data.json

{
    "1": [9, 11, 12],
    "2": [6, 14, 12, 9, 5, 1],
    "3": [1, 6, 8, 20, 22],
    "4": [24, 20, 3, 11],
    "5": [2, 1, 6],
    "6": [2, 18, 17],
    "7": [24, 1, 10, 12, 2],
    "8": [3, 14, 6, 1],
    "9": [3, 21, 10, 2, 18, 1],
    "10": [16, 18, 6, 9],
    "11": [24, 17, 21, 4],
    "12": [5, 23, 17, 19],
    "13": [20, 1, 22, 14],
    "14": [10, 15, 7, 8],
    "15": [1, 18, 19, 12],
    "16": [1, 6, 5, 10, 2],
    "17": [19, 1, 3],
    "18": [20, 1, 19, 6],
    "19": [21, 2, 15, 12, 6],
    "20": [1, 21, 15, 23],
    "21": [19, 20, 3, 14, 22],
    "22": [15, 21, 6, 2, 20, 13, 14],
    "23": [10, 19, 1, 12, 22, 6],
    "24": []
}

finder.py

import json

data = json.loads(open('./data.json').read())
data = {int(k): v for k,v in data.iteritems()}

def nxt(path):
    here = path[-1]
    valid = (i for i in data[here] if i not in path)
    for dest in valid:
        potential = path + [dest]
        for v in nxt(potential):
            yield v
    else:
        yield tuple(path)

found = set()
for path in nxt([1,]):
    if path in found:
        continue
    if path[-1] == 24:
        print '{:3}: {}'.format(len(path), path)
    if len(path) == 24:
        break
    found.add(path)