How to solve the 36 Cube puzzle – hints & solution

Daniel Hepper — Sun, 10 Jan 2010 19:00:51 +0000

For Christmas, I got the 36 Cube puzzle. It is consists of 36 towers in 6 colors and 6 different sizes and a base plate with 6 by 6 slots to plug in the towers. These slots are of different heights. The goal is to place one towers of every color in each row and column. And the towers must fit to form a level cube.

After some tries, I came to the conclusion that this puzzle is the work of the devil and that I should not waste more brain cycles on solving it. So I wrote a little python script to solve the puzzle for me.

Show sourcecode

My program quickly came up with a correct placement for 34 towers – but it failed to find the complete solution.

[('P', 5), ('Y', 3), ('O', 2), ('B', 1), ('R', 4), ('G', 6)]
[('Y', 4), ('O', 1), ('P', 6), ('R', 2), ('G', 5), ('B', 3)]
[('O', 6), ('B', 5), ('R', 3), ('G', 4), ('P', 1), ('Y', 2)]
[('R', 1), ('G', 2), ('Y', 5), ('P', 3), ('B', 6), ('O', 4)]
[('B', 2), ('P', 4), ('G', 1), ('Y', 6), ('O', 3), ('R', 5)]
[('G', 3), ('R', 6), ('B', 4), ('O', 5), ('X', 2), ('X', 1)]

Legend:
P = Purple, Y = Yellow, O = Orange, B = Blue, R = Red, G = Green, X = Empty
The number is the size of the tower.
As you can see, I didn’t waste much time on making the output pretty

So close and yet so far

After spending lots of time verifying that my program was working correctly, I became impatient and googled for help. I found an answer, but it revealed to much, taking all the fun.

Therefore, I split my solution into multiple hints. If you are stuck, reveal just one of them at a time and try to figure it out by yourself. It is way more rewarding!

Hint #1 (show):

Hint #2 (show):

Hint #3 (show):

Hint #4 (show):

Hint #5 (show):

Even if you uncovered all hints, the puzzle is still far from solved. You can still tinker with it forever.

Spoiler alert: Don’t uncover the solution, unless your are really desperate!

Solution (show):

After I had found the two culprits, I adjusted my program by changing the array and inserting two special conditions. Within seconds I had a complete solution of the 36 Cube:

[('P', 5), ('Y', 3), ('B', 2), ('O', 1), ('R', 4), ('G', 6)]
[('B', 4), ('P', 1), ('Y', 5), ('G', 2), ('O', 5), ('R', 3)]
[('R', 6), ('G', 5), ('P', 3), ('Y', 4), ('B', 1), ('O', 2)]
[('G', 1), ('R', 2), ('O', 6), ('B', 3), ('Y', 6), ('P', 4)]
[('Y', 2), ('O', 4), ('R', 1), ('P', 6), ('G', 3), ('B', 5)]
[('O', 3), ('B', 6), ('G', 4), ('R', 5), ('P', 2), ('Y', 1)]

Finally solved

#!/usr/bin/python
 
COLORS = dict(zip((1 &lt;&lt; i for i in range(6)), ('P','Y','O','B','R','G')))
COLORS[False] = 'X'
 
used_towers = [0 for i in range(6)]
 
# definition with (1,2) and (3,2) switched
cube = ((1,3,4,5,2,0),
        (2,5,1,4,1,3),
        (0,1,3,2,5,4),
        (5,4,0,3,0,2),
        (4,2,5,0,3,1),
        (3,0,2,1,4,5))
 
solution = [[False for x in range(6)] for x in range(6)]
used = tuple(set() for i in range(6))
 
def test_color(pos, solution, used_towers, color, size):
 #check if tower of this size and color has already been used
 if color &amp; used_towers[size]:
  return False
 #check if tower has already been used in row or column:
 for i in range(6):
  if color == solution[pos/6][i]: return False
  if color == solution[i][pos%6]: return False
 # special conditions for the two special towers
 if (pos/6, pos%6) == (1,2) and color != 2: return False
 if (pos/6, pos%6) == (3,2) and color != 4: return False
 return True
 
def print_solution(solution):
 for i in range(6):
  print zip((COLORS[c] for c in solution[i]), (6 - s for s in cube[i]))
 
def solve(pos, solution, used_towers, best):
 if pos == 36: return best, solution
 size = cube[pos/6][pos%6]
 for i in range(6):
  color = 1 &lt;&lt; i
  if test_color(pos, solution, used_towers, color, size):
   used_towers[size] = used_towers[size] | color
   solution[pos/6][pos%6] = color
   if pos &gt; best:
    print "#######", pos + 1, "towers placed"
    print_solution(solution)
    best = pos
   best, solution = solve(pos+1, solution, used_towers, best)
   if best &gt;= 35:
    return best, solution
   solution[pos/6][pos%6] = False
   used_towers[size] = used_towers[size] ^ color
 return best, solution
 
solve(0, solution, used_towers, 0)

Daniel Hepper » Python

How to solve the 36 Cube puzzle – hints & solution