Find the maximum distance covered using n bikes

16 Aug, 2021
There are n bikes and each can cover 100 km when fully fueled. What is the maximum amount of distance you can go using n bikes? You may assume that all bikes are similar and a bike takes 1 litre to cover 1 km.You have n bikes and using one bike you can only cover 100 km. so if n bikes start from same point and run simultaneously you can go only 100 km. Let’s think bit differently, trick is when you want to cover maximum distance, you should always try to waste minimum fuel. Minimum wastage of fuel means to run minimum number of bikes. Instead of parallel running of n bikes, you can think of serially running them. That means if you transfer some amount of fuel from last bike to another bikes and throw the last bike i.e., don’t run the last bike after certain point. But the question is, after what distance the fuel transfer has to be done so that the maximum distance is covered and fuel tank of remaining bikes do not overflow.  
Let us take following base cases and then generalize the solution.  
Base Case 1: There is one bike: This is simple, we can cover 100 kms only.
Base Case 2: There are two bikes: What is the maximum distance we can cover when there are 2 bikes? To maximize the distance, we must drop second bike at some point and transfer its fuel to first bike. Let we do the transfer after x kms.
Total distance covered = Distance covered by 100 ltr in first bike +  
                         Distance covered by fuel transferred from 
                         first bike. 
Remaining fuel in second bike is 100 – x. If we transfer this much fuel to first bike, then the total distance would become 100 + 100 – x which is 200 – x. So our task is to maximize 200-x. The constraint is, 100 – x must be less than or equal to the space created in first bike after x kms, i.e., 100 – x <= x. The value of 200-x becomes maximum when x is minimum. The minimum possible value of x is 50. So we are able to travel 150 kms.  
Base Case 3: There are three bikes : Let the first transfer is done after x kms. After x distance all bikes contain 100-x amount of fuel. If we take 100-x amount of fuel from 3rd bike and distribute it among 1st and 2nd bike so that fuel tanks of 1st and 2nd bikes get full. So 100-x <= 2*x; or, x=33.333 so we should transfer the remaining fuel of third bike and distribute that amount of fuel among 1st and 2nd bike after exactly 33.33 km.
Let us generalize it. If we take a closer look at above cases, we can observe that if there are n bikes, then the first transfer is done (or a bike is dropped) after 100/n kms. To generalize it more, when we have x litre remaining fuel in every bike and n remaining bikes, we drop a bike after x/n kms.
Following is the implementation of a general function. 
C++
Java
Python3
C#
PHP
Javascript
#include <stdio.h> // Returns maximum distance that can be traveled by n bikes and given fuel// in every bikedouble maxDistance(int n, int fuel){    // dist_covered is the result of this function    double dist_covered = 0;     while (n > 0)    {        // after ever fuel/n km we are discarding one bike and filling        // all the other bikes with fuel/n liters of fuel i.e. to their        // maximum limit (100 litre)         dist_covered += (double)fuel / n;         n -= 1; // reduce number of bikes    }    return dist_covered;} // Driver program to test above functionint main(){    int n = 3; // number of bikes    int fuel = 100;    printf("Maximum distance possible with %d bikes is %f",            n, maxDistance(n, fuel));    return 0;}
// Java program to find the maximum// distance covered using n bikesimport java.io.*; class GFG{    // Function that returns maximum distance that can be traveled by n bikes    // and given fuel in every bike    static double maxDistance(int n, int fuel)    {         // dist_covered is the result of this function        double dist_covered = 0;          while (n > 0)        {            // after ever fuel/n km we are discarding one bike and filling            // all the other bikes with fuel/n liters of fuel i.e. to their            // maximum limit (100 litre)              dist_covered += (double)fuel / n;              n -= 1;  // reduce number of bikes        }        return dist_covered;    }         // driver program    public static void main (String[] args)    {        int n = 3; // number of bikes        int fuel = 100;        System.out.println("Maximum distance possible with "                                   + n + " bikes is " + maxDistance(n, fuel));    }} // Contributed by Pramod Kumar
# Python 3 program to find the maximum# distance covered using n bikes # Returns maximum distance that can be# traveled by n bikes and given fuel# in every bikedef maxDistance(n, fuel):         # dist_covered is the result    # of this function    dist_covered = 0     while (n > 0):                 # after ever fuel/n km we are        # discarding one bike and filling        # all the other bikes with fuel/n        # liters of fuel i.e. to their        # maximum limit (100 litre)        dist_covered = dist_covered + (fuel / n)                 # reduce number of bikes        n = n - 1     return dist_covered # Driver Codeif __name__ =='__main__':    n = 3         # number of bikes    fuel = 100    print("Maximum distance possible with",       n, "bikes is", maxDistance(n, fuel)) # This code is contributed by# Surendra_Gangwar
// C# program to find the maximum// distance covered using n bikesusing System; class GFG {     // Function that returns maximum distance    // that can be travelled by n bikes    // and given fuel in every bike    static double maxDistance(int n, int fuel)    {        // dist_covered is the result of this function        double dist_covered = 0;         while (n > 0) {                         // after ever fuel/n km we are discarding            // one bike and filling all the other bikes            // with fuel/n liters of fuel i.e. to their            // maximum limit (100 litre)            dist_covered += (double)fuel / n;             n -= 1; // reduce number of bikes        }        return dist_covered;    }     // driver program    public static void Main()    {        // number of bikes        int n = 3;        int fuel = 100;        Console.WriteLine("Maximum distance possible with " + n +                          " bikes is " + maxDistance(n, fuel));    }} // This code is contributed by Sam007
<?php// Returns maximum distance that can// be traveled by n bikes and given// fuel in every bike function maxDistance($n, $fuel){    // dist_covered is the result    // of this function    $dist_covered = 0;     while ($n > 0)    {        // after ever fuel/n km we are        // discarding one bike and filling        // all the other bikes with fuel/n        // liters of fuel i.e. to their        // maximum limit (100 litre)         $dist_covered += (double)$fuel / $n;         // reduce number of bikes        $n -= 1;    }    return $dist_covered;} // Driver Code // number of bikes$n = 3;$fuel = 100;echo "Maximum distance possible with ",                      $n, " bikes is ",                maxDistance($n, $fuel); // This code is contributed by ajit?>
<script> // Javascript program to find the maximum// distance covered using n bikes // Function that returns maximum distance// that can be travelled by n bikes// and given fuel in every bikefunction maxDistance(n, fuel){         // dist_covered is the result of this function    let dist_covered = 0;     while (n > 0)    {                 // After ever fuel/n km we are discarding        // one bike and filling all the other bikes        // with fuel/n liters of fuel i.e. to their        // maximum limit (100 litre)        dist_covered += fuel / n;                 // Reduce number of bikes        n -= 1;    }    return dist_covered.toFixed(6);} // Driver code // Number of bikeslet n = 3;let fuel = 100; document.write("Maximum distance possible with " + n +               " bikes is " + maxDistance(n, fuel));                // This code is contributed by divyesh072019 </script>
Output : 
Maximum distance possible with 3 bikes is 183.333333
Time Complexity: O(N)Auxiliary Space: O(1) This article is contributed by Shamik Mitra. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above 
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