D. Clock Breaking
After numerous unfortunate freak fatalities and the lawsuits, settlements, protests, and boycotts that naturally followed, the beleaguered executives at ACME Clock Manufacturers have decided they need to finally fix their disastrous quality control issues....
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
After numerous unfortunate freak fatalities and the lawsuits, settlements, protests, and boycotts that naturally followed, the beleaguered executives at ACME Clock Manufacturers have decided they need to finally fix their disastrous quality control issues. It has been known for years that the digital clocks they manufacture have an unacceptably high ratio of faulty liquid-crystal display (LCD) screens, and yet these heartless souls have repeatedly failed to address the issue, or even warn their hapless consumers! You have been called in as a quality consultant to finally put a stop to the madness. Your job is to write an automated program that can test a clock and find faults in its display. These clocks use a standard 7-segment LCD display for all digits (shown on the left in Figure D.1), plus two small segments for the ‘:’, and show all times in a 24-hour format. The minute before midnight is 23:59, and midnight is 0:00. The ‘:’ segments of a working clock are on at all times. The representation of each digit using the seven segments is shown on the right in Figure D.1.
Figure D.1: LCD display of each digit.Your program will be given the display of a clock at several consecutive minutes, although you do not know exactly what time these displays start. Some of the LCD segments are burnt out (permanently off) and some are burnt in (permanently on). Your program must determine, where possible, which segments are definitely malfunctioning and which are definitely in working order.
Input
The first input line contains a single integer n (1 ≤ n ≤ 100), which is the number of consecutive minutes of a clock’s display. The next 8n − 1 lines contain n ASCII images of these clock displays of size 7 × 21, with a single blank line separating the representations. All digit segments are represented by two characters, and each colon segment is represented by one character. The character ‘X’ indicates a segment that is on. The character ‘.’ indicates anything else (segments that are off or non-segment portions of the display). See the sample input/output for details; the first output shows every possible LCD segment along with the smaller segments used to represent the ‘:’. No clock representation has an ‘X’ in a non-segment position or only half of a segment showing.
Output
Display a 7 × 21 ASCII image with a ‘0’ for every segment that is burnt out, a ‘1’ for every segment that is burnt in, a ‘W’ for every segment that is definitely working, and a ‘?’ for every segment for which the status cannot be determined. Use ‘.’ for non-segments. If the given displays cannot come from consecutive minutes, display impossible.
Sample Tests
3
......XX.....XX...XX.
.....X..X...X..X....X
.....X..X.X.X..X....X
.............XX...XX.
.....X..X......X.X..X
.....X..X......X.X..X
......XX.....XX...XX.
......XX.....XX...XX.
.....X..X...X..X....X
.....X..X.X.X..X....X
.............XX...XX.
.....X..X......X.X..X
.....X..X......X.X..X
......XX.....XX...XX.
.............XX...XX.
........X...X..X....X
........X.X.X..X....X
.............XX......
........X...X..X.X..X
........X...X..X.X..X
......XX.....XX...XX..??...WW.....??...??.
?..?.W..?...?..1.0..?
?..?.W..?.?.?..1.0..?
.??...??.....11...WW.
?..?.W..?.0.W..?.1..?
?..?.W..?...W..?.1..?
.??...11.....??...??.2
......XX.....XX...XX.
...X....X...X..X.X..X
...X....X.X.X..X.X..X
......XX..........XX.
...X.X....X.X..X.X..X
...X.X......X..X.X..X
......XX.....XX...XX.
......XX.....XX......
...X....X...X..X.....
...X....X.X.X..X.....
......XX.............
...X.X....X.X..X.....
...X.X......X..X.....
......XX.....XX......impossibleEditorial
The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.
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
Describe the data structures and the state maintained by the algorithm.
Explain the processing order and why it is sufficient.
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.
Code
C++ solution used for this page.
#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;
}
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