Binary Watch

Binary Watch - Problem

A binary watch has 4 LEDs on the top to represent the hours (0-11), and 6 LEDs on the bottom to represent the minutes (0-59). Each LED represents a zero or one, with the least significant bit on the right.

Given an integer turnedOn which represents the number of LEDs that are currently on, return all possible times the watch could represent. You may return the answer in any order.

Formatting rules:

The hour must not contain a leading zero. For example, "01:00" is not valid. It should be "1:00".
The minute must consist of two digits and may contain a leading zero. For example, "10:2" is not valid. It should be "10:02".

Input & Output

Example 1 — Single LED

$ Input: turnedOn = 1

› Output: ["0:01","0:02","0:04","0:08","0:16","0:32","1:00","2:00","4:00","8:00"]

💡 Note: With 1 LED on, we can have either 1 bit set in hours (1,2,4,8) with 0 minutes, or 0 bits in hours with 1 bit set in minutes (1,2,4,8,16,32).

Example 2 — No LEDs

$ Input: turnedOn = 9

› Output: []

💡 Note: With 9 LEDs on, no valid time exists since maximum LEDs for valid time is 8 (hour 11 = 3 bits + minute 59 = 5 bits = 8 total).

Example 3 — All LEDs Off

$ Input: turnedOn = 0

› Output: ["0:00"]

💡 Note: With 0 LEDs on, only "0:00" is possible (0 hours, 0 minutes, both have 0 bits set).

Constraints

0 ≤ turnedOn ≤ 10

Visualization

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Asked in

G Google 25 a Amazon 18 M Microsoft 12

The key insight is to count set bits in binary representations of hours and minutes. The brute force approach checks all 720 possible times (12 hours × 60 minutes) and uses bit counting to filter valid combinations. Time: O(1), Space: O(1).

Common Approaches

✓ Backtracking - Try LED Positions

⏱️ Time: O(1) Space: O(1)

Use backtracking to systematically try placing the required number of LEDs across the 10 positions (4 for hours, 6 for minutes), ensuring hours stay ≤11 and minutes ≤59.

Brute Force - Check All Times

⏱️ Time: O(1) Space: O(1)

Iterate through all 12 hours and 60 minutes, count the number of 1 bits in their binary representations, and collect times where the total matches turnedOn.

Backtracking - Try LED Positions — Algorithm Steps

Start with 0 LEDs placed
Try placing LED at each position
Recursively place remaining LEDs
Check if hour ≤ 11 and minute ≤ 59

Visualization

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Step-by-Step Walkthrough

LED Positions

10 positions: 4 for hours, 6 for minutes

Place/Skip Decision

At each position, try placing LED or skipping

Valid Times

Collect times where hour<12 and minute<60

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

char result[1000][10];
int resultCount = 0;

void backtrack(int pos, int hour, int minute, int count, int turnedOn) {
    if (count == turnedOn) {
        if (hour < 12 && minute < 60) {
            sprintf(result[resultCount], "%d:%02d", hour, minute);
            resultCount++;
        }
        return;
    }
    
    if (pos >= 10 || count > turnedOn) {
        return;
    }
    
    // Try placing LED at current position
    if (pos < 4) {  // Hour positions
        backtrack(pos + 1, hour | (1 << pos), minute, count + 1, turnedOn);
    } else {  // Minute positions
        backtrack(pos + 1, hour, minute | (1 << (pos - 4)), count + 1, turnedOn);
    }
    
    // Try not placing LED at current position
    backtrack(pos + 1, hour, minute, count, turnedOn);
}

int main() {
    int turnedOn;
    scanf("%d", &turnedOn);
    
    resultCount = 0;
    backtrack(0, 0, 0, 0, turnedOn);
    
    printf("[");
    for (int i = 0; i < resultCount; i++) {
        printf("\"%s\"", result[i]);
        if (i < resultCount - 1) printf(",");
    }
    printf("]\n");
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(1)

Fixed number of LED positions (10) and combinations, constant time

✓ Linear Growth

Space Complexity

O(1)

Recursion depth is limited by number of LEDs (max 10), constant space

✓ Linear Space

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Input & Output

Constraints

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Related Problems

Common Approaches

Backtracking - Try LED Positions — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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