i, -1));
var expressions = expressionsPlus.Concat(expressionsMinus);
var results = new Dictionary<int, List<Expression>>();
foreach (var e in expressions)
{
if (results.Keys.Contains(e.Value))
results[e.Value].Add(e);
else
results[e.Value] = new List<Expression>() { e };
}
Console.WriteLine("Show all solutions that sum to 100");
foreach (Expression e in results[100])
Console.WriteLine(" " + e);
Console.WriteLine("Show the sum that has the maximum number of solutions (from zero to infinity)");
var summary = results.Keys.Select(k => new Tuple<int, int>(k, results[k].Count));
var maxSols = summary.Aggregate((a, b) => a.Item2 > b.Item2 ? a : b);
Console.WriteLine(" The sum " + maxSols.Item1 + " has " + maxSols.Item2 + " solutions.");
Console.WriteLine("Show the lowest positive sum that can't be expressed (has no solutions), using the rules for this task");
var lowestPositive = Enumerable.Range(1, int.MaxValue).First(x => !results.Keys.Contains(x));
Console.WriteLine(" " + lowestPositive);
Console.WriteLine("Show the ten highest numbers that can be expressed using the rules for this task (extra credit)");
var highest = from k in results.Keys
orderby k descending
select k;
foreach (var x in highest.Take(10))
Console.WriteLine(" " + x);
}
}
public enum Operations { Plus, Minus, Join };
public class Expression
{
protected Operations[] Gaps;
// 123456789 => there are 8 "gaps" between each number
/// with 3 possibilities for each gap: plus, minus, or join
public int Value; // What this expression sums up to
protected int _one;
 
public Expression(int serial, int one)
{
_one = one;
Gaps = new Operations[8];
// This represents "serial" as a base 3 number, each Gap expression being a base-three digit
int divisor = 2187; // == Math.Pow(3,7)
int times;
for (int i = 0; i < 8; i++)
{
times = Math.DivRem(serial, divisor, out serial);
divisor /= 3;
if (times == 0)
Gaps[i] = Operations.Join;
else if (times == 1)
Gaps[i] = Operations.Minus;
else
Gaps[i] = Operations.Plus;
}
// go ahead and calculate the value of this expression
// because this is going to be done in a parallel thread (save time)
Value = Evaluate();
}
public override string ToString()
{
string ret = _one.ToString();
for (int i = 0; i < 8; i++)
{
switch (Gaps[i])
{
case Operations.Plus:
ret += "+";
break;
case Operations.Minus:
ret += "-";
break;
}
ret += (i + 2);
}
return ret;
}
private int Evaluate()
/* Calculate what this expression equals */
{
var numbers = new int[9];
int nc = 0;
var operations = new List<Operations>();
int a = 1;
for (int i = 0; i < 8; i++)
{
if (Gaps[i] == Operations.Join)
a = a * 10 + (i + 2);
else
{
if (a > 0)
{
if (nc == 0)
a *= _one;
numbers[nc++] = a;
a = i + 2;
}
operations.Add(Gaps[i]);
}
}
if (nc == 0)
a *= _one;
numbers[nc++] = a;
int ni = 0;
int left = numbers[ni++];
foreach (var operation in operations)
{
int right = numbers[ni++];
if (operation == Operations.Plus)
left = left + right;
else
left = left - right;
}
return left;
}
}
Output:
Show all solutions that sum to 100
  123-45-67+89
  123-4-5-6-7+8-9
  123+45-67+8-9
  123+4-5+67-89
  12-3-4+5-6+7+89
  12+3-4+5+67+8+9
  12+3+4+5-6-7+89
  1+23-4+5+6+78-9
  1+23-4+56+7+8+9
  1+2+34-5+67-8+9
  1+2+3-4+5+6+78+9
  -1+2-3+4+5+6+78+9
Show the sum that has the maximum number of solutions (from zero to infinity)
  The sum 9 has 46 solutions.
Show the lowest positive sum that can't be expressed (has no solutions), using the rules for this task
  211
Show the ten highest numbers that can be expressed using the rules for this task (extra credit)
  123456789
  23456790
  23456788
  12345687
  12345669
  3456801
  3456792
  3456790
  3456788
  3456786

Elixir[edit]

defmodule Sum do
def to(val) do
generate
|> Enum.map(&{eval(&1), &1})
|> Enum.filter(fn {v, _s} -> v==val end)
|> Enum.each(&IO.inspect &1)
end
 
def max_solve do
generate
|> Enum.group_by(&eval &1)
|> Enum.filter_map(fn {k,_} -> k>=0 end, fn {k,v} -> {length(v),k} end)
|> Enum.max
|> fn {len,sum} -> IO.puts "sum of #{sum} has the maximum number of solutions : #{len}" end.()
end
 
def min_solve do
solve = generate |> Enum.group_by(&eval &1)
Stream.iterate(1, &(&1+1))
|> Enum.find(fn n -> solve[n]==nil end)
|> fn sum -> IO.puts "lowest positive sum that can't be expressed : #{sum}" end.()
end
 
def highest_sums(n\\10) do
IO.puts "highest sums :"
generate
|> Enum.map(&eval &1)
|> Enum.uniq
|> Enum.sort_by(fn sum -> -sum end)
|> Enum.take(n)
|> IO.inspect
end
 
defp generate do
x = ["+", "-", ""]
for a <- ["-", ""], b <- x, c <- x, d <- x, e <- x, f <- x, g <- x, h <- x, i <- x,
do: "#{a}1#{b}2#{c}3#{d}4#{e}5#{f}6#{g}7#{h}8#{i}9"
end
 
defp eval(str), do: Code.eval_string(str) |> elem(0)
end
 
Sum.to(100)
Sum.max_solve
Sum.min_solve
Sum.highest_sums
Output:
{100, "-1+2-3+4+5+6+78+9"}
{100, "1+2+3-4+5+6+78+9"}
{100, "1+2+34-5+67-8+9"}
{100, "1+23-4+5+6+78-9"}
{100, "1+23-4+56+7+8+9"}
{100, "12+3+4+5-6-7+89"}
{100, "12+3-4+5+67+8+9"}
{100, "12-3-4+5-6+7+89"}
{100, "123+4-5+67-89"}
{100, "123+45-67+8-9"}
{100, "123-4-5-6-7+8-9"}
{100, "123-45-67+89"}
sum of 9 has the maximum number of solutions : 46
lowest positive sum that can't be expressed : 211
highest sums :
[123456789, 23456790, 23456788, 12345687, 12345669, 3456801, 3456792, 3456790,
 3456788, 3456786]

F#[edit]

 
(*
Generate the data set
Nigel Galloway February 22nd., 2017
*)

type N = {n:string; g:int}
let N = seq {
let rec fn n i g e l = seq {
match i with
|9 -> yield {n=l + "-9"; g=g+e-9}
yield {n=l + "+9"; g=g+e+9}
yield {n=l + "9"; g=g+e*10+9*n}
|_ -> yield! fn -1 (i+1) (g+e) -i (l + string -i)
yield! fn 1 (i+1) (g+e) i (l + "+" + string i)
yield! fn n (i+1) g (e*10+i*n) (l + string i)
}
yield! fn 1 2 0 1 "1"
yield! fn -1 2 0 -1 "-1"
}
 
Output:
 
N |> Seq.filter(fun n->n.g=100) |> Seq.iter(fun n->printfn "%s" n.n)
 
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
123-4-5-6-7+8-9
123-45-67+89
123+4-5+67-89
123+45-67+8-9
-1+2-3+4+5+6+78+9
 
let n,g = N |> Seq.filter(fun n->n.g>=0) |> Seq.countBy(fun n->n.g) |> Seq.maxBy(snd)
printfn "%d has %d solutions" n g
 
9 has 46 solutions
 
match N |> Seq.filter(fun n->n.g>=0) |> Seq.distinctBy(fun n->n.g) |> Seq.sortBy(fun n->n.g) |> Seq.pairwise |> Seq.tryFind(fun n->(snd n).g-(fst n).g > 1) with
|Some(n) -> printfn "least non-value is %d" ((fst n).g+1)
|None -> printfn "No non-values found"
 
least non-value is 211
 
N |> Seq.filter(fun n->n.g>=0) |> Seq.distinctBy(fun n->n.g) |> Seq.sortBy(fun n->(-n.g)) |> Seq.take 10 |> Seq.iter(fun n->printfn "%d" n.g )
 
123456789
23456790
23456788
12345687
12345669
3456801
3456792
3456790
3456788
3456786

Haskell[edit]

import Data.Monoid ((<>))
import Data.Ord (comparing)
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 =
putStrLn $
unlines
[ "Sums to 100:"
, unlines $ asString <$> filter ((== 100) . asSum) universe
, "\n10 commonest sums [sum, number of routes to it]:"
, show
((head &&& length) <$>
take 10 (sortBy (flip (comparing length)) (group allNonNegativeSums)))
, "\nFirst positive integer not expressible as a sum of this kind:"
, maybeReport (find (uncurry (/=)) (zip [0 ..] uniqueNonNegativeSums))
, "\n10 largest sums:"
, show $ take 10 $ sortBy (flip compare) uniqueNonNegativeSums
]
where
maybeReport
:: Show a
=> Maybe (a, b) -> String
maybeReport (Just (x, _)) = show x
maybeReport _ = "No gaps found"
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 1.204s]

JavaScript[edit]

ES5[edit]

Translation of: Haskell
(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);
};
};
};
 
// flip :: (a -> b -> c) -> b -> a -> c
var flip = function (f) {
return function (a, b) {
return f.apply(null, [b, a]);
};
};
 
// 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] : []);
};
 
// compare :: a -> a -> Ordering
var compare = function (a, b) {
return a < b ? -1 : a > b ? 1 : 0;
};
 
// on :: (b -> b -> c) -> (a -> b) -> a -> a -> c
var on = function (f, g) {
return function (a, b) {
return f(g(a), g(b));
};
};
 
// 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;
};
 
// Int -> [a] -> [a]
var take = function (n, xs) {
return xs.slice(0, n);
};
 
// 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;
};
 
// tail :: [a] -> [a]
var tail = function (xs) {
return xs.length ? xs.slice(1) : undefined;
};
 
// length :: [a] -> Int
var length = function (xs) {
return xs.length;
};
 
// 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] === '+' ? tail(ns) : ns;
};
 
// SUM T0 100 ------------------------------------------------------------
 
// universe :: [[Sign]]
var universe = permutationsWithRepetition(9, [0, 1, -1])
.filter(function (x) {
return x[0] !== 1;
});
 
// allNonNegativeSums :: [Int]
var allNonNegativeSums = universe.map(asSum)
.filter(function (x) {
return x >= 0;
})
.sort();
 
// uniqueNonNegativeSums :: [Int]
var uniqueNonNegativeSums = nub(allNonNegativeSums);
 
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(take(10, group(allNonNegativeSums)
.sort(on(flip(compare), length))
.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;
}, uniqueNonNegativeSums.sort(compare)) - 1), // zero-based index
 
"\n10 largest sums:\n",
show(take(10, uniqueNonNegativeSums.sort(flip(compare))))
].join('\n') + '\n';
})();
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 0.381s]

ES6[edit]

Translation of: Haskell
(() => {
'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);
 
// flip :: (a -> b -> c) -> b -> a -> c
const flip = f => (a, b) => f.apply(null, [b, a]);
 
// 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] : []);
};
 
// compare :: a -> a -> Ordering
const compare = (a, b) => a < b ? -1 : (a > b ? 1 : 0);
 
// on :: (b -> b -> c) -> (a -> b) -> a -> a -> c
const on = (f, g) => (a, b) => f(g(a), g(b));
 
// 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;
}
 
// Int -> [a] -> [a]
const take = (n, xs) => xs.slice(0, n);
 
// 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;
 
// tail :: [a] -> [a]
const tail = xs => xs.length ? xs.slice(1) : undefined;
 
// length :: [a] -> Int
const length = xs => xs.length;
 
 
// 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] === '+' ? tail(ns) : ns;
};
 
 
// SUM T0 100 ------------------------------------------------------------
 
// universe :: [[Sign]]
const universe = permutationsWithRepetition(9, [0, 1, -1])
.filter(x => x[0] !== 1);
 
// allNonNegativeSums :: [Int]
const allNonNegativeSums = universe.map(asSum)
.filter(x => x >= 0)
.sort();
 
// 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(take(10, group(allNonNegativeSums)
.sort(on(flip(compare), length))
.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,
uniqueNonNegativeSums.sort(compare)
) - 1), // i is the the zero-based Array index.
 
"\n10 largest sums:\n",
show(take(10, uniqueNonNegativeSums.sort(flip(compare))))
].join('\n') + '\n';
})();
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 0.382s]

Mathematica[edit]

This example is incorrect. Please fix the code and remove this message.
Details:
I think the least non-value is 211 not 221

221 is the 3rd value that has a zero solutions.

Defining all possible sums and counting them:

operations = 
DeleteCases[Tuples[{"+", "-", ""}, 9], {x_, y__} /; x == "+"];
 
allsums =
Map[StringJoin[Riffle[#, CharacterRange["1", "9"]]] &, operations];
 
counts = CountsBy[allsums, ToExpression];

Sums to 100:

 [email protected][allsums, ToExpression@# == 100 &] 
Output:
-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

Maximum number of solutions:

 MaximalBy[counts, Identity] 
Output:
 <|9 -> 46, -9 -> 46|> 

First unsolvable:

 i = 1; While[KeyExistsQ[counts, i], ++i]; i 
Output:
221

Ten largest sums:

 TakeLargest[[email protected], 10] 
Output:
 {123456789, 23456790, 23456788, 12345687, 12345669, 3456801, 3456792, 3456790, 3456788, 3456786} 

Nim[edit]

 
import strutils
 
var
ligne: string = ""
sum: int
opera: array[0..9, int] = [0,0,1,1,1,1,1,1,1,1]
curseur: int = 9
boucle: bool
tot: array[1..123456789, int]
pG: int
plusGrandes: array[1..10, string]
 
let
ope: array[0..3, string] = ["-",""," +"," -"]
aAtteindre = 100
 
proc calcul(li: string): int =
var liS: seq[string]
liS = split(li," ")
for i in liS:
result += parseInt(i)
 
echo "Valeur à atteindre : ",aAtteindre
 
while opera[1]<2:
ligne.add(ope[opera[1]])
ligne.add("1")
for i in 2..9:
ligne.add(ope[opera[i]])
ligne.add($i)
sum = calcul(ligne)
if sum == aAtteindre:
stdout.write(ligne)
echo " = ",sum
if sum>0:
tot[sum] += 1
pG = 1
while pG<10:
if sum>calcul(plusGrandes[pG]):
for k in countdown(10,pG+1):
plusGrandes[k]=plusGrandes[k-1]
plusGrandes[pG]=ligne
pG = 11
pG += 1
ligne = ""
boucle = true
while boucle:
opera[curseur] += 1
if opera[curseur] == 4:
opera[curseur]=1
curseur -= 1
else:
curseur = 9
boucle = false
 
echo "Valeur atteinte ",tot[aAtteindre]," fois."
echo ""
 
var
min0: int = 0
max: int = 0
valmax: int = 0
 
for i in 1..123456789:
if tot[i]==0 and min0 == 0:
min0 = i
if tot[i]>max:
max = tot[i]
valmax = i
 
echo "Plus petite valeur ne pouvant pas être atteinte : ",min0
echo "Valeur atteinte le plus souvent : ",valmax,", atteinte ",max," fois."
echo ""
echo "Plus grandes valeurs pouvant être atteintes :"
for i in 1..10:
echo calcul(plusGrandes[i])," = ",plusGrandes[i]
Output:
Valeur à atteindre : 100
-1 +2 -3 +4 +5 +6 +78 +9 = 100
123 +45 -67 +8 -9 = 100
123 +4 -5 +67 -89 = 100
123 -45 -67 +89 = 100
123 -4 -5 -6 -7 +8 -9 = 100
12 +3 +4 +5 -6 -7 +89 = 100
12 +3 -4 +5 +67 +8 +9 = 100
12 -3 -4 +5 -6 +7 +89 = 100
1 +23 -4 +56 +7 +8 +9 = 100
1 +23 -4 +5 +6 +78 -9 = 100
1 +2 +34 -5 +67 -8 +9 = 100
1 +2 +3 -4 +5 +6 +78 +9 = 100
Valeur atteinte 12 fois.

Plus petite valeur ne pouvant pas être atteinte : 211
Valeur atteinte le plus souvent : 9, atteinte 46 fois.

Plus grandes valeurs pouvant être atteintes :
123456789 = 123456789
23456790 = 1 +23456789
23456788 = -1 +23456789
12345687 = 12345678 +9
12345669 = 12345678 -9
3456801 = 12 +3456789
3456792 = 1 +2 +3456789
3456790 = -1 +2 +3456789
3456788 = 1 -2 +3456789
3456786 = -1 -2 +3456789

Perl 6[edit]

Works with: Rakudo version 2016.12
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;
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"])


Phix[edit]

This is just a trivial count in base 3, with a leading '+' being irrelevant, so from 0(3)000_000_000 to 0(3)122_222_222 which is only (in decimal) 13,122 ...
Admittedly, categorising them into 3429 bins is slightly more effort, but otherwise I am somewhat bemused by all the applescript/javascript/Haskell shenanegins.

enum SUB=-1, NOP=0, ADD=1
 
function eval(sequence s)
integer res = 0, this = 0, op = ADD
for i=1 to length(s) do
if s[i]=NOP then
this = this*10+i
else
res += op*this
this = i
op = s[i]
end if
end for
return res + op*this
end function
 
procedure show(sequence s)
string res = ""
for i=1 to length(s) do
if s[i]!=NOP then
res &= ','-s[i]
end if
res &= '0'+i
end for
puts(1,res&" = ")
end procedure
 
-- Logically this intersperses -/nop/+ between each digit, but you do not actually need the digit.
sequence s = repeat(SUB,9) -- (==> ..nop+add*8)
 
bool done = false
integer maxl = 0, maxr
integer count = 0
while not done do
count += 1
integer r = eval(s), k = getd_index(r)
sequence solns = iff(k=0?{s}:append(getd_by_index(k),s))
setd(r,solns)
if r>0 and maxl<length(solns) then
maxl = length(solns)
maxr = r
end if
for i=length(s) to 1 by -1 do
if i=1 and s[i]=NOP then
done = true
exit
elsif s[i]!=ADD then
s[i] += 1
exit
end if
s[i] = SUB
end for
end while
 
printf(1,"%d solutions considered (dictionary size: %d)\n",{count,dict_size()})
 
sequence s100 = getd(100)
printf(1,"There are %d sums to 100:\n",{length(s100)})
for i=1 to length(s100) do
show(s100[i])
 ?100
end for
 
printf(1,"The positive sum of %d has the maximum number of solutions: %d\n",{maxr,maxl})
 
integer prev = 0
function missing(integer key, sequence /*data*/, integer /*pkey*/, object /*user_data=-2*/)
if key!=prev+1 then
return 0
end if
prev = key
return 1
end function
traverse_dict_partial_key(routine_id("missing"),1)
printf(1,"The lowest positive sum that cannot be expressed: %d\n",{prev+1})
 
sequence highest = {}
function top10(integer key, sequence /*data*/, object /*user_data*/)
highest &= key
return length(highest)<10
end function
traverse_dict(routine_id("top10"),rev:=1)
printf(1,"The 10 highest sums: ") ?highest
Output:
13122 solutions considered (dictionary size: 3429)
There are 12 sums to 100:
-1+2-3+4+5+6+78+9 = 100
12-3-4+5-6+7+89 = 100
123-4-5-6-7+8-9 = 100
123-45-67+89 = 100
123+4-5+67-89 = 100
123+45-67+8-9 = 100
12+3-4+5+67+8+9 = 100
12+3+4+5-6-7+89 = 100
1+23-4+56+7+8+9 = 100
1+23-4+5+6+78-9 = 100
1+2+3-4+5+6+78+9 = 100
1+2+34-5+67-8+9 = 100
The positive sum of 9 has the maximum number of solutions: 46
The lowest positive sum that cannot be expressed: 211
The 10 highest sums: {123456789,23456790,23456788,12345687,12345669,3456801,3456792,3456790,3456788,3456786}

Python[edit]

This example does not show the output mentioned in the task description on this page (or a page linked to from here). Please ensure that it meets all task requirements and remove this message.
Note that phrases in task descriptions such as "print and display" and "print and show" for example, indicate that (reasonable length) output be a part of a language's solution.


 
from itertools import product, islice
 
 
def expr(p):
return "{}1{}2{}3{}4{}5{}6{}7{}8{}9".format(*p)
 
 
def gen_expr():
op = ['+', '-', '']
return [expr(p) for p in product(op, repeat=9) if p[0] != '+']
 
 
def all_exprs():
values = {}
for expr in gen_expr():
val = eval(expr)
if val not in values:
values[val] = 1
else:
values[val] += 1
return values
 
 
def sum_to(val):
for s in filter(lambda x: x[0] == val, map(lambda x: (eval(x), x), gen_expr())):
print(s)
 
 
def max_solve():
print("Sum {} has the maximum number of solutions: {}".
format(*max(all_exprs().items(), key=lambda x: x[1])))
 
 
def min_solve():
values = all_exprs()
for i in range(123456789):
if i not in values:
print("Lowest positive sum that can't be expressed: {}".format(i))
return
 
 
def highest_sums(n=10):
sums = map(lambda x: x[0],
islice(sorted(all_exprs().items(), key=lambda x: x[0], reverse=True), n))
print("Highest Sums: {}".format(list(sums)))
 
 
sum_to(100)
max_solve()
min_solve()
highest_sums()
 
 

Racket[edit]

#lang racket
 
(define list-partitions
(match-lambda
[(list) (list null)]
[(and L (list _)) (list (list L))]
[(list L ...)
(for*/list
((i (in-range 1 (add1 (length L))))
(r (in-list (list-partitions (drop L i)))))
(cons (take L i) r))]))
 
(define digits->number (curry foldl (λ (dgt acc) (+ (* 10 acc) dgt)) 0))
 
(define partition-digits-to-numbers
(let ((memo (make-hash)))
(λ (dgts)
(hash-ref! memo dgts
(λ ()
(map (λ (p) (map digits->number p))
(list-partitions dgts)))))))
 
(define (fold-sum-to-ns digits kons k0)
(define (get-solutions nmbrs acc chain k)
(match nmbrs
[(list)
(kons (cons acc (let ((niahc (reverse chain)))
(if (eq? '+ (car niahc)) (cdr niahc) niahc)))
k)]
[(cons a d)
(get-solutions d (- acc a) (list* a '- chain)
(get-solutions d (+ acc a) (list* a '+ chain) k))]))
(foldl (λ (nmbrs k) (get-solutions nmbrs 0 null k)) k0 (partition-digits-to-numbers digits)))
 
(define sum-to-ns/hash-promise
(delay (fold-sum-to-ns
'(1 2 3 4 5 6 7 8 9)
(λ (a.s d) (hash-update d (car a.s) (λ (x) (cons (cdr a.s) x)) list))
(hash))))
 
(module+ main
(define S (force sum-to-ns/hash-promise))
(displayln "Show all solutions that sum to 100")
(pretty-print (hash-ref S 100))
 
(displayln "Show the sum that has the maximum number of solutions (from zero to infinity*)")
(let-values (([k-max v-max]
(for/fold ((k-max #f) (v-max 0))
(([k v] (in-hash S)) #:when (> (length v) v-max))
(values k (length v)))))
(printf "~a has ~a solutions~%" k-max v-max))
 
(displayln "Show the lowest positive sum that can't be expressed (has no solutions),
using the rules for this task")
(for/first ((n (in-range 1 (add1 123456789))) #:unless (hash-has-key? S n)) n)
 
(displayln "Show the ten highest numbers that can be expressed using the rules for this task")
(take (sort (hash-keys S) >) 10))
 
(module+ test
(require rackunit)
(check-equal? (list-partitions null) '(()))
(check-equal? (list-partitions '(1)) '(((1))))
(check-equal? (list-partitions '(1 2)) '(((1) (2)) ((1 2))))
(check-equal? (partition-digits-to-numbers '()) '(()))
(check-equal? (partition-digits-to-numbers '(1)) '((1)))
(check-equal? (partition-digits-to-numbers '(1 2)) '((1 2) (12))))
Output:
Show all solutions that sum to 100
'((123 - 45 - 67 + 89)
  (123 + 45 - 67 + 8 - 9)
  (123 + 4 - 5 + 67 - 89)
  (123 - 4 - 5 - 6 - 7 + 8 - 9)
  (12 + 3 - 4 + 5 + 67 + 8 + 9)
  (12 - 3 - 4 + 5 - 6 + 7 + 89)
  (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))
Show the sum that has the maximum number of solutions (from zero to infinity*)
9 has 46 solutions
Show the lowest positive sum that can't be expressed (has no solutions),
 using the rules for this task
211
Show the ten highest numbers that can be expressed using the rules for this task
'(123456789 23456790 23456788 12345687 12345669 3456801 3456792 3456790 3456788 3456786)

REXX[edit]

/*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 0.*/
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 # of solutions 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=# /* [↓] adjust the number of solutions?*/
if y==0 then y='no' /* [↓] left justify plural of solution*/
if LO\==00 then say right(y, 9) 'solution's(#, , " ") 'found for' ,
right(j, length(HI) ) left('', #, "─")
return # /*return the number of solutions found.*/

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

Ruby[edit]

Translation of: Elixir
def gen_expr
x = ['-', '']
y = ['+', '-', '']
x.product(y,y,y,y,y,y,y,y)
.map do |a,b,c,d,e,f,g,h,i|
"#{a}1#{b}2#{c}3#{d}4#{e}5#{f}6#{g}7#{h}8#{i}9"
end
end
 
def sum_to(val)
gen_expr.map{|expr| [eval(expr), expr]}.select{|v,expr| v==val}.each{|x| p x}
end
 
def max_solve
n,size = gen_expr.group_by{|expr| eval(expr)}
.select{|val,_| val>=0}
.map{|val,exprs| [val, exprs.size]}
.max_by{|_,size| size}
puts "sum of #{n} has the maximum number of solutions : #{size}"
end
 
def min_solve
solves = gen_expr.group_by{|expr| eval(expr)}
n = 0.step{|i| break i unless solves[i]}
puts "lowest positive sum that can't be expressed : #{n}"
end
 
def highest_sums(n=10)
n = gen_expr.map{|expr| eval(expr)}.uniq.sort.reverse.take(n)
puts "highest sums : #{n}"
end
 
sum_to(100)
max_solve
min_solve
highest_sums
Output:
[100, "-1+2-3+4+5+6+78+9"]
[100, "1+2+3-4+5+6+78+9"]
[100, "1+2+34-5+67-8+9"]
[100, "1+23-4+5+6+78-9"]
[100, "1+23-4+56+7+8+9"]
[100, "12+3+4+5-6-7+89"]
[100, "12+3-4+5+67+8+9"]
[100, "12-3-4+5-6+7+89"]
[100, "123+4-5+67-89"]
[100, "123+45-67+8-9"]
[100, "123-4-5-6-7+8-9"]
[100, "123-45-67+89"]
sum of 9 has the maximum number of solutions : 46
lowest positive sum that can't be expressed : 211
highest sums : [123456789, 23456790, 23456788, 12345687, 12345669, 3456801, 3456792, 3456790, 3456788, 3456786]

Tcl[edit]

proc sum_to_100 {} {
for {set i 0} {$i <= 13121} {incr i} {
set i3 [format %09d [dec2base 3 $i]]
set form ""
set subs {"" - +}
foreach a [split $i3 ""] b [split 123456789 ""] {
append form [lindex $subs $a] $b
}
lappend R([expr $form]) $form
}
puts "solutions for sum=100:\n[join [lsort $R(100)] \n]"
set max -1
foreach key [array names R] {
if {[llength $R($key)] > $max} {
set max [llength $R($key)]
set maxkey $key
}
}
puts "max solutions: $max for $maxkey"
for {set i 0} {$i <= 123456789} {incr i} {
if ![info exists R($i)] {
puts "first unsolvable: $i"
break
}
}
puts "highest 10:\n[lrange [lsort -integer -decr [array names R]] 0 9]"
}
proc dec2base {base dec} {
set res ""
while {$dec > 0} {
set res [expr $dec%$base]$res
set dec [expr $dec/$base]
}
if {$res eq ""} {set res 0}
return $res
}
sum_to_100
~ $ ./sum_to_100.tcl
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 for 9
first unsolvable: 211
highest 10:
123456789 23456790 23456788 12345687 12345669 3456801 3456792 3456790 3456788 3456786

zkl[edit]

Taking a big clue from Haskell and just calculate the world.

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") }
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();
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))