<< problem 190 - Maximising a weighted product Squarefree Numbers - problem 193 >>

# Problem 191: Prize Strings

A particular school offers cash rewards to children with good attendance and punctuality.
If they are absent for three consecutive days or late on more than one occasion then they forfeit their prize.

During an n-day period a trinary string is formed for each child consisting of L's (late), O's (on time), and A's (absent).

Although there are eighty-one trinary strings for a 4-day period that can be formed, exactly forty-three strings would lead to a prize:

OOOO OOOA OOOL OOAO OOAA OOAL OOLO OOLA OAOO OAOA
OAOL OAAO OAAL OALO OALA OLOO OLOA OLAO OLAA AOOO
AOOA AOOL AOAO AOAA AOAL AOLO AOLA AAOO AAOA AAOL
AALO AALA ALOO ALOA ALAO ALAA LOOO LOOA LOAO LOAA
LAOO LAOA LAAO

How many "prize" strings exist over a 30-day period?

# My Algorithm

Another dynamic programming problem ... my function count has 3 parameters:

• day stands for the number of days to be evaluated, initially 30
• absent counts the consecutive absent days
• late represents the total number of days where the pupil was late
There are two important things:
1. absent has to be reset to zero everytime the pupil shows up (it doesn't matter whether on time or late)
2. A massive amount of situations produce the same parameter set. That means, some parameter sets are evaluated several thousand times.

A simple memoization scheme brings down the computation time from several seconds to less than 10 milliseconds:
There are at most 30*2*3 = 1800 different parameter sets:
• 30 days
• child was 0 or 1 days late so far
• he/she was 0, 1 or 2 days absent (only counting the most recent days)
A simple hash days * 2 * 3 + absent * 2 + late is unique for each parameter set and used as a index for a small cache.

## Note

My cache can hold the results for up to 80 days.
The result fits in unsigned int for 30 days, too.

# 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 number of days

This is equivalent to
echo 4 | ./191

Output:

Note: the original problem's input 30 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++ and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       #include <iostream>

// memoize intermediate results
unsigned long long cache[80*2*3] = { 0 };

// recursively evaluate a day:
// days   - number of days left
// absent - number of consecutive absent day (only consider up to three past days)
// late   - total number of late days
unsigned long long count(unsigned int days, unsigned int absent = 0, unsigned int late = 0)
{
// too many consecutive absent days ?
if (absent == 3)
return 0;
// too late to often ?
if (late > 1)
return 0;

// all days passed ? => collect prize
if (days == 0)
return 1;

// unique ID of current parameter set
unsigned int hash = days * 2 * 3 + absent * 2 + late;
if (cache[hash] != 0)
return cache[hash];

unsigned long long result;
// assume pupil is today neither late nor absent
result  = count(days - 1, 0, late);
// assume pupil is absent today
result += count(days - 1, absent + 1, late);
// assume pupil is late today
result += count(days - 1, 0, late + 1);

// store result
cache[hash] = result;
return result;
}

int main()
{
unsigned int days;
std::cin  >> days;
std::cout << count(days) << std::endl;
return 0;
}


This solution contains 7 empty lines, 14 comments and 1 preprocessor command.

# Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 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 25, 2017 submitted solution

# Difficulty

Project Euler ranks this problem at 45% (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 190 - Maximising a weighted product Squarefree Numbers - problem 193 >>
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