Object: Fill the cube with a given sequence of 'Sticks'.
 
Place a Stick (a row or file of tokens) by clicking the start and end position. To click Stick of length 1, click the same position twice.
The next Stick must be adjacent to the Stick placed previously.
 
The length of each Stick is given and indicated by the numbers 3-2-1-2: the first Stick has length 3, the next 2, then 1, then 2, then 3 and so on.
The sequence is different for the different variants.
 
Variant 2 plays sequence 4-3-2-1-2-3-4... on a 4x4x4 board.
Variant 3 plays sequence 5-4-3-2-1-2-3-4-5... on a 5x5x5 board.
 
 
Note that a similar problem can be stated for any cube size: For any given natural number n, its cube n^3 is the sum of the sequence
n, n-1, n-2,...,2,1,2,...n-1,n,n-1,...,2,1,2,...,n-1,n
of numbers. Exactly n times we have to go from n down to 1 and back up to n. (Can you prove this mathematically?)
 
This begs the question whether a sequence of Sticks having these lengths can fill such a cube. It is easy to prove that this is indeed the case for any cube size.
In this game 'Sticks Sequence', however, the problem is more difficult since consecutive Sticks have to be adjacent. It might be interesting to find out whether the problem has a solution for all cube sizes n > 5.
 
More freeware as well as real puzzles and games at my homepage http://karl.kiwi.gen.nz.