Question

Suppose I will go on a n miles road trip, and I could choose to go 1, 3 or 5 miles in each stage. Thus, my stages will look like a sequence such as 1, 3, 3, 5, 3, 1, 1,....., which add up to n. Give a dynamic programming algorithm that calculates how many different ways are there of doing this trip. Note that, you don't need to output sequences of each trip, just the number of different ways.

Answer 1

#Function to calculate number of ways for walking 'n' miles with 1,3 or 5 steps
def calculate_path(n) :
   paths = [1] * (n+1)    #Initializing an array of size n+1 with 1. paths[i] will give number of ways to walk "i" miles.
   paths[0]=paths[1]=paths[2]=1   #Base cases : For 0/1/2 miles number of ways is only 1
   paths[3] = 2   #Base case : For 3 miles, number of ways is 2 (111,3)
   paths[4] = 3   #Base case : For 4 miles, number of ways is 3 (1111,13,31)
   for i in range(5,n+1) :  
       #Since the last stage for any mile will be either 1 step or 3 steps or 5 steps
       #So, we can calculate total steps by adding number of ways for (n-1)miles , (n-3)miles and (n-5)miles.
       paths[i] = paths[i-1] + paths[i-3] + paths[i-5]  
   return paths[n]       #Final number of ways will be stored at paths[n]
      

Sample Input 1 : 6
Sample Output 1 : 8  
Sample Input 2 : 10
Sample Output 2 : 47