Examples of admissible simplification of mathematical theories

cation of mathematical models, questionnaire " Mathematicians, like physicists, are pushed by a strong fascination. Research in mathematics is hard, it is intellectually painful even if it is rewarding , and you wouldnt do it without some strong urge. " [D. Ruelle]. We shall give some examples from our experience, when we were able to simplify some serious mathematical models to make them understandable by children, preserving both aesthetic and intellectual value. The latter is in particularly measured by whether a given simplification allows setting a sufficient list of problems feasible for school students. For a more evident demonstration of our method we chose primary school students (6-9 year) as a target group. We give examples of Turing machine, Cellular automata, Minesweeper, Graph theory. Nowadays, it is thought of importance to introduce current scientific achievements to children by telling them of the black holes or demonstrating some impressive chemical experiments. Can this approach however satisfy us? A. Zvonkin in his acclaimed work [J.Math. Behaviour, 1992] on early child development translates Poincares theory on the role of subconscious into a practical recommendation: questions are more important than answers. He depicted an experiment that targeted to find out whether it is using attractive materials that engaged children in his lessons or the lessons per se. Then I say, All right, I have to finish the lesson, but you may play with mosaic. My words are met with an unanimous yell of indignation, No-o, we want a problem!. Thats how I understood what the truth was. Children