According to the flight instructor and the student pilot, they flew from their home airport to another airport where the student pilot completed four landings without the flight instructor touching the controls. After the fourth landing, the flight instructor got out of the airplane so the student pilot could complete three solo takeoffs and landings. The first two landings were uneventful; however, according to the student pilot, just before the landing flare for the third landing, the airplane was "pushed" to the left side of the runway, "possibly by a wind gust," then landed hard. The flight instructor noted that during the approach, the airplane appeared to be slower than normal, followed by a hard landing. The main landing gear touched down first, then there was a "gallop on the nose wheel," and the airplane bounced twice before all landing gear were on the ground. The flight instructor subsequently performed a partial inspection the airplane, determined there was no damage, and the two flew it back to their home airport. During a subsequent rental, at an en route stop, another pilot noted propeller damage, and further inspection revealed firewall damage. Neither the flight instructor nor the student pilot reported any preexisting mechanical anomalies or damage to the airplane. Wind, recorded about the time of the accident, was 10 degrees from the right of runway heading, at 6 knots.