The purposes of this research are to develop algorithms for contradiction-based problem solving and to prove the educational effects of these algorithms. The Butterfly model includes the algorithms that solve problems by defining specific ideal solutions with given trade-offs and contradiction relationships of the problems. The Butterfly model is a contradiction resolution model that can satisfy the both-and way of thinking. This research proposes the methods that consider (1) time, (2) combination, (3) the whole and its parts, (4) version,(5) extraction, (6) dimension change as the both-and fulfilling methods. In order to verify the educational effects of the proposed model, cannonball, needle, and compass problems as well as Duncker's radiation problem are given to middle school students and university students. Compared to the previous research that showed about 10% problem solving ratio for Duncker's radiation problem, this research had 33% problem solving ratio when the students solved the same problem without any hint. The problem solving ratio of the students who had learned the proposed model one year ago maintained their problem solving ability without decrease. According to the results of regression analysis, studying the Butterfly model increases the contradictory problem solving ability. With the outcomes from the Butterfly model learning,32 students won prizes in invention festivals and 17 students got 13 patents, 2 students won Korea Outstanding Individual Prize. The registered patent ratio achieved by the Butterfly Model learners, 100%, is higher than 72% (reported in 2011. 12), the average registered patent ratio of top 10 organizations. Finall