Eight cards, ±1, ±i, ±j, ±k. Multiply the pile on the left or on the right to hit the target before the computer does. Left and right give different answers, and that is the whole game.
Play. On your turn, pick a card and put it on the left of the pile (card × pile) or on the right (pile × card). If the new pile equals the target, you score and a new target is turned up. Then you draw back to five cards. The computer plays the same way.
The group. The cards form the quaternion group Q₈, of order 8. It is not commutative: ij = k, ji = −k. So the same card can land on k or −k depending on the side you play it on.
Rotations. Each unit quaternion q rotates space by v ↦ q v q⁻¹. The card i is a half-turn about the x-axis, j about y, k about z. The cards q and −q give the same rotation, so −1 is a full 360° turn that is not the identity card: Dirac's belt trick. Aiming for −1 versus +1 is aiming for a 360° versus a 720° turn.
Defects. Q₈ is also the group that classifies line defects in a biaxial nematic liquid crystal. Two defect lines labeled i and j cannot pass through each other freely: their commutator iji⁻¹j⁻¹ = −1 leaves a tether behind.