Some individuals are cross-dominant or have no dominant hemisphere. Gestalt thinkers do well at games or activities which require pattern analysis or pattern matching. The oriental board game, Go, is an example of a game where the strategies are best developed using a whole to parts method. When I play Go, I picture the board as I want it to be and then work towards that goal. This type of thinker is more likely to try and solve problems by using parallel solutions or groups of steps.
Linear thinkers do well at games or activities which can be broken down into sequential steps.
Chess is an example of a game where the strategies are commonly broken down into linear sequences. It isn't hard to find activities which are organized in a linear sequential fashion since the vast majority of people think this way. Recipes and navigating from a set of written directions are two common examples of linear sequential activities. Tony Buzan in his book, Using Both Sides of Your Brain, presents ideas for incorporating both linear and gestalt thinking strategies in your learning and teaching. Concepts are written in bubbles or on lines which connect the idea to related ideas or concepts.