Problem:
Let's consider operations on monochrome images that consist of hexagonal pixels, each of which is colored in either black or white. Because of the shape of pixels, each of them has exactly six neighbors (e.g. pixels that share an edge with it.)  "Filtering" is an operation to determine the color of a pixel from the colors of itself and its six neighbors. Examples of filterings are shown below.  Example 1: Color a pixel in white when all of its neighboring pixels are white. Otherwise the color will not change.    Performing this operation on all the pixels simultaneously results in "noise canceling," which removes isolated black pixels.  Example 2: Color a pixel in white when its all neighboring pixels are black. Otherwise the color will not change.    Performing this operation on all the pixels simultaneously results in "edge detection," which leaves only the edges of filled areas.  Example 3: Color a pixel with the color of the pixel just below it, ignoring any other neighbors.    Performing this operation on all the pixels simultaneously results in "shifting up" the whole image by one pixel.  Applying some filter, such as "noise canceling" and "edge detection," twice to any image yields the exactly same result as if they were applied only once. We call such filters _idempotent_. The "shifting up" filter is not idempotent since every repeated application shifts the image up by one pixel.  Your task is to determine whether the given filter is idempotent or not.

Incorrect submission (verdict: Time Limit Exceeded):
import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,random,time,copy,functools

sys.setrecursionlimit(10**7)
inf = 10**20
eps = 1.0 / 10**10
mod = 10**9+7
dd = [(-1,0),(0,1),(1,0),(0,-1)]
ddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]

def LI(): return [int(x) for x in sys.stdin.readline().split()]
def LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]
def LF(): return [float(x) for x in sys.stdin.readline().split()]
def LS(): return sys.stdin.readline().split()
def I(): return int(sys.stdin.readline())
def F(): return float(sys.stdin.readline())
def S(): return input()
def pf(s): return print(s, flush=True)

#     18
#   17  16
# 15  06  14
#   05  04
# 13  03  12
#   02  01
# 11  00  10
#   09  08
#     07

def main():
    rr = []

    tt = [
            [3, 0, 2, 5, 6, 4, 1],
            [0, 7, 9, 2, 3, 1, 8],
            [2, 9, 11, 13, 5, 3, 0],
            [5, 2, 13 ,15, 17, 6, 3],
            [6, 3, 5, 17, 18, 16, 4],
            [4, 1, 3, 6, 16, 14, 12],
            [1, 8, 0, 3, 4, 12, 10]
    ]

    while True:
        s = S()
        if s == '#':
            break

        fil = int(s[::-1], 2)
        ii = [2**i for i in range(128)]
        m = {}
        def u(a):
            key = tuple(a)
            if key in m:
                return m[key]
            t = 0
            for i in range(7):
                if a[i]:
                    t |= ii[tt[0][i]]
            if fil & ii[t]:
                m[key] = 1
                return 1
            m[key] = 0
            return 0

        f = True
        for k in range(ii[19]):
            
            #     print(k)
            a = [1 if k & ii[i] else 0 for i in range(19)]
            b = [u(list(map(lambda x: a[x], ta))) for ta in tt]
            if u(b) != b[0]:
                # print('false', a, b, u(b))
                f = False
                break
        if f:
            rr.append('yes')
        else:
            rr.append('no')
        # print(rr)

    return '\n'.join(rr)




print(main())



Fixed, accepted submission:
import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,random,time,copy,functools

sys.setrecursionlimit(10**7)
inf = 10**20
eps = 1.0 / 10**10
mod = 10**9+7
dd = [(-1,0),(0,1),(1,0),(0,-1)]
ddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]

def LI(): return [int(x) for x in sys.stdin.readline().split()]
def LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]
def LF(): return [float(x) for x in sys.stdin.readline().split()]
def LS(): return sys.stdin.readline().split()
def I(): return int(sys.stdin.readline())
def F(): return float(sys.stdin.readline())
def S(): return input()
def pf(s): return print(s, flush=True)

#     13
#   12  11
# 13  06  13
#   05  04
# 10  03  09
#   02  01
# 13  00  13
#   08  07
#     13

def main():
    rr = []

    tt = [
            [13, 7, 8, 0, 1, 2, 3],
            [7, 13, 0, 1, 9, 3, 4],
            [8, 0, 13, 2, 3, 10, 5],
            [0, 1, 2, 3, 4, 5, 6],
            [1, 9, 3, 4, 13, 6, 11],
            [2, 3, 10, 5, 6, 13, 12],
            [3, 4, 5, 6, 11, 12, 13]
    ]

    while True:
        s = S()
        if s == '#':
            break

        fil = [int(c) for c in s]
        i2 = [2**i for i in range(128)]
        f3 = {}
        for k in range(3**7):
            a = []
            kk = k
            for _ in range(7):
                a.append(k%3)
                k //= 3
            if 2 in a:
                continue
            a.reverse()
            e = 0
            for c in a:
                e *= 2
                e += c
            f3[kk] = fil[e]

        for k in range(3**7):
            a = []
            kk = k
            for _ in range(7):
                a.append(k%3)
                k //= 3
            if 2 not in a:
                continue
            a.reverse()
            ki = a.index(2)
            e = 0
            y = 0
            for i in range(7):
                e *= 3
                y *= 3
                if i == ki:
                    y += 1
                else:
                    e += a[i]
                    y += a[i]

            fe = f3[e]
            fy = f3[y]
            if fe == fy:
                f3[kk] = fe
            else:
                f3[kk] = 2

        f = True
        for k in range(2**13):
            ba = [1 if i2[i] & k else 0 for i in range(13)] + [2]
            ca = []
            for i in range(7):
                ti = tt[i]
                e = 0
                for c in ti:
                    e *= 3
                    e += ba[c]
                ca.append(f3[e])
            y = 0
            for c in ca:
                y *= 3
                y += c
            if ca[3] != f3[y]:
                f = False
                break

        if f:
            rr.append('yes')
        else:
            rr.append('no')

    return '\n'.join(rr)




print(main())