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There are a total of 3 Mini Games in Sonic X. They primarily involve use of the stylus to solve mathematical problems.
Eggman's Super Sucky Machine
This level involves Eggman's Super Sucky Machine. Numbers must be dragged into the tube of this machine with the stylus and they will be sucked to the top of the screen. There are three different mini-games involving this which are Counting, Number Sequencing, and Operations. Counting involves counting the number of certain objects which appear on-screen. Number Sequencing calls for numbers to be placed in the correct order. Operations has players solve addition and subtraction problems.
Math Robot Round-up
Solve addition problems by circling the correct number of on-screen objects with the stylus. These objects move so it's not as easy as it sounds. After circling the correct number of objects shake the box with the stylus and collect the golden rings.
Solve subscription problems by tapping the on-screen objects with the stylus. The objects that are touched remain on-screen while the others fly away.