IBM Ponder This #195 · July 2014

32 pebbles are placed on the six edges of a tetrahedron, as shown in the following figure.

The image is now missing but it used to be image of a tetrahedron, a four-sided, three-dimensional triangular pyramid, has 32 pebbles distributed on its six edges. The three edges that make up its base have 1, 3, and 5 pebbles respectively. The edge emanating from the vertex of the edges with 1 and 5 pebbles contains 2 pebbles; the edge emanating from the vertex of the edges with 1 and 3 pebbles contains 13 pebbles; and the last edge emanating from the vertex of the edges with 3 and 5 pebbles contains 8 pebbles.

Two players are playing a game. Each player, on her turn, can take as many pebbles as she chooses, but a minimum of one pebble must be taken each turn, and all the pebbles taken must be on the same face. If you cannot play (i.e., if all the pebbles are taken) - you lose.

What are the winning moves, for the first player, in this situation?

For example, if the first player plays 0,0,3,0,8,13 i.e., taking the whole northeast face, leaving the tetrahedron as 1,2,0,5,0,0; then the second player can win by playing 0,1,0,4,0,0 on the bottom face which leaves 1,1,0,1,0,0; and for every possible move of the first player, the second player can immediately win.

Please supply your answer as a list of lines, each line is a possible move.

Solution

Best opened after a real attempt

To be added.