EP Ethan Pham Project Euler, competitive programming, and notes on the things I keep studying.
ICPC 2014
ICPC 2014

H. Pachinko

You have been hired by Addictive Coin Machines to help design the next hit in their line of eye-catching, coin-guzzling, just-one-more-try Pachinko machines for casinos around the world. Playing a Pachinko machine involves launching balls into a rectangular...

Updated May 21, 2026
Track ICPC
Year 2014
Statement Text + PDF
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Problem statement

Problem Statement

Formatted from the contest statement text, with sample tests broken out into copyable blocks.

Statement text Statement PDF
Time limit 6 seconds

You have been hired by Addictive Coin Machines to help design the next hit in their line of eye-catching, coin-guzzling, just-one-more-try Pachinko machines for casinos around the world. Playing a Pachinko machine involves launching balls into a rectangular grid filled with pegs, obstacles, and targets. The ball bounces around the grid until it eventually hits one of the targets. The player earns a certain number of points depending on which target is hit. The grid pattern for the next Pachinko machine has already been designed, but point values for the targets have not been assigned. These must be set so that like all casino machines, the machine is profitable but not too profitable. Thus it is important to figure out the probability of a ball hitting any particular target. That’s your job! For simplicity, the grid is modeled as a tall rectangle filled with mostly-open spaces (each represented by ‘.’), impassable obstacles (each represented by ‘X’), and targets (each represented by ‘T’). A ball is launched randomly with uniform probability into one of the mostly-open spaces on the top row of the grid. From that point on, collisions with pegs cause the ball to randomly bounce up, down, left, or right, with various given probabilities. For simplicity, assume these probabilities are the same for every space in the grid. If the ball bounces into an obstacle or attempts to move off the grid, it won’t actually move from its current space. When the ball moves into a target it is removed from play. You can safely assume that the average number of spaces visited by a ball before hitting a target will not exceed 109 . It would not make for a very enjoyable game if the ball just bounces forever! For each target, calculate the probability that it is the one hit by a launched ball.

Input

The input consists of a single test case. The first line contains integers w and h, which are the width and height of the Pachinko grid (1 ≤ w ≤ 20 and 2 ≤ h ≤ 10 000). The next line contains four non-negative integers u, d, l, and r, which sum to 100 and are the percentage probabilities of the ball bouncing up, down, left, or right from any open space. Each of the next h lines contains w characters, each of which is ‘.’, ‘X’, or ‘T’. These lines describe the Pachinko grid. The first line, which describes the top row of the grid, contains at least one ‘.’ and no ‘T’s.

Output

Display one line for each ‘T’ in the grid, in order from top to bottom, breaking ties left to right. For each target, display the probability that a launched ball will hit it. Give the answer with an absolute error of at most 10−6 .

Sample Tests

Sample 1
Sample Input 
3 2
20 20 20 40
X.X
T.T
Sample Output 
0.333333333
0.666666667
Sample 2
Sample Input 
4 5
12 33 28 27
....
.XX.
....
T..T
XTTX
Sample Output 
0.435853889
0.403753221
0.081202502
0.079190387
Editorial

Editorial

The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.

Raw TeX

Key Observations

  • Write the structural observations that make the problem tractable.

  • State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.

  • If the constraints matter, explain exactly which part of the solution they enable.

Algorithm

  1. Describe the data structures and the state maintained by the algorithm.

  2. Explain the processing order and why it is sufficient.

  3. Mention corner cases explicitly if they affect the implementation.

Correctness Proof

We prove that the algorithm returns the correct answer.

Lemma 1.

State the first key claim.

Proof.

Provide a concise proof.

Lemma 2.

State the next claim if needed.

Proof.

Provide a concise proof.

Theorem.

The algorithm outputs the correct answer for every valid input.

Proof.

Combine the lemmas and finish the argument.

Complexity Analysis

State the running time and memory usage in terms of the input size.

Implementation Notes

  • Mention any non-obvious implementation detail that is easy to get wrong.

  • Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.

Implementation

Code

C++ solution used for this page.

C++

Clean code view with a raw-file link when you want the original source.

Raw file 
#include <bits/stdc++.h>
using namespace std;

namespace {

void solve() {
    // Fill in the full solution logic for the problem here.
}

}  // namespace

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    solve();
    return 0;
}
Resources

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TeX write-upcompetitive_programming/icpc/2014/H-pachinko/solution.texC++ implementationcompetitive_programming/icpc/2014/H-pachinko/solution.cppStatement textcompetitive_programming/icpc/2014/H-pachinko/statement.txtStatement PDFcompetitive_programming/icpc/2014/H-pachinko/statement.pdfMetadatacompetitive_programming/icpc/2014/H-pachinko/meta.jsonYear packetcompetitive_programming/icpc/2014/contest_problems.pdf