<< problem 171 - Finding numbers for which the sum of the squares ... Using up to one million tiles how many different ... - problem 173 >>

# Problem 172: Investigating numbers with few repeated digits

How many 18-digit numbers n (without leading zeros) are there such that no digit occurs more than three times in n?

# My Algorithm

A straight-forward recursive solution:

• append a digit (0 to 9) if that digit's limit isn't exceeded (maxUse = 3)
• stop if enough digits where appended (maxDigit = 18)
The number of used digits is converted to a fingerprint, which doesn't care about the order of digits because appending something
to 12345 will produce the same number of combinations as appending to 54321.

Since many fingerprints will occur repeatedly (fingerprint of 12345 is the same as the fingerprint of 54321) they are stored in a cache.

# Interactive test

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

Input data (separated by spaces or newlines):
Note: Enter the total number of digits followed by how often each digit can be used at most

This is equivalent to
echo "10 2" | ./172

Output:

Note: the original problem's input 18 3 cannot be entered
because just copying results is a soft skill reserved for idiots.

(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. Or just jump to my GitHub repository.

       #include <iostream>

// identify a combination of digits, order doesn't matter
union Fingerprint
{
struct
{
unsigned char zero  : 2;
unsigned char one   : 2;
unsigned char two   : 2;
unsigned char three : 2;
unsigned char four  : 2;
unsigned char five  : 2;
unsigned char six   : 2;
unsigned char seven : 2;
unsigned char eight : 2;
unsigned char nine  : 2;
};
unsigned int id;
};
// 10 digits with 2 bits each => fingerprint's id is below 1 << 20
unsigned long long cache[1 << 20] = { 0 };

unsigned int maxDigits = 18; // a total of 18 digits
unsigned int maxUse    =  3; // each digit at most 3 times

// compute number of possible numbers
unsigned long long search(Fingerprint current, unsigned int digits)
{
// done ?
if (digits == maxDigits)
return 1;

// use memoized results if possible
if (cache[current.id] > 0)
return cache[current.id];

// count combinations
unsigned long long result = 0;

// must not place a zero at the first position
if (digits > 0 && current.zero < maxUse)
{
auto next = current;
next.zero++;
result += search(next, digits + 1);
}
// the following if's are all the same, except that they increment next.one, next.two, next.three, ...
if (current.one < maxUse)
{
auto next = current;
next.one++;
result += search(next, digits + 1);
}
if (current.two < maxUse)
{
auto next = current;
next.two++;
result += search(next, digits + 1);
}
if (current.three < maxUse)
{
auto next = current;
next.three++;
result += search(next, digits + 1);
}
if (current.four < maxUse)
{
auto next = current;
next.four++;
result += search(next, digits + 1);
}
if (current.five < maxUse)
{
auto next = current;
next.five++;
result += search(next, digits + 1);
}
if (current.six < maxUse)
{
auto next = current;
next.six++;
result += search(next, digits + 1);
}
if (current.seven < maxUse)
{
auto next = current;
next.seven++;
result += search(next, digits + 1);
}
if (current.eight < maxUse)
{
auto next = current;
next.eight++;
result += search(next, digits + 1);
}
if (current.nine < maxUse)
{
auto next = current;
next.nine++;
result += search(next, digits + 1);
}

cache[current.id] = result;
return result;
}

int main()
{
std::cin >> maxDigits >> maxUse;

// live test only: catch invalid input
if (maxDigits == 0 || maxDigits > 29 ||
maxUse    == 0 || maxUse    >  3)
return 1;

Fingerprint start;
start.id = 0;
std::cout << search(start, 0) << std::endl;
return 0;
}


This solution contains 10 empty lines, 10 comments and 1 preprocessor command.

# Benchmark

The correct solution to the original Project Euler problem was found in 0.14 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 10 MByte.

(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

June 15, 2017 submitted solution

# Difficulty

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

# 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

I stopped working on Project Euler problems around the time they released 617.
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The 310 solved problems (that's level 12) had an average difficulty of 32.6% 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.

 << problem 171 - Finding numbers for which the sum of the squares ... Using up to one million tiles how many different ... - problem 173 >>
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