Dice stack

Here is a puzzle from one of the Professor Layton games.

Puzzle

You have three regular six-sided dice. You are to stack the dice in a single column. The bottom die should be positioned so that the single dot is facing out, towards you. The only other condition as that whenever two dice touch, the numbers on the touching sides must sum to 5. What number will be on the top of the top-most die?

Help

Hint

You need to know that the sides of a regular six-sided die are arranged so that opposite sides always sum to seven. Now consider all the possibilities.

Answer

6

Solution

The sides touching between the bottom and middle dice must be 2 and 3, in some order (it cannot be 1 and 4, since the bottom die has 1 facing out, and if 1 was facing down on the middle die, then 6 would be facing up and this cannot be since the sum between the middle and top die must be 5). It must be that the bottom die has 2 up, since if the middle die had 2 down, then it would have 5 up, making the sum between the middle and top die impossible. Thus the middle die is positioned with 3 down, so 4 facing up. The top die then must have 1 facing down, which makes 6 facing up.

Extensions

How far can this idea go? Could you do something similar with more dice?

References

Professor Layton and the Diabolical Box - sequel to the Curious Village. Another great Nintendo DS puzzle game.