Problem 185: Number Mind

(see projecteuler.net/problem=185)

The game Number Mind is a variant of the well known game Master Mind.

Instead of coloured pegs, you have to guess a secret sequence of digits. After each guess you're only told in how many places you've guessed the correct digit.
So, if the sequence was 1234 and you guessed 2036, you'd be told that you have one correct digit; however, you would NOT be told that you also have another digit in the wrong place.

For instance, given the following guesses for a 5-digit secret sequence,

90342 ;2 correct
70794 ;0 correct
39458 ;2 correct
34109 ;1 correct
51545 ;2 correct
12531 ;1 correct

The correct sequence 39542 is unique.

Based on the following guesses,

5616185650518293 ;2 correct
3847439647293047 ;1 correct
5855462940810587 ;3 correct
9742855507068353 ;3 correct
4296849643607543 ;3 correct
3174248439465858 ;1 correct
4513559094146117 ;2 correct
7890971548908067 ;3 correct
8157356344118483 ;1 correct
2615250744386899 ;2 correct
8690095851526254 ;3 correct
6375711915077050 ;1 correct
6913859173121360 ;1 correct
6442889055042768 ;2 correct
2321386104303845 ;0 correct
2326509471271448 ;2 correct
5251583379644322 ;2 correct
1748270476758276 ;3 correct
4895722652190306 ;1 correct
3041631117224635 ;3 correct
1841236454324589 ;3 correct
2659862637316867 ;2 correct

Find the unique 16-digit secret sequence.

My Algorithm

I attempted to solve this problem with constraint-based programming but failed miserably: my ideas were either too slow or plain wrong ...

Then I managed to solve it using simulated annealing:

The last point is very important because otherwise you can get stuck in a local optimum.

distance compares the current sequence against all 22 guesses.
My error metric is the sum of all differences between the actually matching digits and the expected number.
For example, comparing 6666666666666666 against the first guess 5616185650518293 finds 3 matching digits, but only 2 were expected → the error is 1.

I added a simple pseudo-random generator to ensure that my solution terminates successfully on various compilers (with different implementations of rand()).
It takes about 0.2 seconds on my machine to find the correct solution.

Let's be honest: the number of iterations without improvement (MaxRoundsWithoutImprovement = 20) is fine-tuned to minimize execution time.
Changing it to 19 or 21 makes the program significantly slower (more than a second).
And changing the constants of my random-number generator (a few candidates are listed on Wikipedia) may have the same effect.

Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This live test is based on the Hackerrank problem.

Input data (separated by spaces or newlines):
Note: Enter the number of guesses and then each guess followed by the number of correct digits.

This is equivalent to
echo "6 90342 2 70794 0 39458 2 34109 1 51545 2 12531 1" | ./185

Output:

(please click 'Go !')

(this interactive test is still under development, computations will be aborted after one second)

My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

The code contains #ifdefs to switch between the original problem and the Hackerrank version.
Enable #ifdef ORIGINAL to produce the result for the original problem (default setting for most problems).

#include <iostream>
#include <vector>
#include <string>
 
// all 22 guesses
typedef std::vector<unsigned char> Sequence;
std::vector<Sequence> sequences;
// and how many of their digits match the secret number
std::vector<unsigned int> hits;
 
// a simple pseudo-random number generator, result in 0 .. modulo - 1
// (produces the same result no matter what compiler you have - unlike rand() from math.h)
unsigned int myrand(unsigned int modulo)
{
static unsigned int seed = 0;
seed = 1103515245 * seed + 12345;
return seed % modulo;
}
 
// replace reference by a new random digit (0..9)
void shuffle(unsigned char& digit)
{
auto old = digit;
do
digit = myrand(10);
while (digit == old);
}
 
// a player's guess and how many digits were correct
void add(const std::string& guess, unsigned int matches)
{
// convert from ASCII to int
Sequence s;
for (auto c : guess)
s.push_back(c - '0');
sequences.push_back(s);
 
hits.push_back(matches);
}
 
// compute how many digits of the guesses don't match to the currently analyzed number
// a perfect match returns 0, "mismatches" return > 0
unsigned int distance(const Sequence& current)
{
unsigned int errors = 0;
 
for (unsigned int i = 0; i < sequences.size(); i++)
{
// count number of matching digits
unsigned int same = 0;
for (unsigned int j = 0; j < current.size(); j++)
if (current[j] == sequences[i][j])
same++;
 
// too many identical digits ?
if (same > hits[i])
errors += same - hits[i];
else // or too few ?
errors += hits[i] - same;
}
 
return errors;
}
 
int main()
{
//#define ORIGINAL
#ifdef ORIGINAL
// guesses of the problem
add("5616185650518293", 2);
add("3847439647293047", 1);
add("5855462940810587", 3);
add("9742855507068353", 3);
add("4296849643607543", 3);
add("3174248439465858", 1);
add("4513559094146117", 2);
add("7890971548908067", 3);
add("8157356344118483", 1);
add("2615250744386899", 2);
add("8690095851526254", 3);
add("6375711915077050", 1);
add("6913859173121360", 1);
add("6442889055042768", 2);
add("2321386104303845", 0);
add("2326509471271448", 2);
add("5251583379644322", 2);
add("1748270476758276", 3);
add("4895722652190306", 1);
add("3041631117224635", 3);
add("1841236454324589", 3);
add("2659862637316867", 2);
#else
unsigned int numGuesses;
std::cin >> numGuesses;
while (numGuesses--)
{
std::string guess;
unsigned int correct;
std::cin >> guess >> correct;
add(guess.c_str(), correct);
}
#endif
 
// initially a purely random guess
const auto NumDigits = sequences.front().size();
Sequence current(NumDigits, 0);
for (auto& x : current)
shuffle(x);
 
// shuffle a random digit when stuck in a local optimum, too
const auto MaxRoundsWithoutImprovement = 20; // "sweet spot" for my random number generator
auto quietRounds = 0;
 
auto errors = distance(current);
auto previous = errors;
while (errors != 0)
{
// replace every digit by a different random number, keep those that minimize the error metric
for (auto& digit : current)
{
// replace by a new random digit
auto previousDigit = digit;
do
shuffle(digit);
while (digit == previousDigit);
 
// compute error metric
auto modified = distance(current);
if (modified <= errors)
{
// better than before, adjust error level and keep new digit
errors = modified;
}
else
// mutation is equal or worse, restore old digit
digit = previousDigit;
}
 
// unchanged score ? we didn't improve on the previous solution ...
if (errors == previous)
{
// stuck too long ? try to escape local optimum
quietRounds++;
if (quietRounds == MaxRoundsWithoutImprovement)
{
// change a random number
shuffle(current[myrand(current.size())]);
errors = distance(current);
 
// reset counter
quietRounds = 0;
}
}
else
{
// we got closer to the goal ...
quietRounds = 0;
previous = errors;
}
}
 
// show solution
for (auto c : current)
std::cout << int(c);
std::cout << std::endl;
 
return 0;
}

This solution contains 18 empty lines, 26 comments and 6 preprocessor commands.

Benchmark

The correct solution to the original Project Euler problem was found in 0.18 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

Changelog

May 30, 2017 submitted solution
May 30, 2017 added comments
August 8, 2017 modified to solve Hackerrank, too

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler185

My code solves 21 out of 21 test cases (score: 100%)

Difficulty

55% Project Euler ranks this problem at 55% (out of 100%).

Hackerrank describes this problem as medium.

Note:
Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.
In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is rarely an option.

Heatmap

Please click on a problem's number to open my solution to that problem:

green   solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too
yellow solutions score less than 100% at Hackerrank (but still solve the original problem easily)
gray problems are already solved but I haven't published my solution yet
blue solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much
orange problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte
red problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too
black problems are solved but access to the solution is blocked for a few days until the next problem is published
[new] the flashing problem is the one I solved most recently
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The 306 solved problems (that's level 12) had an average difficulty of 32.5% at Project Euler and
I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Look at my progress and performance pages to get more details.

more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !