A basic (and slow) modulo chain search program

us_states.py

from collections import defaultdict
from pprint import pprint

state_list = '''
Alabama         AL
Alaska          AK
Arizona         AZ
Arkansas        AR
California      CA
Colorado        CO
Connecticut     CT
Delaware        DE
Florida         FL
Georgia         GA
Hawaii          HI
Idaho           ID
Illinois        IL
Indiana         IN
Iowa            IA
Kansas          KS
Kentucky        KY
Louisiana       LA
Maine           ME
Maryland        MD
Massachusetts   MA
Michigan        MI
Minnesota       MN
Mississippi     MS
Missouri        MO
Montana         MT
Nebraska        NE
Nevada          NV
New Hampshire   NH
New Jersey      NJ
New Mexico      NM
New York        NY
North Carolina  NC
North Dakota    ND
Ohio            OH
Oklahoma        OK
Oregon          OR
Pennsylvania    PA
Rhode Island    RI
South Carolina  SC
South Dakota    SD
Tennessee       TN
Texas           TX
Utah            UT
Vermont         VT
Virginia        VA
Washington      WA
West Virginia   WV
Wisconsin       WI
Wyoming         WY
'''.strip()

states = [line.rsplit(maxsplit=1) for line in state_list.splitlines()]

numbers_by_abbr = defaultdict(set)

for name, abbr in states:
    n = int(name[0]+name[-2:],36)
    numbers_by_abbr[abbr[1]].add(n)

sets = list(numbers_by_abbr.values())
pprint(sets)


def find_mod_chain(sets, iterations=2, limit=None):
    m = max(max(x)for x in sets)
    if iterations==0:
        # if the last iteration is reached, return the maximum number
        return (m,)

    if limit is None:
        limit = m
    best_chain = (limit, )

    # if there are n sets the lowest possible modulo value without intersection is n
    for i in range(len(sets), limit):
        # apply modulo candidate i to each set
        n = [{k % i for k in x} for x in sets]
        # the resulting sets are disjoint if there are as many elements in the union as in all sets
        if len(set.union(*n)) == sum(len(x) for x in n):
            r = find_mod_chain(n, iterations - 1)
            if r[-1] < best_chain[-1]:
                best_chain = (i,) + r
    return best_chain

def display_mod_chain(chain):
    *c, m = chain
    print('x', *c, sep='%')
    print(f'Highest remaining value is {m}.')

display_mod_chain(find_mod_chain(sets, 5, 500))