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)


‡   (where   infinity   would be a relatively small   123,456,789)



An example of a sum that can't be expressed   (within the rules of this task)   is:   5074

(which,   of course,   isn't the lowest positive sum that can't be expressed).




/* 
 * RossetaCode: Sum to 100, C++, STL, OOP. 
 * Works with: MSC 16.0 (MSVS2010); GCC 5.1 (use -std=c++11 or -std=c++14 etc.).
 *
 * Find solutions to the "sum to one hundred" puzzle.
 */
#include <iostream>
#include <iomanip>
#include <algorithm>
#include <string>
#include <set>
#include <map>
 
using namespace std;
 
class Expression{
    private:
        enum { NUMBER_OF_DIGITS = 9 }; // hack for C++98, use const int in C++11
        enum Op { ADD, SUB, JOIN };
        int code[NUMBER_OF_DIGITS];
    public:
        static const int NUMBER_OF_EXPRESSIONS;
        Expression(){
            for ( int i = 0; i < NUMBER_OF_DIGITS; i++ )
                code[i] = ADD;
        }
        Expression& operator++(int){ // post incrementation
            for ( int i = 0; i < NUMBER_OF_DIGITS; i++ )
                if ( ++code[i] > JOIN ) code[i] = ADD; 
                else break;
            return *this;        
        }
        operator int() const{
            int value = 0, number = 0, sign = (+1);
            for ( int digit = 1; digit <= 9; digit++ )
                switch ( code[NUMBER_OF_DIGITS - digit] ){
                case ADD: value += sign*number; number = digit; sign = (+1); break;
                case SUB: value += sign*number; number = digit; sign = (-1); break;
                case JOIN:                      number = 10*number + digit;  break;
            }
            return value + sign*number;
        }
        operator string() const{
            string s;
            for ( int digit = 1; digit <= NUMBER_OF_DIGITS; digit++ ){
                switch( code[NUMBER_OF_DIGITS - digit] ){
                    case ADD: if ( digit > 1 ) s.push_back('+'); break;
                    case SUB:                  s.push_back('-'); break;
                }
                s.push_back('0' + digit);
            }
            return s;
        }
};
const int Expression::NUMBER_OF_EXPRESSIONS = 2 * 3*3*3*3 * 3*3*3*3;
 
ostream& operator<< (ostream& os, Expression& ex){
    ios::fmtflags oldFlags(os.flags());
    os << setw(9) << right << static_cast<int>(ex)    << " = " 
       << setw(0) << left  << static_cast<string>(ex) << endl;
    os.flags(oldFlags);
    return os;
}
 
struct Stat{
    map<int,int> countSum;
    map<int, set<int> > sumCount;
    Stat(){
        Expression expression;
        for ( int i = 0; i < Expression::NUMBER_OF_EXPRESSIONS; i++, expression++ )
            countSum[expression]++;
        for ( auto it = countSum.begin(); it != countSum.end(); it++ )
            sumCount[it->second].insert(it->first);
    }
};
 
void print(int givenSum){
    Expression expression;
    for ( int i = 0; i < Expression::NUMBER_OF_EXPRESSIONS; i++, expression++ )
        if ( expression == givenSum ) 
            cout << expression;
}
 
void comment(string commentString){
    cout << endl << commentString << endl << endl;
}
 
int main(){
    Stat stat;
 
    comment( "Show all solutions that sum to 100" );
    const int givenSum = 100;
    print(givenSum);
 
    comment( "Show the sum that has the maximum number of solutions" );
    auto maxi = max_element(stat.sumCount.begin(),stat.sumCount.end());
    auto it = maxi->second.begin(); 
    while ( *it < 0 ) it++;
    cout << static_cast<int>(*it) << " has " << maxi->first << " solutions" << endl;
 
    comment( "Show the lowest positive number that can't be expressed" );
    int value = 0; 
    while(stat.countSum.count(value) != 0) value++;
    cout << value << endl;
 
    comment( "Show the ten highest numbers that can be expressed" );
    auto rit = stat.countSum.rbegin();
    for ( int i = 0; i < 10; i++, rit++ ) print(rit->first);
 
    return 0;
}