I'm stuck on this verbal arithmetic algorithm

Question

Given the format string (11 characters, without space) in the form of ABC+DEF=GHI, where A, B, C, D, E, F, G, H, I are representing decimal digit (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) "placeholders" (not necessarily distinct!), develop and implement an algorithm to find the 3-digit positive numbers (ABC, DEF, and GHI) that give maximum GHI according to the equality given. The letters (A, B, C, D, E, F, G, H, I) are "placeholders", so you may have repeating digits. For example in the formula ABA+BBB=GGB, all B’s represent same decimal digit. Your algorithm should also check if such numbers are not possible to find. For example, for the formula AAB+AAB=AAB, it is not possible to satisfy it with decimal digits. Your program will tell "No solution!".There might be more than one solution. In this case just output the first one your algorithm finds. Continue till the user enter -1 for the formula for which your program will exit by saying "Bye!".

Here is a sample run:

Enter the formula:AAB+AAB=AAB

No solution! Enter the formula:AAA+BBB=AAA

999+000=999

Enter the formula: -1

Bye!

Here is the code I've written before:

#include<iostream>
#include<cmath>
#include<string>
using namespace std;
int main()
{
    string str = "abcdefghi";
    string num = "0123456789";
    string formula;
    int var1[3];
    int var2[3];
    int var3[3];
    while (1)
{
    cout << "Enter the formula: ";
    cin >> formula;
    if (formula == "-1")
    {
        cout << "Bye" << endl;
        break;
    }
    else
    {
        for (int i = 0; i < str.length(); i++)
        {
            if (formula.at(0) == str.at(i))
            {
                var1[0] = i;
            }
            if (formula.at(1) == str.at(i))
            {
                var1[1] = i;
            }
            if (formula.at(2) == str.at(i))
            {
                var1[2] = i;
            }
        }
        for (int i = 0; i < str.length(); i++)
        {
            if (formula.at(4) == str.at(i))
            {
                var2[0] = i;
            }
            if (formula.at(5) == str.at(i))
            {
                var2[1] = i;
            }
            if (formula.at(6) == str.at(i))
            {
                var2[2] = i;
            }
        }
        for (int i = 0; i < str.length(); i++)
        {
            if (formula.at(8) == str.at(i))
            {
                var3[0] = i;
            }
            if (formula.at(9) == str.at(i))
            {
                var3[1] = i;
            }
            if (formula.at(10) == str.at(i))
            {
                var3[2] = i;
            }
        }
        if (var1[0] + var2[0] == var3[0] && var1[1] + var2[1] == var3[1] && var1[2] + var2[2] == var3[2])
            cout << var1[0] << var1[1] << var1[2] << "+" << var2[0] << var2[1] << var2[2] << "=" << var3[0] << var3[1] << var3[2] << endl << endl;
        else
            cout << "No Solution!" << endl << endl;
    }
}
system("pause");
return 0;
}

But it doesn't seem to solve the problem exactly. How can I modify my code to get what this question wants.

Answer 1

The answer can be found by brute force. Assume you have an array or vector of 10 digits. With 10 possible digit values there are 10! = 3628800 possible permutations. Of each permutation you can assign the first 9 elements to each of the 9 possible letters check if that provides a new max value.

If your letter values are in int letter_vals[10]; for example then you populate the initial values with:

std::iota(std::begin(letter_vals), std::end(letter_vals), 0);

Then you can iterate through the possible letter assignments:

while (std::next_permutation(std::begin(letter_vals), std::end(letter_vals) )) {

A string can be translated to a value through the permutation e.g.

val1 = letter_vals[(var1[0] - 'A')]*100;
val1 += letter_vals[(var1[1] - 'A')]*10;
val1 += letter_vals[(var1[2] - 'A')]*1;

---

If different letters may represent the same digit then you could also have applied brute force, but then you would have 10 possible values per letter and 9 letters for a total number of 10^9 possible combinations. Perhaps something like:

for(size_t i{0}; i <= 999999999; ++i) {
    auto current_num{i};
    for(auto& letter_val : letter_vals) {
        letter_val = current_num % 10;
        current_num /= 10;
    }
    std::reverse(std::begin(letter_vals), std::end(letter_vals));

---

The answers differ depending on whether or not repeating digits are allowed. Without repeating digits the program is much faster, with fewer answers: Evaluating: AAC+BBF=GHI

New max answer found: 345
116+229=345
New max answer found: 346
117+229=346
New max answer found: 347
118+229=347
New max answer found: 450
112+338=450
New max answer found: 456
117+339=456
New max answer found: 457
118+339=457
New max answer found: 560
112+448=560
New max answer found: 562
113+449=562
New max answer found: 567
118+449=567
New max answer found: 670
112+558=670
New max answer found: 672
113+559=672
New max answer found: 673
114+559=673
New max answer found: 782
113+669=782
New max answer found: 783
114+669=783
New max answer found: 784
115+669=784
New max answer found: 890
114+776=890
New max answer found: 891
224+667=891
Maxval was: 891
224+667=891

With repeating digits there are more possible combinations:

Evaluating: AAC+BBF=GHI
New max answer found: 1
000+001=001
New max answer found: 2
000+002=002
New max answer found: 3
000+003=003
New max answer found: 4
000+004=004
New max answer found: 5
000+005=005
New max answer found: 6
000+006=006
New max answer found: 7
000+007=007
New max answer found: 8
000+008=008
New max answer found: 9
000+009=009
New max answer found: 10
001+009=010
New max answer found: 11
002+009=011
New max answer found: 12
003+009=012
New max answer found: 13
004+009=013
New max answer found: 14
005+009=014
New max answer found: 15
006+009=015
New max answer found: 16
007+009=016
New max answer found: 17
008+009=017
New max answer found: 18
009+009=018
New max answer found: 110
000+110=110
New max answer found: 111
000+111=111
New max answer found: 112
000+112=112
New max answer found: 113
000+113=113
New max answer found: 114
000+114=114
New max answer found: 115
000+115=115
New max answer found: 116
000+116=116
New max answer found: 117
000+117=117
New max answer found: 118
000+118=118
New max answer found: 119
000+119=119
New max answer found: 120
001+119=120
New max answer found: 121
002+119=121
New max answer found: 122
003+119=122
New max answer found: 123
004+119=123
New max answer found: 124
005+119=124
New max answer found: 125
006+119=125
New max answer found: 126
007+119=126
New max answer found: 127
008+119=127
New max answer found: 128
009+119=128
New max answer found: 220
000+220=220
New max answer found: 221
000+221=221
New max answer found: 222
000+222=222
New max answer found: 223
000+223=223
New max answer found: 224
000+224=224
New max answer found: 225
000+225=225
New max answer found: 226
000+226=226
New max answer found: 227
000+227=227
New max answer found: 228
000+228=228
New max answer found: 229
000+229=229
New max answer found: 230
001+229=230
New max answer found: 231
002+229=231
New max answer found: 232
003+229=232
New max answer found: 233
004+229=233
New max answer found: 234
005+229=234
New max answer found: 235
006+229=235
New max answer found: 236
007+229=236
New max answer found: 237
008+229=237
New max answer found: 238
009+229=238
New max answer found: 330
000+330=330
New max answer found: 331
000+331=331
New max answer found: 332
000+332=332
New max answer found: 333
000+333=333
New max answer found: 334
000+334=334
New max answer found: 335
000+335=335
New max answer found: 336
000+336=336
New max answer found: 337
000+337=337
New max answer found: 338
000+338=338
New max answer found: 339
000+339=339
New max answer found: 340
001+339=340
New max answer found: 341
002+339=341
New max answer found: 342
003+339=342
New max answer found: 343
004+339=343
New max answer found: 344
005+339=344
New max answer found: 345
006+339=345
New max answer found: 346
007+339=346
New max answer found: 347
008+339=347
New max answer found: 348
009+339=348
New max answer found: 440
000+440=440
New max answer found: 441
000+441=441
New max answer found: 442
000+442=442
New max answer found: 443
000+443=443
New max answer found: 444
000+444=444
New max answer found: 445
000+445=445
New max answer found: 446
000+446=446
New max answer found: 447
000+447=447
New max answer found: 448
000+448=448
New max answer found: 449
000+449=449
New max answer found: 450
001+449=450
New max answer found: 451
002+449=451
New max answer found: 452
003+449=452
New max answer found: 453
004+449=453
New max answer found: 454
005+449=454
New max answer found: 455
006+449=455
New max answer found: 456
007+449=456
New max answer found: 457
008+449=457
New max answer found: 458
009+449=458
New max answer found: 550
000+550=550
New max answer found: 551
000+551=551
New max answer found: 552
000+552=552
New max answer found: 553
000+553=553
New max answer found: 554
000+554=554
New max answer found: 555
000+555=555
New max answer found: 556
000+556=556
New max answer found: 557
000+557=557
New max answer found: 558
000+558=558
New max answer found: 559
000+559=559
New max answer found: 560
001+559=560
New max answer found: 561
002+559=561
New max answer found: 562
003+559=562
New max answer found: 563
004+559=563
New max answer found: 564
005+559=564
New max answer found: 565
006+559=565
New max answer found: 566
007+559=566
New max answer found: 567
008+559=567
New max answer found: 568
009+559=568
New max answer found: 660
000+660=660
New max answer found: 661
000+661=661
New max answer found: 662
000+662=662
New max answer found: 663
000+663=663
New max answer found: 664
000+664=664
New max answer found: 665
000+665=665
New max answer found: 666
000+666=666
New max answer found: 667
000+667=667
New max answer found: 668
000+668=668
New max answer found: 669
000+669=669
New max answer found: 670
001+669=670
New max answer found: 671
002+669=671
New max answer found: 672
003+669=672
New max answer found: 673
004+669=673
New max answer found: 674
005+669=674
New max answer found: 675
006+669=675
New max answer found: 676
007+669=676
New max answer found: 677
008+669=677
New max answer found: 678
009+669=678
New max answer found: 770
000+770=770
New max answer found: 771
000+771=771
New max answer found: 772
000+772=772
New max answer found: 773
000+773=773
New max answer found: 774
000+774=774
New max answer found: 775
000+775=775
New max answer found: 776
000+776=776
New max answer found: 777
000+777=777
New max answer found: 778
000+778=778
New max answer found: 779
000+779=779
New max answer found: 780
001+779=780
New max answer found: 781
002+779=781
New max answer found: 782
003+779=782
New max answer found: 783
004+779=783
New max answer found: 784
005+779=784
New max answer found: 785
006+779=785
New max answer found: 786
007+779=786
New max answer found: 787
008+779=787
New max answer found: 788
009+779=788
New max answer found: 880
000+880=880
New max answer found: 881
000+881=881
New max answer found: 882
000+882=882
New max answer found: 883
000+883=883
New max answer found: 884
000+884=884
New max answer found: 885
000+885=885
New max answer found: 886
000+886=886
New max answer found: 887
000+887=887
New max answer found: 888
000+888=888
New max answer found: 889
000+889=889
New max answer found: 890
001+889=890
New max answer found: 891
002+889=891
New max answer found: 892
003+889=892
New max answer found: 893
004+889=893
New max answer found: 894
005+889=894
New max answer found: 895
006+889=895
New max answer found: 896
007+889=896
New max answer found: 897
008+889=897
New max answer found: 898
009+889=898
New max answer found: 990
000+990=990
New max answer found: 991
000+991=991
New max answer found: 992
000+992=992
New max answer found: 993
000+993=993
New max answer found: 994
000+994=994
New max answer found: 995
000+995=995
New max answer found: 996
000+996=996
New max answer found: 997
000+997=997
New max answer found: 998
000+998=998
New max answer found: 999
000+999=999
Maxval was: 999
000+999=999