Number of ways to cut pizza into identical-shaped pieces such that some do not touch the centre.

parker_pizza.py

import itertools
import math
import typing
from collections import Counter

from sympy import factorint


def submultisets(multiset: Counter) -> typing.Iterable[Counter]:
    for counts in itertools.product(*[range(count + 1) for count in multiset.values()]):
        yield Counter({element: count for element, count in zip(multiset, counts)})


def factors_to_number(factors: Counter) -> int:
    return math.prod(factors.elements())


def parker_ways_to_cut_pizza(pieces: int):
    assert pieces % 2 == 0, "The number of pieces must be even."

    factors = Counter(factorint(pieces))
    del factors[2]  # We can only make odd-sized grins.

    for submultiset in submultisets(factors):
        grin_size = factors_to_number(submultiset)
        number_of_grins = 2 * grin_size
        pieces_per_grin = pieces // number_of_grins

        # A 1-grin doesn't exist (or is a circle), and we need to cut each grin in more
        # than one piece so that some of them do not touch the centre (Parker cut).
        if grin_size == 1 or pieces_per_grin == 1:
            continue

        print(
            f"{number_of_grins} {grin_size}-grins cut into {pieces_per_grin} pieces each"
        )