placed beside the board or on the group elements (graph vertices in Version III), in which case we say a group element is pebbled. Pebbles $p_i$ are placed on elements of $G$ (vertices of $\Gamma_G$) and pebbles $p'_i$ on $H$ ($\Gamma_H$). At any point in time the pebbles $(p_1,\ldots,p_k)$ give us a pebbled tuple in $(G\cup \{\bot\})^k$ (or $(\Gamma_G\cup\{\bot\})^k$), where $\bot$ indicates that the pebble is placed besides the board.

#### Version I:

All $k$ pairs of pebbles are initially placed beside the board. A round of the game consists of these steps:

1.  Spoiler picks up a pair of pebbles $(p_i;p_i')$.

2.  Duplicator chooses a bijection $f\colon G\to H$.

3.  Spoiler pebbles some element $g\in G$ with $p_i$, the corresponding pebble $p_i'$ is placed on $f(g)$.

The winning condition is always checked right after Step 1. At that moment, the pebbles not in Spoiler’s hand then pebble a $k$-tuple over $G\cup\{\bot\}$ and a corresponding $k$-tuple over $H\cup\{\bot\}$. Spoiler wins if the pebbled tuples differ with respect to the initial coloring of Version I. (This implies that no more pebbles are placed beside the graph.)

#### Version II:

Version II differs from Version I only in that the winning condition 