chain algo | Python Fiddle

class Node:
    def __init__(self, number, parent):
        self.number = number
        self.parent = parent

# get a list of all the numbers to reach this node
    def getPath(self):
        path = [self.number]
        parent = self.parent
        while parent != None:
            path.append(parent.number)
            parent = parent.parent
        return path

def solve(start, target):    
    currentList = []                                    # List of visited nodes in the previous round
    nextList = [Node(start, None)]                      # List of visited nodes in the next round (start with at least one number)
    seen = set([start])                                 # store all number that have already been seen in the previous round

while nextList:                                     # continue until the final number has reached, on each iteration the complexity grows
        currentList = nextList                          # swap the lists around
        nextList = []

for n in currentList:
            path = n.getPath()                          # fetch all the number that are needed to reach this point
            if n.number == target:                      # check of we have reach our target
                return path[::-1]                       # the path is in reverse at this point, so reverse it back
            for a in path:                              # for any combination of the last number and a previous number (including the last, which is the same as multiplying it by 2)
                newnumber = a + path[0]
                if newnumber <= target and newnumber not in seen:   # only evaluate the new number if is is not yet seen already on a lower complexity 
                    nextList.append(Node(newnumber, n))

for n in nextList:                              # update the seen list
            seen.add(n.number)
    return []                                           # if the destination could not be reached

print "path to 25  = ", solve(1, 25)
print "path to 8   = ", solve(1, 8)
print "path to 15  = ", solve(1, 15)
print "path to 500 = ", solve(1, 500)

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