Tangram is one of the oldest puzzle games in China. It can make up volatile and wonderful patterns. To master the game, it would be ideal if the players can go through several complexity levels from the easiest to the most complicated. However, tangram today does not have the difficulty grade. Only a similar puzzle game—Japanese jigsaw puzzle has the difficulty grade which is not very reasonable. And the grading process is not provided. Therefore, we try to provide the theoretical basis of dividing the difficulty grades by means of probability of the patterns. We hope we can make some complements to this ancient game. Among the complicated influential factors, we sum up three key elements: the number of answers，the length of side, the basic figures. Based on these points, we have devised three plans, the procedural thinking of computer, the length-of-side principle, division principle, to calculate the probability. Then, we made comprehensive analysis on the probability. Finally, we got the difficulty grade of some classical tangram patterns. Moreover, we summarize some skills of piecing together the patterns.

We present the first computational tool to help ordinary users create transforming pop-up books. In each transforming pop-up, when the user pulls a tab, an initial flat 2D pattern, i.e. a 2D shape with a superimposed picture, such as an airplane, turns into a new 2D pattern, such as a robot, standing up from the page. Given the two 2D patterns, our approach automatically computes a 3D pop-up mechanism that transforms one pattern into the other; it also outputs a design blueprint, allowing the user to easily make the final model. We also present a theoretical analysis of basic transformation mechanisms; combining these basic mechanisms allows more flexibility of final designs. Using our approach, inexperienced users can create models in a short time; previously, even experienced artists often took weeks to manually create them. We demonstrate our method on a variety of real world examples.

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