<< problem 231 - The prime factorisation of binomial coefficients | Semidivisible numbers - problem 234 >> |
Problem 232: The Race
(see projecteuler.net/problem=232)
Two players share an unbiased coin and take it in turns to play "The Race".
On Player 1's turn, he tosses the coin once: if it comes up Heads, he scores one point; if it comes up Tails, he scores nothing.
On Player 2's turn, she chooses a positive integer T and tosses the coin T times: if it comes up all Heads, she scores 2^{T-1} points; otherwise, she scores nothing.
Player 1 goes first. The winner is the first to 100 or more points.
On each turn Player 2 selects the number, T, of coin tosses that maximises the probability of her winning.
What is the probability that Player 2 wins?
Give your answer rounded to eight decimal places in the form 0.abcdefgh .
My Algorithm
My solution follows the standard Dynamic Programming pattern:
the solution is found recursively with aggressive caching of partial results.
The function twoWins
considers only player 2's move. Its parameter is the number of remaining points:
- if zero points left for player 2, then player 2 won
- if zero points left for player 1, then player 1 won
- in any other case, all possible moves of player 2 are analyzed and the best is chosen
There are four cases:
1. player 2 has all heads up, player 1 has heads up as well
2. player 2 has all heads up, player 1 has tails
3. player 2 has tosses tails at least once, player 1 has heads up
4. both players toss tails
The probabilities of cases 1 to 3 are well defined (
win1
and win2
).The only "tricky" case is case 4 where basically nothing changes.
Since
twoWins
focuses on player 2, I have to manually play the first turn of player 1 in main
andadd the probabilities for both (equally likely) outcomes of player 1's first turn.
Note
I was sure to have a correct solution but was rejected by the Project Euler website.
A simple Monte-Carlo simulation told me that my result should be okay, too (considering a certain error margin).
It took me quite some time to figure out that I have to switch the order of the first lines in twoWins
:
check first whether player 2 won, only then check player 1. That was a nasty mistake.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 10 | ./232
Output:
Note: the original problem's input 100
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>
#include <iomanip>
#include <vector>
// the first player with 100 points wins
unsigned int maxScore = 100;
// fixed probabilities of player one
const double win1 = 0.5;
const double lose1 = 1 - win1; // = 0.5
// return chance of winning for player 2 when it's his/her turn
double twoWins(unsigned int needPointsOne, unsigned int needPointsTwo)
{
// player two won
if (needPointsTwo == 0)
return 1;
// player one won
if (needPointsOne == 0)
return 0;
// memoize
const double Unknown = -1;
static std::vector<double> cache(maxScore * maxScore, Unknown);
auto id = (needPointsOne - 1) * maxScore + needPointsTwo - 1;
if (cache[id] != Unknown)
return cache[id];
// find highest chance of winning
double best = 0;
// current bet (observation: betting more than 2^3 = 8 never seems profitable)
unsigned int bet = 1;
while (true)
{
// probabilities
auto win2 = 0.5 / bet; // 2^(bet-1)
auto lose2 = 1 - win2;
// in case player two scores he shall not exceed the maximum score
auto nextPointsTwo = (needPointsTwo < bet) ? 0 : needPointsTwo - bet;
// at least one player scored a point
auto current = win1 * win2 * twoWins(needPointsOne - 1, nextPointsTwo) +
lose1 * win2 * twoWins(needPointsOne, nextPointsTwo) +
win1 * lose2 * twoWins(needPointsOne - 1, needPointsTwo);
// both players lost, stay in same state
current /= 1 - lose1 * lose2;
// better than before ?
if (best < current)
best = current;
// no use in further increasing the risk of player 2 ?
if (nextPointsTwo == 0)
break;
// twice the risk, twice the reward ...
bet *= 2;
}
cache[id] = best;
return best;
}
int main()
{
std::cin >> maxScore;
// player one moves first
// two options: he/she score one point or not
// => add both states (multiplied by their probability)
auto result = win1 * twoWins(maxScore - 1, maxScore) +
lose1 * twoWins(maxScore, maxScore);
std::cout << std::fixed << std::setprecision(8)
<< result << std::endl;
return 0;
}
This solution contains 16 empty lines, 18 comments and 3 preprocessor commands.
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
July 21, 2017 submitted solution
July 21, 2017 added comments
Difficulty
Project Euler ranks this problem at 65% (out of 100%).
Links
projecteuler.net/thread=232 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/Meng-Gen/ProjectEuler/blob/master/232.py (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/232.cpp (written by Yuping Luo)
C++ github.com/smacke/project-euler/blob/master/cpp/232.cpp (written by Stephen Macke)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own.
Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
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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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.
Copyright
I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.
All of my solutions can be used for any purpose and I am in no way liable for any damages caused.
You can even remove my name and claim it's yours. But then you shall burn in hell.
The problems and most of the problems' images were created by Project Euler.
Thanks for all their endless effort !!!
<< problem 231 - The prime factorisation of binomial coefficients | Semidivisible numbers - problem 234 >> |