# Examples from the paper "Determining unit groups and K1 of finite rings"
# 
# We give the elementary divisors of the underling abelian group R^+ and
# the matrices describing the map R^+ -> R^+, y -> yx for each of the standard
# generators x

# Example 1
elem_divs = [6, 6, 18, 18, 36, 36, 36, 36, 72, 72, 72, 72]
mats =
[[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24], [4, 0, 0, 0, 6, 0, 0, 0, 36, 0, 60, 0], [0, 4, 0, 0, 6, 6, 0, 0, 12, 36, 60, 12], [4, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 24, 24, 0, 0, 48, 0, 24, 24], [1, 0, 0, 0, 30, 0, 0, 0, 48, 0, 36, 0], [3, 1, 0, 0, 30, 30, 0, 0, 0, 48, 36, 60], [0, 0, 0, 0, 12, 0, 0, 0, 24, 0, 36, 0], [0, 0, 0, 0, 12, 12, 0, 0, 12, 60, 24, 36], [4, 0, 0, 0, 0, 0, 0, 0, 12, 0, 12, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, 12, 12, 0]],
 [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0], [0, 4, 0, 0, 0, 30, 0, 0, 0, 36, 0, 12], [2, 0, 0, 0, 6, 0, 0, 0, 36, 0, 12, 24], [0, 4, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 24, 0, 0, 0, 0, 0, 24, 24], [0, 1, 0, 0, 0, 6, 0, 0, 0, 48, 0, 36], [5, 0, 0, 0, 30, 0, 0, 0, 24, 0, 60, 24], [0, 0, 0, 0, 0, 24, 0, 0, 36, 24, 36, 36], [0, 0, 0, 0, 12, 0, 0, 0, 48, 36, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 36, 12, 36, 60], [2, 0, 0, 0, 0, 0, 0, 0, 60, 36, 36, 60]],
 [[4, 0, 0, 0, 6, 0, 0, 0, 36, 0, 60, 0], [0, 4, 0, 0, 6, 6, 0, 0, 60, 36, 12, 60], [2, 0, 0, 0, 20, 0, 2, 0, 40, 0, 12, 0], [2, 2, 16, 0, 8, 20, 2, 2, 0, 40, 52, 60], [1, 0, 0, 0, 22, 0, 0, 0, 32, 0, 12, 0], [3, 1, 0, 0, 26, 22, 0, 0, 40, 32, 4, 60], [4, 0, 17, 0, 32, 0, 2, 0, 24, 0, 16, 0], [2, 4, 17, 17, 30, 32, 2, 2, 68, 24, 20, 64], [4, 0, 0, 0, 10, 0, 0, 0, 64, 0, 8, 0], [2, 4, 0, 0, 10, 10, 0, 0, 16, 28, 20, 8], [2, 0, 0, 0, 0, 0, 0, 0, 60, 0, 64, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 60, 60, 4]],
 [[0, 2, 0, 0, 0, 6, 0, 0, 0, 36, 24, 60], [4, 0, 0, 0, 30, 0, 0, 0, 36, 24, 12, 24], [0, 4, 0, 2, 0, 16, 0, 34, 0, 32, 48, 60], [2, 4, 16, 0, 20, 28, 2, 0, 40, 48, 36, 56], [0, 1, 0, 0, 0, 34, 0, 0, 0, 64, 24, 36], [5, 4, 0, 0, 2, 4, 0, 0, 8, 48, 60, 56], [0, 2, 0, 1, 0, 4, 0, 34, 0, 48, 48, 16], [4, 0, 17, 0, 32, 34, 2, 0, 24, 68, 8, 68], [0, 4, 0, 0, 0, 34, 0, 0, 0, 32, 12, 56], [2, 4, 0, 0, 2, 0, 0, 0, 4, 52, 52, 48], [0, 4, 0, 0, 0, 0, 0, 0, 0, 12, 12, 64], [2, 0, 0, 0, 0, 0, 0, 0, 60, 0, 68, 0]],
 [[4, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 24, 24, 0, 0, 24, 0, 48, 0], [1, 0, 0, 0, 22, 0, 0, 0, 32, 0, 12, 0], [0, 1, 0, 0, 26, 22, 0, 0, 4, 32, 28, 36], [0, 0, 0, 0, 34, 0, 0, 0, 8, 0, 24, 0], [0, 0, 0, 0, 32, 34, 0, 0, 16, 8, 40, 48], [5, 0, 0, 0, 33, 0, 0, 0, 46, 0, 46, 0], [1, 5, 0, 0, 34, 33, 0, 0, 0, 46, 44, 70], [4, 0, 0, 0, 4, 0, 0, 0, 22, 0, 20, 0], [2, 4, 0, 0, 4, 4, 0, 0, 28, 58, 8, 20], [4, 0, 0, 0, 0, 0, 0, 0, 12, 0, 10, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, 12, 12, 70]],
 [[0, 2, 0, 0, 0, 24, 0, 0, 0, 0, 24, 48], [4, 0, 0, 0, 12, 0, 0, 0, 0, 24, 24, 24], [0, 5, 0, 0, 0, 16, 0, 0, 0, 4, 48, 24], [1, 1, 0, 0, 20, 4, 0, 0, 68, 12, 48, 56], [0, 0, 0, 0, 0, 10, 0, 0, 0, 64, 24, 24], [0, 4, 0, 0, 26, 34, 0, 0, 44, 48, 12, 56], [0, 5, 0, 0, 0, 3, 0, 0, 36, 2, 48, 10], [1, 0, 0, 0, 33, 1, 0, 0, 70, 38, 26, 14], [0, 4, 0, 0, 0, 28, 0, 0, 18, 2, 12, 56], [2, 4, 0, 0, 8, 0, 0, 0, 52, 28, 16, 12], [0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 12, 70], [0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 20, 66]],
 [[1, 0, 0, 0, 30, 0, 0, 0, 48, 0, 36, 0], [0, 1, 0, 0, 30, 30, 0, 0, 12, 48, 60, 36], [4, 0, 17, 0, 32, 0, 2, 0, 24, 0, 16, 0], [2, 4, 16, 17, 28, 32, 2, 2, 64, 24, 20, 64], [5, 0, 0, 0, 33, 0, 0, 0, 46, 0, 46, 0], [3, 5, 0, 0, 34, 33, 0, 0, 44, 10, 56, 58], [2, 0, 17, 0, 34, 0, 1, 0, 56, 0, 66, 0], [0, 2, 17, 17, 34, 34, 2, 1, 68, 56, 64, 6], [1, 0, 0, 0, 2, 0, 0, 0, 6, 0, 10, 0], [4, 1, 0, 0, 2, 2, 0, 0, 14, 60, 4, 10], [2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 4, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 6, 6, 70]],
 [[0, 5, 0, 0, 0, 0, 0, 0, 0, 12, 24, 48], [1, 3, 0, 0, 0, 0, 0, 0, 60, 60, 24, 24], [0, 2, 0, 1, 0, 4, 0, 0, 0, 48, 48, 20], [4, 0, 17, 1, 32, 0, 0, 0, 24, 0, 4, 68], [0, 1, 0, 0, 0, 35, 0, 0, 0, 18, 24, 50], [5, 4, 0, 0, 1, 1, 0, 0, 18, 30, 46, 50], [0, 4, 0, 1, 0, 2, 0, 35, 0, 16, 12, 62], [2, 0, 17, 0, 34, 0, 1, 1, 56, 0, 70, 2], [0, 5, 0, 0, 0, 0, 0, 0, 0, 70, 66, 6], [1, 1, 0, 0, 0, 0, 0, 0, 56, 58, 48, 12], [0, 4, 0, 0, 0, 0, 0, 0, 0, 66, 66, 4], [2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0]],
 [[0, 0, 0, 0, 12, 0, 0, 0, 24, 0, 36, 0], [0, 0, 0, 0, 12, 12, 0, 0, 12, 60, 24, 36], [4, 0, 0, 0, 10, 0, 0, 0, 64, 0, 8, 0], [0, 4, 0, 0, 10, 10, 0, 0, 68, 64, 8, 20], [4, 0, 0, 0, 4, 0, 0, 0, 22, 0, 20, 0], [0, 4, 0, 0, 4, 4, 0, 0, 2, 4, 14, 14], [1, 0, 0, 0, 2, 0, 0, 0, 6, 0, 10, 0], [5, 1, 0, 0, 2, 2, 0, 0, 0, 6, 10, 4], [0, 0, 0, 0, 32, 0, 0, 0, 64, 0, 61, 0], [1, 3, 0, 0, 32, 32, 0, 0, 71, 1, 64, 61], [5, 0, 0, 0, 0, 0, 0, 0, 69, 0, 69, 0], [0, 5, 0, 0, 0, 0, 0, 0, 0, 69, 69, 0]],
 [[0, 0, 0, 0, 0, 24, 0, 0, 36, 24, 36, 36], [0, 0, 0, 0, 12, 0, 0, 0, 48, 36, 0, 0], [0, 2, 0, 0, 0, 26, 0, 0, 36, 64, 36, 64], [4, 0, 0, 0, 10, 0, 0, 0, 8, 36, 56, 12], [0, 2, 0, 0, 0, 32, 0, 0, 36, 22, 36, 52], [4, 0, 0, 0, 4, 0, 0, 0, 50, 0, 50, 66], [0, 3, 0, 0, 0, 34, 0, 0, 18, 6, 18, 62], [3, 0, 0, 0, 2, 0, 0, 0, 66, 18, 22, 66], [3, 2, 0, 0, 0, 4, 0, 0, 63, 64, 63, 11], [4, 3, 0, 0, 32, 0, 0, 0, 44, 9, 52, 0], [3, 5, 0, 0, 0, 0, 0, 0, 63, 69, 63, 3], [1, 3, 0, 0, 0, 0, 0, 0, 3, 63, 63, 3]],
 [[4, 0, 0, 0, 0, 0, 0, 0, 12, 0, 12, 0], [0, 4, 0, 0, 0, 0, 0, 0, 36, 48, 0, 12], [2, 0, 0, 0, 0, 0, 0, 0, 60, 0, 64, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 60, 60, 4], [4, 0, 0, 0, 0, 0, 0, 0, 12, 0, 10, 0], [0, 4, 0, 0, 0, 0, 0, 0, 18, 66, 6, 4], [2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 4, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 6, 6, 70], [5, 0, 0, 0, 0, 0, 0, 0, 69, 0, 69, 0], [3, 2, 0, 0, 0, 0, 0, 0, 63, 6, 0, 69], [5, 0, 0, 0, 0, 0, 0, 0, 69, 0, 70, 0], [0, 5, 0, 0, 0, 0, 0, 0, 0, 69, 69, 1]],
 [[0, 2, 0, 0, 0, 0, 0, 0, 0, 60, 60, 12], [4, 0, 0, 0, 0, 0, 0, 0, 48, 36, 24, 24], [0, 4, 0, 0, 0, 0, 0, 0, 0, 12, 12, 64], [2, 0, 0, 0, 0, 0, 0, 0, 60, 0, 68, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 60, 60, 10], [4, 0, 0, 0, 0, 0, 0, 0, 66, 54, 50, 12], [0, 4, 0, 0, 0, 0, 0, 0, 0, 66, 66, 4], [2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 3, 69], [2, 3, 0, 0, 0, 0, 0, 0, 6, 9, 12, 66], [0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 3, 70], [5, 0, 0, 0, 0, 0, 0, 0, 69, 0, 71, 0]]]

# Example 2

elem_divs = [6, 6, 6, 6, 12, 12, 60, 60, 120, 120, 120, 120, 120, 120, 120, 120]
mats =
[[[0, 4, 4, 0, 8, 0, 0, 0, 0, 40, 40, 80, 0, 80, 0, 0], [4, 0, 4, 0, 8, 8, 0, 0, 0, 0, 40, 80, 0, 80, 80, 0], [0, 2, 0, 0, 2, 0, 0, 0, 80, 80, 0, 80, 0, 20, 0, 0], [0, 2, 0, 4, 10, 2, 0, 0, 0, 40, 0, 0, 40, 60, 20, 0], [2, 0, 0, 4, 0, 4, 0, 0, 40, 0, 0, 0, 40, 80, 80, 0], [2, 2, 0, 2, 0, 8, 0, 0, 40, 40, 0, 0, 80, 80, 0, 80], [1, 0, 0, 0, 2, 0, 0, 0, 20, 0, 0, 0, 0, 20, 0, 0], [1, 1, 0, 0, 2, 2, 0, 0, 20, 20, 0, 0, 0, 20, 20, 0], [0, 4, 4, 0, 4, 0, 0, 0, 0, 100, 100, 20, 0, 40, 0, 0], [0, 0, 4, 0, 4, 4, 0, 0, 40, 0, 100, 20, 0, 40, 40, 0], [2, 2, 0, 0, 10, 0, 0, 0, 60, 80, 0, 20, 0, 100, 0, 0], [4, 4, 4, 0, 0, 0, 0, 0, 100, 100, 100, 20, 0, 0, 0, 0], [0, 4, 0, 4, 6, 10, 0, 0, 0, 80, 0, 0, 40, 80, 100, 0], [2, 0, 0, 2, 8, 8, 0, 0, 40, 0, 0, 0, 20, 60, 100, 0], [2, 2, 0, 4, 8, 0, 0, 0, 40, 40, 0, 0, 40, 60, 80, 100], [0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0]],
 [[2, 4, 4, 0, 0, 4, 0, 0, 80, 80, 40, 80, 0, 0, 40, 0], [2, 4, 0, 0, 8, 0, 0, 0, 40, 40, 0, 80, 0, 80, 0, 0], [0, 0, 2, 0, 0, 10, 0, 0, 0, 0, 80, 40, 0, 0, 100, 0], [0, 0, 0, 4, 2, 4, 0, 0, 0, 0, 0, 0, 40, 20, 80, 40], [0, 4, 0, 0, 8, 8, 0, 0, 0, 80, 0, 0, 0, 40, 80, 80], [2, 0, 0, 0, 4, 4, 0, 0, 40, 0, 0, 0, 0, 40, 80, 40], [0, 5, 0, 0, 0, 10, 0, 0, 0, 100, 0, 0, 0, 0, 100, 0], [1, 0, 0, 0, 2, 0, 0, 0, 20, 0, 0, 0, 0, 20, 0, 0], [2, 2, 4, 0, 0, 8, 0, 0, 80, 40, 100, 20, 0, 0, 80, 0], [4, 4, 0, 0, 4, 0, 0, 0, 20, 100, 0, 20, 0, 40, 0, 0], [0, 0, 2, 0, 0, 2, 0, 0, 60, 60, 80, 40, 0, 0, 20, 0], [2, 4, 4, 0, 0, 0, 0, 0, 80, 100, 100, 20, 0, 0, 0, 0], [0, 0, 0, 4, 10, 4, 0, 0, 0, 0, 0, 0, 40, 100, 20, 100], [0, 4, 0, 0, 4, 8, 0, 0, 0, 80, 0, 0, 60, 20, 80, 100], [2, 0, 0, 0, 4, 8, 0, 0, 40, 0, 0, 0, 60, 100, 40, 80], [0, 0, 0, 4, 4, 4, 0, 0, 0, 0, 0, 0, 40, 20, 20, 100]],
 [[0, 2, 0, 0, 2, 0, 0, 0, 80, 80, 0, 80, 0, 20, 0, 0], [2, 2, 2, 0, 2, 2, 0, 0, 40, 40, 80, 40, 0, 20, 20, 0], [4, 0, 4, 0, 8, 0, 50, 0, 40, 0, 40, 80, 0, 80, 0, 0], [4, 0, 4, 0, 0, 0, 50, 50, 80, 0, 80, 0, 0, 40, 40, 80], [5, 0, 0, 4, 6, 8, 0, 0, 100, 0, 0, 0, 40, 60, 40, 0], [5, 5, 0, 0, 6, 6, 0, 0, 100, 100, 0, 0, 0, 20, 60, 80], [2, 0, 1, 0, 4, 0, 10, 0, 40, 0, 20, 0, 0, 40, 0, 0], [2, 2, 1, 1, 6, 4, 10, 10, 40, 40, 20, 0, 20, 60, 40, 0], [2, 2, 0, 0, 10, 0, 0, 0, 60, 80, 0, 20, 0, 100, 0, 0], [4, 4, 2, 0, 10, 10, 0, 0, 20, 20, 80, 40, 0, 100, 100, 0], [4, 0, 4, 0, 4, 0, 10, 0, 100, 60, 100, 20, 0, 40, 0, 0], [4, 2, 4, 0, 0, 0, 0, 0, 100, 80, 100, 20, 0, 0, 0, 0], [2, 0, 2, 0, 8, 8, 10, 10, 40, 0, 40, 0, 0, 0, 0, 20], [2, 0, 0, 2, 0, 4, 0, 0, 40, 0, 0, 0, 20, 0, 80, 0], [2, 2, 0, 0, 8, 0, 0, 0, 40, 40, 0, 0, 60, 60, 0, 100], [0, 0, 0, 2, 0, 8, 0, 0, 0, 0, 0, 0, 20, 0, 100, 0]],
 [[0, 0, 0, 0, 8, 10, 0, 0, 0, 0, 0, 0, 0, 40, 60, 80], [0, 4, 0, 2, 10, 8, 0, 0, 0, 80, 0, 0, 80, 100, 40, 80], [0, 0, 0, 0, 4, 4, 0, 10, 0, 0, 0, 0, 80, 80, 40, 80], [2, 4, 4, 0, 8, 4, 50, 0, 80, 0, 80, 0, 0, 80, 40, 0], [2, 3, 4, 0, 0, 6, 0, 0, 80, 100, 40, 0, 0, 0, 60, 0], [3, 2, 4, 0, 6, 4, 0, 0, 20, 80, 40, 40, 0, 60, 40, 0], [0, 4, 0, 5, 0, 8, 0, 50, 0, 80, 0, 0, 100, 0, 80, 0], [2, 0, 1, 0, 4, 2, 10, 0, 40, 0, 20, 0, 0, 40, 20, 0], [0, 0, 0, 0, 8, 6, 0, 0, 0, 0, 0, 0, 0, 100, 80, 20], [0, 2, 0, 2, 2, 8, 0, 0, 0, 40, 0, 0, 80, 20, 100, 20], [0, 0, 0, 2, 4, 8, 0, 50, 0, 0, 0, 0, 60, 80, 80, 20], [0, 0, 0, 4, 8, 8, 0, 0, 0, 0, 0, 0, 100, 100, 100, 20], [2, 0, 2, 0, 4, 8, 10, 0, 80, 100, 40, 0, 0, 40, 80, 0], [4, 2, 2, 0, 0, 0, 0, 0, 100, 60, 20, 0, 0, 0, 0, 0], [4, 4, 2, 0, 0, 4, 0, 0, 0, 100, 80, 20, 0, 0, 40, 0], [4, 0, 0, 0, 0, 0, 0, 0, 100, 0, 0, 0, 0, 0, 0, 0]],
 [[2, 0, 0, 4, 4, 4, 0, 0, 40, 0, 0, 0, 40, 40, 80, 40], [4, 2, 0, 0, 8, 4, 0, 0, 80, 40, 0, 0, 0, 0, 40, 40], [5, 0, 0, 0, 6, 8, 0, 0, 100, 0, 0, 0, 0, 60, 40, 80], [4, 5, 2, 0, 10, 6, 0, 0, 40, 100, 80, 40, 0, 100, 60, 0], [4, 4, 0, 0, 10, 0, 0, 0, 20, 100, 0, 20, 0, 100, 0, 0], [4, 4, 2, 0, 8, 10, 0, 0, 20, 20, 80, 40, 0, 80, 100, 0], [5, 0, 0, 0, 5, 0, 0, 0, 100, 0, 0, 0, 0, 110, 0, 0], [5, 5, 0, 0, 10, 5, 0, 0, 100, 100, 0, 0, 0, 100, 110, 0], [4, 0, 0, 4, 8, 4, 0, 0, 80, 0, 0, 0, 100, 80, 110, 10], [2, 4, 0, 0, 0, 8, 0, 0, 40, 80, 0, 0, 90, 70, 80, 10], [1, 0, 0, 0, 6, 8, 0, 0, 20, 0, 0, 0, 90, 30, 100, 20], [0, 0, 0, 4, 4, 4, 0, 0, 0, 0, 0, 0, 100, 110, 110, 10], [4, 1, 2, 0, 2, 6, 0, 0, 100, 20, 110, 10, 0, 20, 60, 0], [4, 2, 0, 0, 4, 0, 0, 0, 50, 110, 0, 10, 0, 40, 0, 0], [4, 4, 4, 0, 8, 4, 0, 0, 50, 50, 100, 20, 0, 80, 40, 0], [4, 2, 2, 0, 0, 0, 0, 0, 100, 110, 110, 10, 0, 0, 0, 0]],
 [[0, 4, 0, 2, 8, 0, 0, 0, 0, 80, 0, 0, 80, 40, 0, 40], [2, 4, 0, 0, 0, 8, 0, 0, 40, 80, 0, 0, 0, 0, 40, 80], [0, 1, 0, 2, 0, 0, 0, 0, 0, 20, 0, 0, 80, 0, 40, 40], [5, 1, 2, 0, 4, 4, 0, 0, 100, 60, 80, 80, 0, 40, 40, 0], [4, 2, 4, 0, 0, 2, 0, 0, 100, 40, 40, 0, 0, 0, 20, 0], [4, 4, 0, 0, 10, 10, 0, 0, 20, 100, 0, 20, 0, 100, 100, 0], [0, 1, 0, 0, 0, 7, 0, 0, 0, 20, 0, 0, 0, 0, 10, 0], [5, 0, 0, 0, 5, 5, 0, 0, 100, 0, 0, 0, 0, 110, 110, 0], [0, 2, 0, 2, 8, 0, 0, 0, 0, 40, 0, 0, 110, 100, 0, 10], [4, 2, 0, 0, 0, 8, 0, 0, 80, 40, 0, 0, 90, 90, 100, 20], [0, 5, 0, 2, 0, 8, 0, 0, 0, 100, 0, 0, 110, 90, 30, 10], [0, 0, 0, 2, 8, 4, 0, 0, 0, 0, 0, 0, 110, 100, 110, 10], [1, 3, 2, 0, 8, 8, 0, 0, 20, 40, 20, 110, 0, 80, 80, 0], [2, 2, 2, 0, 0, 8, 0, 0, 110, 40, 20, 0, 0, 0, 80, 0], [4, 2, 0, 0, 4, 4, 0, 0, 50, 110, 0, 10, 0, 40, 40, 0], [2, 0, 4, 0, 0, 0, 0, 0, 110, 0, 10, 0, 0, 0, 0, 0]],
 [[1, 0, 0, 0, 2, 0, 0, 0, 20, 0, 0, 0, 0, 20, 0, 0], [2, 1, 0, 0, 2, 2, 0, 0, 40, 20, 0, 0, 0, 20, 20, 0], [2, 0, 1, 0, 4, 0, 10, 0, 40, 0, 20, 0, 0, 40, 0, 0], [2, 2, 2, 1, 8, 4, 10, 10, 40, 40, 40, 0, 20, 80, 40, 0], [5, 0, 0, 0, 5, 0, 0, 0, 100, 0, 0, 0, 0, 110, 0, 0], [5, 5, 0, 0, 10, 5, 0, 0, 100, 100, 0, 0, 0, 100, 110, 0], [0, 0, 5, 0, 10, 0, 1, 0, 0, 0, 4, 0, 0, 4, 0, 0], [0, 0, 5, 5, 10, 10, 2, 1, 0, 0, 4, 0, 4, 4, 4, 0], [5, 0, 0, 0, 10, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0], [4, 5, 0, 0, 10, 10, 0, 0, 8, 4, 0, 0, 0, 4, 4, 0], [4, 0, 5, 0, 8, 0, 2, 0, 8, 0, 4, 0, 0, 8, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, 4, 5, 4, 8, 2, 2, 8, 8, 8, 0, 4, 16, 8, 0], [2, 0, 0, 0, 2, 0, 0, 0, 112, 0, 0, 0, 0, 116, 0, 0], [2, 2, 0, 0, 4, 2, 0, 0, 112, 112, 0, 0, 0, 112, 116, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]],
 [[0, 5, 0, 0, 0, 0, 0, 0, 0, 100, 0, 0, 0, 0, 0, 0], [1, 5, 0, 0, 0, 0, 0, 0, 20, 100, 0, 0, 0, 0, 0, 0], [0, 4, 0, 5, 0, 8, 0, 0, 0, 80, 0, 0, 100, 0, 80, 0], [2, 0, 1, 5, 4, 0, 0, 0, 40, 0, 20, 0, 100, 40, 0, 0], [0, 1, 0, 0, 0, 7, 0, 0, 0, 20, 0, 0, 0, 0, 10, 0], [5, 0, 0, 0, 5, 5, 0, 0, 100, 0, 0, 0, 0, 110, 110, 0], [0, 0, 0, 1, 0, 2, 0, 59, 0, 0, 0, 0, 116, 0, 116, 0], [0, 0, 5, 0, 10, 0, 1, 1, 0, 0, 4, 0, 0, 4, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 116, 0, 0, 0, 0, 0, 0], [5, 1, 0, 0, 0, 0, 0, 0, 4, 116, 0, 0, 0, 0, 0, 0], [0, 2, 0, 1, 0, 4, 0, 0, 0, 112, 0, 0, 116, 0, 112, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 0, 5, 1, 8, 0, 0, 0, 8, 0, 4, 0, 116, 8, 0, 0], [0, 4, 0, 0, 0, 10, 0, 0, 0, 8, 0, 0, 0, 0, 4, 0], [2, 0, 0, 0, 2, 2, 0, 0, 112, 0, 0, 0, 0, 116, 116, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]],
 [[0, 4, 4, 0, 4, 0, 0, 0, 0, 100, 100, 20, 0, 40, 0, 0], [0, 0, 4, 0, 4, 4, 0, 0, 40, 0, 100, 20, 0, 40, 40, 0], [2, 2, 0, 0, 10, 0, 0, 0, 60, 80, 0, 20, 0, 100, 0, 0], [0, 4, 0, 4, 6, 10, 0, 0, 0, 80, 0, 0, 40, 80, 100, 0], [4, 0, 0, 4, 4, 4, 0, 0, 80, 0, 0, 0, 10, 90, 110, 0], [4, 4, 0, 2, 4, 0, 0, 0, 80, 80, 0, 0, 20, 90, 100, 110], [5, 0, 0, 0, 10, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0], [5, 5, 0, 0, 10, 10, 0, 0, 4, 4, 0, 0, 0, 4, 4, 0], [0, 1, 1, 0, 8, 0, 0, 0, 0, 115, 115, 5, 0, 8, 0, 0], [4, 0, 1, 0, 8, 8, 0, 0, 6, 0, 115, 5, 0, 8, 8, 0], [3, 2, 0, 0, 2, 0, 0, 0, 11, 110, 0, 5, 0, 20, 0, 0], [1, 1, 1, 0, 0, 0, 0, 0, 115, 115, 115, 5, 0, 0, 0, 0], [0, 2, 0, 4, 4, 2, 0, 0, 24, 16, 0, 0, 10, 15, 20, 0], [4, 0, 0, 5, 6, 2, 0, 0, 104, 0, 0, 0, 5, 109, 115, 0], [4, 4, 0, 4, 6, 4, 0, 0, 104, 104, 0, 0, 10, 109, 114, 115], [0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0]],
 [[2, 2, 4, 0, 0, 8, 0, 0, 80, 40, 100, 20, 0, 0, 80, 0], [4, 4, 0, 0, 4, 0, 0, 0, 20, 100, 0, 20, 0, 40, 0, 0], [0, 0, 2, 0, 0, 2, 0, 0, 60, 60, 80, 40, 0, 0, 20, 0], [0, 0, 0, 4, 10, 4, 0, 0, 0, 0, 0, 0, 40, 100, 20, 100], [0, 2, 0, 0, 8, 4, 0, 0, 0, 40, 0, 0, 30, 10, 40, 110], [4, 0, 0, 0, 8, 4, 0, 0, 80, 0, 0, 0, 30, 110, 20, 100], [0, 1, 0, 0, 0, 2, 0, 0, 0, 116, 0, 0, 0, 0, 116, 0], [5, 0, 0, 0, 10, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0], [2, 4, 1, 0, 0, 4, 0, 0, 110, 104, 115, 5, 0, 0, 112, 0], [5, 1, 0, 0, 8, 0, 0, 0, 1, 115, 0, 5, 0, 8, 0, 0], [3, 3, 2, 0, 0, 10, 0, 0, 105, 81, 110, 10, 0, 0, 100, 0], [2, 1, 1, 0, 0, 0, 0, 0, 110, 115, 115, 5, 0, 0, 0, 0], [0, 0, 0, 4, 2, 10, 0, 0, 24, 0, 0, 0, 10, 20, 5, 115], [0, 2, 0, 3, 10, 4, 0, 0, 0, 16, 0, 0, 15, 5, 16, 115], [4, 0, 0, 3, 2, 8, 0, 0, 104, 0, 0, 0, 15, 119, 10, 110], [0, 0, 0, 4, 10, 10, 0, 0, 0, 0, 0, 0, 10, 5, 5, 115]],
 [[2, 2, 0, 0, 10, 0, 0, 0, 60, 80, 0, 20, 0, 100, 0, 0], [4, 4, 2, 0, 10, 10, 0, 0, 20, 20, 80, 40, 0, 100, 100, 0], [4, 0, 4, 0, 4, 0, 10, 0, 100, 60, 100, 20, 0, 40, 0, 0], [2, 0, 2, 0, 8, 8, 10, 10, 40, 0, 40, 0, 0, 0, 0, 20], [1, 0, 0, 4, 6, 8, 0, 0, 20, 0, 0, 0, 10, 60, 100, 0], [1, 1, 0, 0, 10, 6, 0, 0, 20, 20, 0, 0, 30, 30, 60, 110], [4, 0, 5, 0, 8, 0, 2, 0, 8, 0, 4, 0, 0, 8, 0, 0], [4, 4, 5, 5, 6, 8, 2, 2, 8, 8, 4, 0, 4, 12, 8, 0], [3, 2, 0, 0, 2, 0, 0, 0, 11, 110, 0, 5, 0, 20, 0, 0], [5, 5, 2, 0, 2, 2, 0, 0, 25, 1, 110, 10, 0, 20, 20, 0], [1, 3, 1, 0, 8, 0, 2, 0, 43, 105, 115, 5, 0, 32, 0, 0], [1, 2, 1, 0, 0, 0, 0, 0, 115, 110, 115, 5, 0, 0, 0, 0], [4, 0, 4, 0, 0, 0, 2, 2, 56, 48, 8, 0, 0, 46, 22, 5], [4, 0, 0, 5, 0, 4, 0, 0, 80, 0, 0, 0, 5, 96, 110, 0], [4, 4, 0, 3, 6, 0, 0, 0, 80, 80, 0, 0, 15, 85, 96, 115], [0, 0, 0, 5, 0, 2, 0, 0, 0, 0, 0, 0, 5, 0, 115, 0]],
 [[4, 4, 4, 0, 0, 0, 0, 0, 100, 100, 100, 20, 0, 0, 0, 0], [2, 4, 4, 0, 0, 0, 0, 0, 80, 100, 100, 20, 0, 0, 0, 0], [4, 2, 4, 0, 0, 0, 0, 0, 100, 80, 100, 20, 0, 0, 0, 0], [0, 0, 0, 2, 8, 0, 0, 0, 0, 0, 0, 0, 20, 100, 0, 0], [0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 10, 0, 110, 0], [0, 0, 0, 2, 8, 8, 0, 0, 0, 0, 0, 0, 20, 10, 10, 110], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 0, 0, 0, 0, 0, 115, 115, 115, 5, 0, 0, 0, 0], [2, 1, 1, 0, 0, 0, 0, 0, 110, 115, 115, 5, 0, 0, 0, 0], [1, 2, 1, 0, 0, 0, 0, 0, 115, 110, 115, 5, 0, 0, 0, 0], [1, 1, 1, 0, 0, 0, 0, 0, 115, 115, 115, 4, 0, 0, 0, 0], [0, 0, 0, 5, 2, 0, 0, 0, 0, 0, 0, 0, 5, 115, 0, 0], [0, 0, 0, 5, 0, 2, 0, 0, 0, 0, 0, 0, 5, 0, 115, 0], [0, 0, 0, 4, 10, 10, 0, 0, 0, 0, 0, 0, 10, 5, 5, 115], [0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 119]],
 [[0, 0, 0, 0, 8, 6, 0, 0, 0, 0, 0, 0, 0, 100, 80, 20], [0, 2, 0, 2, 2, 8, 0, 0, 0, 40, 0, 0, 80, 20, 100, 20], [0, 0, 0, 2, 4, 8, 0, 50, 0, 0, 0, 0, 60, 80, 80, 20], [2, 0, 2, 0, 4, 8, 10, 0, 80, 100, 40, 0, 0, 40, 80, 0], [2, 1, 4, 0, 0, 6, 0, 0, 110, 90, 10, 0, 0, 0, 60, 0], [5, 2, 4, 0, 6, 8, 0, 0, 0, 110, 100, 10, 0, 60, 80, 0], [0, 2, 0, 1, 0, 4, 0, 58, 0, 112, 0, 0, 116, 0, 112, 0], [4, 0, 5, 0, 8, 10, 2, 0, 8, 0, 4, 0, 0, 8, 4, 0], [0, 0, 0, 0, 2, 4, 0, 0, 0, 96, 0, 0, 0, 115, 111, 5], [0, 4, 0, 2, 10, 2, 0, 0, 24, 104, 0, 0, 110, 4, 115, 5], [0, 0, 0, 3, 4, 4, 0, 58, 0, 72, 0, 0, 107, 110, 88, 5], [0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 0, 115, 115, 115, 5], [2, 1, 4, 0, 8, 4, 2, 0, 38, 117, 8, 0, 0, 32, 16, 0], [1, 3, 5, 0, 0, 0, 0, 0, 115, 35, 5, 0, 0, 0, 24, 0], [0, 1, 2, 0, 0, 8, 0, 0, 70, 115, 110, 5, 0, 96, 104, 0], [1, 0, 0, 0, 0, 0, 0, 0, 115, 0, 0, 0, 0, 0, 0, 0]],
 [[2, 0, 0, 2, 4, 8, 0, 0, 40, 0, 0, 0, 80, 40, 100, 20], [4, 2, 0, 0, 0, 4, 0, 0, 80, 40, 0, 0, 60, 20, 40, 20], [2, 0, 0, 0, 0, 4, 0, 0, 40, 0, 0, 0, 60, 60, 80, 40], [2, 2, 4, 0, 4, 0, 0, 0, 80, 40, 100, 20, 0, 40, 0, 0], [4, 2, 0, 0, 4, 0, 0, 0, 50, 110, 0, 10, 0, 40, 0, 0], [4, 4, 4, 0, 8, 4, 0, 0, 50, 50, 100, 20, 0, 80, 40, 0], [2, 0, 0, 0, 2, 0, 0, 0, 112, 0, 0, 0, 0, 116, 0, 0], [2, 2, 0, 0, 4, 2, 0, 0, 112, 112, 0, 0, 0, 112, 116, 0], [4, 0, 0, 2, 8, 2, 0, 0, 104, 0, 0, 0, 110, 104, 115, 5], [2, 4, 0, 3, 10, 8, 0, 0, 88, 104, 0, 0, 105, 99, 104, 5], [4, 0, 0, 3, 6, 4, 0, 0, 80, 0, 0, 0, 105, 81, 110, 10], [0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0, 0, 110, 115, 115, 5], [2, 4, 1, 0, 8, 0, 0, 0, 62, 80, 115, 5, 0, 80, 96, 0], [1, 1, 0, 0, 4, 0, 0, 0, 17, 115, 0, 5, 0, 16, 0, 0], [1, 1, 2, 0, 8, 4, 0, 0, 17, 17, 110, 10, 0, 32, 16, 0], [2, 1, 1, 0, 0, 0, 0, 0, 110, 115, 115, 5, 0, 0, 0, 0]],
 [[0, 4, 0, 4, 4, 0, 0, 0, 0, 80, 0, 0, 100, 80, 0, 20], [2, 4, 0, 0, 0, 4, 0, 0, 40, 80, 0, 0, 60, 60, 80, 40], [0, 4, 0, 4, 0, 4, 0, 0, 0, 80, 0, 0, 100, 60, 60, 20], [2, 0, 4, 0, 4, 4, 0, 0, 40, 80, 40, 100, 0, 40, 40, 0], [2, 2, 2, 0, 0, 8, 0, 0, 110, 40, 20, 0, 0, 0, 80, 0], [4, 2, 0, 0, 4, 4, 0, 0, 50, 110, 0, 10, 0, 40, 40, 0], [0, 4, 0, 0, 0, 10, 0, 0, 0, 8, 0, 0, 0, 0, 4, 0], [2, 0, 0, 0, 2, 2, 0, 0, 112, 0, 0, 0, 0, 116, 116, 0], [0, 2, 0, 1, 4, 0, 0, 0, 0, 16, 0, 0, 115, 110, 0, 5], [4, 2, 0, 3, 6, 4, 0, 0, 104, 16, 0, 0, 105, 105, 110, 10], [0, 2, 0, 1, 6, 6, 0, 0, 0, 40, 0, 0, 115, 105, 11, 5], [0, 0, 0, 1, 4, 2, 0, 0, 0, 0, 0, 0, 115, 110, 115, 5], [4, 2, 4, 0, 8, 8, 0, 0, 80, 18, 10, 115, 0, 104, 104, 0], [1, 2, 4, 0, 0, 8, 0, 0, 115, 88, 10, 0, 0, 0, 104, 0], [1, 1, 0, 0, 4, 4, 0, 0, 17, 115, 0, 5, 0, 16, 16, 0], [1, 0, 5, 0, 0, 0, 0, 0, 115, 0, 5, 0, 0, 0, 0, 0]],
 [[0, 0, 0, 4, 8, 8, 0, 0, 0, 0, 0, 0, 100, 100, 100, 20], [0, 0, 0, 2, 8, 8, 0, 0, 0, 0, 0, 0, 80, 100, 100, 20], [0, 0, 0, 4, 4, 8, 0, 0, 0, 0, 0, 0, 100, 80, 100, 20], [4, 2, 0, 0, 0, 0, 0, 0, 100, 20, 0, 0, 0, 0, 0, 0], [2, 0, 4, 0, 0, 0, 0, 0, 110, 0, 10, 0, 0, 0, 0, 0], [4, 2, 2, 0, 0, 0, 0, 0, 100, 110, 110, 10, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 0, 115, 115, 115, 5], [0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0, 0, 110, 115, 115, 5], [0, 0, 0, 1, 4, 2, 0, 0, 0, 0, 0, 0, 115, 110, 115, 5], [0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 0, 115, 115, 115, 4], [1, 5, 0, 0, 0, 0, 0, 0, 115, 5, 0, 0, 0, 0, 0, 0], [1, 0, 5, 0, 0, 0, 0, 0, 115, 0, 5, 0, 0, 0, 0, 0], [2, 1, 1, 0, 0, 0, 0, 0, 110, 115, 115, 5, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 115, 0, 0, 1, 0, 0, 0, 0]]]

# Example 3

elem_divs = [24, 24, 24, 24, 24, 24, 24, 24]
mats = 
[[[2, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [23, 1, 1, 1, 0, 0, 0, 0]],
 [[0, 2, 0, 0, 0, 0, 0, 0], [23, 2, 0, 0, 0, 0, 0, 0], [0, 1, 1, 23, 0, 0, 0, 0], [23, 1, 1, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [23, 1, 1, 0, 0, 0, 0, 0]],
 [[0, 0, 2, 0, 0, 0, 0, 0], [23, 1, 1, 1, 0, 0, 0, 0], [23, 0, 2, 0, 0, 0, 0, 0], [0, 23, 1, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [23, 0, 1, 1, 0, 0, 0, 0]],
 [[0, 0, 0, 2, 0, 0, 0, 0], [0, 1, 23, 1, 0, 0, 0, 0], [23, 1, 1, 1, 0, 0, 0, 0], [23, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [23, 1, 0, 1, 0, 0, 0, 0]],
 [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1]],
 [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 23, 0, 0, 0], [22, 1, 1, 1, 1, 1, 1, 22], [22, 1, 1, 1, 0, 1, 1, 23]],
 [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [3, 23, 23, 23, 23, 23, 23, 2], [1, 0, 0, 0, 23, 0, 0, 0], [1, 0, 0, 0, 23, 23, 0, 1]],
 [[23, 1, 1, 1, 0, 0, 0, 0], [23, 1, 0, 1, 0, 0, 0, 0], [23, 1, 1, 0, 0, 0, 0, 0], [23, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 23, 0, 23, 1], [22, 1, 1, 1, 0, 1, 1, 23], [0, 0, 0, 0, 23, 0, 0, 1]]]