Puzzle | 1000 light bulbs switched on/off by 1000 persons passing by

There are 1000 light bulbs and 1000 persons. All light bulbs are initially off. Person 1 goes flipping light bulb 1, 2, 3, 4, … person 2 then flips 2, 4, 6, 8, … person 3 then 3, 6, 9, … etc until all 1000 persons have done this. What is the status of light bulb 25, 93, 576, 132, 605, 26, 45, 37, 36 after all persons have flipped their respective light bulbs? Is there a general solution to predict the status of a light bulb? How many light bulbs are on after all 1000 persons have gone by?

Explanation: The key observations are:

  1. Person 1 flips the light bulb 1, 2, 3, … which are multiples of 1.
  2. Person 2 flips the light bulb 2, 4, 6, … which are multiples of 2.
  3. Person 3 flips the light bulb 3, 6, 9, … which are multiples of 3.
  4. Similarly, Person 1000 flips the light bulb 1000 which is multiple of 1000.
  5. From the above observations, we can say that person i will flip light bulbs which are multiples of i,  \forall i \in \{1, 2, 3, ..., 1000\}.
  6. Thus, a light bulb j will be flipped by all persons for whom j is a multiple of their person number. In other words light bulb j will be flipped by all persons whose person number i is a factor of j,  \forall i, j \in \{1, 2, 3, ..., 1000\}.
  7. Examples:
    • (i) Light Bulb 10 will be flipped by persons 1, 2, 5, 10 whose person numbers are factors of 10.
    • (ii) Light Bulb 12 will be flipped by persons 1, 2, 3, 4, 6, 12 whose person numbers are factors of 12.
  8. Thus, the light bulb 25 will be flipped by persons 1, 5, 25, so it will be flipped 3 times which is odd and since initially all bulbs were “off”, now light bulb 25 will be “on”.
  9. The light bulb 93 will be flipped by persons 1, 3, 31, 93, so it will be flipped 4 times which is even and since initially all bulbs were “off”, now light bulb 93 will be “off”.
  10. The light bulb 576 will be flipped by persons 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576, so it will be flipped 21 times which is odd and since initially all bulbs were “off”, now light bulb 576 will be “on”.
  11. The light bulb 132 will be flipped by persons 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132, so it will be flipped 12 times which is even and since initially all bulbs were “off”, now light bulb 132 will be “off”.
  12. The light bulb 605 will be flipped by persons 1, 5, 11, 55, 121, 605, so it will be flipped 6 times which is even and since initially all bulbs were “off”, now light bulb 605 will be “off”.
  13. The light bulb 26 will be flipped by persons 1, 2, 13, 26, so it will be flipped 4 times which is even and since initially all bulbs were “off”, now light bulb 26 will be “off”.
  14. The light bulb 45 will be flipped by persons 1, 3, 5, 9, 15, 45, so it will be flipped 6 times which is even and since initially all bulbs were “off”, now light bulb 45 will be “off”.
  15. The light bulb 37 being prime numbered bulb will be flipped by persons 1, 37, so it will be flipped 2 times which is even and since initially all bulbs were “off”, now light bulb 37 will be “off”.
  16. The light bulb 36 will be flipped by persons 1, 2, 3, 4, 6, 9, 12, 18, 36, so it will be flipped 9 times which is odd and since initially all bulbs were “off”, now light bulb 36 will be “on”.

To find out the status of a given light bulb:
We count the number of factors of the light bulb number, and as per above observations if the number of factors is odd, then the light bulb will be “on”, and if its even then it will be “off” in the end.

Algorithm to find how many light bulbs will be “on” in the end:
We count the factors of each number from 1 to 1000. If the number of factors for any number is odd, the corresponding light bulb is “on” so we update the result, and finally print it.
Below is the code implementing the above algorithm.

C++

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// C++ implementation of above approach
#include <iostream>
using namespace std;
  
int findOnBulbs(int numberOfBulbs)
{
    // initializing the result
    int onBulbs = 0; 
      
    // to loop over all bulbs from 1 to numberOfBulbs
    int bulb = 1; 
      
    // to loop over persons to check whether their person number
    int person = 1; 
      
      
    // is a factor of light bulb number or not
    for (bulb = 1; bulb <= numberOfBulbs; bulb++) {
          
        // inner loop to find factors of given bulb
        // to count the number of factors of a given bulb
        int factors = 0; 
          
        for (person = 1; person * person <= numberOfBulbs; person++) {
              
            if (bulb % person == 0) // person is a factor
            {
                factors++;
                  
                // bulb != person*person
                if (bulb / person != person) 
                {
                    factors++;
                }
            }
        }
          
        // if number of factors is odd, then the
        if (factors % 2 == 1) 
          
        {
            // light bulb will be "on" in the end
            cout << "Light bulb "
                << bulb
                << " will be on"
                << "\n";
            onBulbs++;
        }
    }
      
      
    return onBulbs;
}
  
  
// Driver program to test above function
int main()
{
    // total number of light bulbs
    int numberOfBulbs = 1000; 
      
    // to find number of on bulbs in
    // the end after all persons have
    // flipped the light bulbs
    int onBulbs = findOnBulbs(numberOfBulbs); 
      
      
  
    cout << "Total "
        << onBulbs
        << " light bulbs will be on in the end out of "
        << numberOfBulbs
        << " light bulbs"
        << "\n";
    return 0;
}

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Java

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// Java implementation of the 
// above given approach
public class GFG
{
  
static int findOnBulbs(int numberOfBulbs)
{
    // initializing the result
    int onBulbs = 0
      
    // to loop over all bulbs from 1 to numberOfBulbs
    int bulb = 1
      
    // to loop over persons to check whether their person number
    int person = 1
      
      
    // is a factor of light bulb number or not
    for (bulb = 1; bulb <= numberOfBulbs; bulb++) {
          
        // inner loop to find factors of given bulb
        // to count the number of factors of a given bulb
        int factors = 0
          
        for (person = 1; person * person <= numberOfBulbs; person++) {
              
            if (bulb % person == 0) // person is a factor
            {
                factors++;
                  
                // bulb != person*person
                if (bulb / person != person) 
                {
                    factors++;
                }
            }
        }
          
        // if number of factors is odd, then the
        if (factors % 2 == 1
          
        {
            // light bulb will be "on" in the end
            System.out.println("Light bulb " + bulb + " will be on");
            onBulbs++;
        }
    }
      
      
    return onBulbs;
}
  
  
// Driver program to test above function
public static void main(String [] args)
{
    // total number of light bulbs
    int numberOfBulbs = 1000
      
    // to find number of on bulbs in
    // the end after all persons have
    // flipped the light bulbs
    int onBulbs = findOnBulbs(numberOfBulbs); 
      
      
  
    System.out.println("Total " + onBulbs
        + " light bulbs will be on in the end out of "
        + numberOfBulbs + " light bulbs");
}
  
// This code is contributed 
// by Ryuga
}

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Python3

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# Python3 code implementing the
# given approach
  
def findOnBulbs(numberOfBulbs): 
  
    # initializing the result 
    onBulbs = 0
      
    # to loop over all bulbs from
    # 1 to numberOfBulbs 
    bulb = 1
      
    # to loop over persons to check 
    # whether their person number 
    person = 1
      
    # Is a factor of light bulb number or not 
    for bulb in range(1, numberOfBulbs + 1): 
          
        # inner loop to find factors of
        # given bulb to count the number 
        # of factors of a given bulb 
        factors = 0
          
        for person in range(1, int(numberOfBulbs**(0.5)) + 1): 
            if bulb % person == 0: # person is a factor 
                factors += 1
                  
                # bulb != person*person 
                if bulb // person != person: 
                    factors += 1
                  
        # if number of factors is odd, then the 
        if factors % 2 == 1
          
            # light bulb will be "on" in the end 
            print("Light bulb", bulb, "will be on")
            onBulbs += 1
          
    return onBulbs 
  
# Driver Code
if __name__ == "__main__"
  
    # total number of light bulbs 
    numberOfBulbs = 1000
      
    # to find number of on bulbs in 
    # the end after all persons have 
    # flipped the light bulbs 
    onBulbs = findOnBulbs(numberOfBulbs) 
      
    print("Total", onBulbs, "light bulbs will",
                     "be on in the end out of"
                  numberOfBulbs, "light bulbs")
      
# This code is contributed 
# by Rituraj Jain

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C#

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// C# implementation of above approach
using System;
class GFG
{
  
static int findOnBulbs(int numberOfBulbs)
{
    // initializing the result
    int onBulbs = 0; 
      
    // to loop over all bulbs from 1 to numberOfBulbs
    int bulb = 1; 
      
    // to loop over persons to check whether their person number
    int person = 1; 
      
      
    // is a factor of light bulb number or not
    for (bulb = 1; bulb <= numberOfBulbs; bulb++) {
          
        // inner loop to find factors of given bulb
        // to count the number of factors of a given bulb
        int factors = 0; 
          
        for (person = 1; person * person <= numberOfBulbs; person++) {
              
            if (bulb % person == 0) // person is a factor
            {
                factors++;
                  
                // bulb != person*person
                if (bulb / person != person) 
                {
                    factors++;
                }
            }
        }
          
        // if number of factors is odd, then the
        if (factors % 2 == 1) 
          
        {
            // light bulb will be "on" in the end
            Console.WriteLine("Light bulb " + bulb + " will be on");
            onBulbs++;
        }
    }
      
      
    return onBulbs;
}
  
  
// Driver program to test above function
public static void Main()
{
    // total number of light bulbs
    int numberOfBulbs = 1000; 
      
    // to find number of on bulbs in
    // the end after all persons have
    // flipped the light bulbs
    int onBulbs = findOnBulbs(numberOfBulbs); 
      
      
  
    Console.WriteLine("Total " + onBulbs
        + " light bulbs will be on in the end out of "
        + numberOfBulbs + " light bulbs");
}
}
  
// This code is contributed 
// by Akanksha Rai

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PHP

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<?php
// PHP implementation of above approach
  
function findOnBulbs($numberOfBulbs)
{
    // initializing the result
    $onBulbs = 0; 
      
    // to loop over all bulbs from 
    // 1 to numberOfBulbs
    $bulb = 1; 
      
    // to loop over persons to check 
    // whether their person number
    $person = 1; 
      
      
    // is a factor of light bulb number or not
    for ($bulb = 1;
         $bulb <= $numberOfBulbs; $bulb++)
    {
          
        // inner loop to find factors of given 
        // bulb to count the number of factors
        // of a given bulb
        $factors = 0; 
          
        for ($person = 1; 
             $person * $person <= $numberOfBulbs; $person++)
        {
              
            if ($bulb % $person == 0) // person is a factor
            {
                $factors++;
                  
                // bulb != person*person
                if ($bulb / $person != $person
                {
                    $factors++;
                }
            }
        }
          
        // if number of factors is odd, then the
        if ($factors % 2 == 1) 
          
        {
            // light bulb will be "on" in the end
            echo "Light bulb " . $bulb
                 " will be on" ."\n";
            $onBulbs++;
        }
    }
      
    return $onBulbs;
}
  
// Driver Code
  
// total number of light bulbs
$numberOfBulbs = 1000; 
  
// to find number of on bulbs in
// the end after all persons have
// flipped the light bulbs
$onBulbs = findOnBulbs($numberOfBulbs); 
  
echo "Total " . $onBulbs . " light bulbs will "
     "be on in the end out of " . $numberOfBulbs
                             " light bulbs" ."\n";
  
// This code is contributed by ita_c
?>

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The previous program is written in O(n*sqrt(n)).
From observation, it is clear that whenever the number of factors will be odd, the bulb will be on.
For any non-square number with each divisor, there is corresponding quotient so number of factors will be even.
For every square number, when we divide with its square root, the quotient will be the same number i.e. its square root. So it has odd number of factor.
Therefore, we can write an efficient code for this problem which computes in O(sqrt(n)).

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import math
root=int(math.sqrt(1000))
for i in range(1,root+1):
    print("Light bulb %d will be on"%(i*i))
print("""Total %d light bulbs will be on
in the end out of 1000 light bulbs"""%root)

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Output:

Light bulb 1 will be on
Light bulb 4 will be on
Light bulb 9 will be on
Light bulb 16 will be on
Light bulb 25 will be on
Light bulb 36 will be on
Light bulb 49 will be on
Light bulb 64 will be on
Light bulb 81 will be on
Light bulb 100 will be on
Light bulb 121 will be on
Light bulb 144 will be on
Light bulb 169 will be on
Light bulb 196 will be on
Light bulb 225 will be on
Light bulb 256 will be on
Light bulb 289 will be on
Light bulb 324 will be on
Light bulb 361 will be on
Light bulb 400 will be on
Light bulb 441 will be on
Light bulb 484 will be on
Light bulb 529 will be on
Light bulb 576 will be on
Light bulb 625 will be on
Light bulb 676 will be on
Light bulb 729 will be on
Light bulb 784 will be on
Light bulb 841 will be on
Light bulb 900 will be on
Light bulb 961 will be on
Total 31 light bulbs will be on in the end out of 1000 light bulbs 


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