In the Queen’s croquet game, the soldiers serve as hoops, or arches.
The Queen of Hearts’ game is a grotesque Wonderland version of croquet. The balls are live hedgehogs, the mallets are flamingos, and the Queen’s soldiers bend themselves into arches for players to strike through.
The arrangement is constantly disrupted because the soldiers must stop acting as hoops whenever the Queen orders an execution. That instability is part of the joke: the game’s equipment is alive, uncooperative, and subordinate to the Queen’s temper.
A common mix-up is to assign the soldiers the role of mallets or balls. Those roles belong to flamingos and hedgehogs respectively. The soldiers are the arches, sometimes called hoops, and the book’s chaotic rules make them particularly unreliable pieces of sporting equipment.