ouroboros.py

#!/usr/bin/env python3
 
def get_init(factor):
    values = []
    for i in range(1, 10):
        s = str(i)
        s = str((i * factor) % 10) + s
        while s[0] == '0':
            s = str((int(s) * factor) // 10) + s
        values += [s]
    return values
 
def comp(factor, i):
    while str(int(i) * factor) != i[-1:] + i[:-1]:
        p = str(int(i) * factor)
        i = p[-len(i):-(len(i)-1)] + i
    if i[0] == '0':
        return None
    return int(i)
 
def get_best(factor):
    best = None
    for v in get_init(factor):
        c = comp(factor, v)
        if not best or c and c < best:
            best = c
    return best
 
for factor in range(2, 10):
    best = get_best(factor)
    print(factor, best, len(str(best)))