From the Super Mario Wiki
Rhythmic Ribbon is a Gymnastics Event that appears in the Wii version of Mario & Sonic at the London 2012 Olympic Games. The objective of the event is to score the most points, with the event taking place in the North Greenwich Arena. Peach can be seen competing in this event in the game's opening. The starting Olympic record in this event is held by Peach at 48.200 points.
After all players have chosen a song, the events start for the first player. The songs that can play are The Blue Danube, Radetzky March and Csikós Post, in order of ascending difficulty. This events starts with the player's character being announced, before showing the starting position of the song with the character performing the intro for the song. To start, the player must swing the Wii Remote down or wait for it to start. After starting, circles appear in various colors. The player is given points based on the timing of swinging down. Being too early or too late always result in zero points. Blue circles are the most common. Red circles allow for tossing and catching. To toss, the player must swing up, but can also swing down. Missing the toss circle causes the catch circle to not appear. Yellow stars can either cause tricks with three swing or be a single toss catch with a symbol. While trick Yellow stars are activated no matter what, toss Yellow stars act like toss red circles with the catch being a red circle. Waning circles are circles that are near the character and the player must swing down when it disappears. Pivoting and Balancing aren't done with circles, but while the player is standing still. While Pivoting, the player must spin the Wii Remote around as fast as possible. While Balancing, the player must tilt the Wii Remote so that it stays as close to the center line as possible. Going into the red area for too long results in the balancing being reset, but not the timer for it. Shaking that appears in the third song requires the player to swing left and right. At the end of the song, the player is given a chance to earn bonus points by spinning then posing. If multiple people are playing, it goes to the next person.
The Ribbon Illusionist challenge is completed by earning perfect timing a number of times in this event. The criteria for unlocking each achievement are as follows: