# Statement

Each of the six faces on a cube has a different digit (0 to 9) written on it; the same is done to a second cube. By placing the two cubes side-by-side in different positions we can form a variety of 2-digit numbers.

In fact, by carefully choosing the digits on both cubes it is possible to display all of the square numbers below one-hundred: 01, 04, 09, 16, 25, 36, 49, 64, and 81.

For example, one way this can be achieved is by placing {0, 5, 6, 7, 8, 9} on one cube and {1, 2, 3, 4, 8, 9} on the other cube.

However, for this problem we shall allow the 6 or 9 to be turned upside-down so that an arrangement like {0, 5, 6, 7, 8, 9} and {1, 2, 3, 4, 6, 7} allows for all nine square numbers to be displayed; otherwise it would be impossible to obtain 09.

In determining a distinct arrangement we are interested in the digits on each cube, not the order.

{1, 2, 3, 4, 5, 6} is equivalent to {3, 6, 4, 1, 2, 5}

{1, 2, 3, 4, 5, 6} is distinct from {1, 2, 3, 4, 5, 9}

But because we are allowing 6 and 9 to be reversed, the two distinct sets in the last example both represent the extended set {1, 2, 3, 4, 5, 6, 9} for the purpose of forming 2-digit numbers.

How many distinct arrangements of the two cubes allow for all of the square numbers to be displayed?

# Solution

The number of different ways of making a cube with all different digits is C(10,6) = 210. 2 cubes = 210^{2} = 44100. Just bruteforce it!

What is needed is not to count repeated combinations.

from itertools import combinations if __name__ == '__main__': result = 0 for dice1 in combinations(range(10), 6): for dice2 in combinations(range(10), 6): if dice2 < dice1: continue if not ((0 in dice1 and 1 in dice2) or (0 in dice2 and 1 in dice1)): continue if not ((0 in dice1 and 4 in dice2) or (0 in dice2 and 4 in dice1)): continue if not ((0 in dice1 and 9 in dice2) or (0 in dice2 and 9 in dice1)) \ and not ((0 in dice1 and 6 in dice2) or (0 in dice2 and 6 in dice1)): continue if not ((1 in dice1 and 6 in dice2) or (1 in dice2 and 6 in dice1)) \ and not ((1 in dice1 and 9 in dice2) or (1 in dice2 and 9 in dice1)): continue if not ((2 in dice1 and 5 in dice2) or (2 in dice2 and 5 in dice1)): continue if not ((3 in dice1 and 6 in dice2) or (3 in dice2 and 6 in dice1)) \ and not ((3 in dice1 and 9 in dice2) or (3 in dice2 and 9 in dice1)): continue if not ((4 in dice1 and 6 in dice2) or (4 in dice2 and 6 in dice1)) \ and not ((4 in dice1 and 9 in dice2) or (4 in dice2 and 9 in dice1)): continue if not ((6 in dice1 and 4 in dice2) or (6 in dice2 and 4 in dice1)) \ and not ((9 in dice1 and 4 in dice2) or (9 in dice2 and 4 in dice1)): continue if not ((8 in dice1 and 1 in dice2) or (8 in dice2 and 1 in dice1)): continue result += 1 print("The result is:", result)

The Python file is available for download here.