Math is typically involved in designing video games when a video game has a physics engine or deals with statistics and probability. Math is also involved in a game's visual design and graphics.
Some first-person shooter or racing games, have complex and realistic physics known as the game's engine. In modern games, the math must be complex to give a game an authentic feel. Determining the path of a bullet or the movement of a car over a race track is a purely mathematical exercise. Mapping the mechanics of real-world physics to the game world requires mathematical knowledge and calculation.
Math is also a critical component of video game graphics. A game world and the things inside it are composed of polygons, and designing ways for these polygons to interact with each other is a complex job requiring lots of math. Geometry is particularly important for this reason as knowledge of planes, vectors and angles is essential.
One of the most complex ways in which mathematics is involved with video game design is in probability, as many games are heavily based on chance. Some have gambling type features and need to have complex statistical underpinnings to make them compelling and enjoyable to play. Computer programming is a mathematical enterprise and programming video games increases the need for the creator to have a strong mathematical basis.