Proof

Let q = (q₁, q₂… *q_p) be a state of the game of Cram and {q_i, 1 ≤ i ≤ p} be its connected components. Then q is a game equivalent to the Marienbad matchstick game nim( q*₁ ) $\oplus \backslash oplus$ … $\oplus \backslash oplus$ nim( *q_p * ), where p represents the number of rows of matches and nim(*q_i *) the number of matches on each of these rows.

In particular, q is losing if and only if nim( q₁ ) $\oplus \backslash oplus$ … $\oplus \backslash oplus$ nim( *q_p * ) = 0.

(i) If there is no remaining move, q is losing, and nim(q) = mex(∅) = 0