Military officials said on Friday that their investigations had found that Army cadets and Navy midshipmen who flashed hand symbols during a nationally televised football pregame show last week were playing a juvenile game known as “the circle game,” and not making hand gestures associated with white nationalists and other hate groups.

The United States Military Academy and the United States Naval Academy announced the finding in separate statements, less than a week after reports that the cadets and midshipmen were making racist gestures during ESPN’s broadcast of the Army-Navy game.

The broadcast showed cadets and midshipmen in the stands flashing the gesture, which looks like an “O.K.” sign, at least five times behind the ESPN broadcaster Rece Davis before the game, which was held on Saturday at Lincoln Financial Field in Philadelphia.

President Trump attended the game and met with players in each team’s locker room before the game.

The Naval Academy said its investigation included a review of video footage, more than two dozen interviews, and background checks by the Naval Criminal Investigative Service and the F.B.I.