Generalized kissing number in the multiple layers plane

L´aszl´o Fejes T´oth and Alad´ar Heppes propose the generalization of the kissing number problem. Given a ball in Rd, we consider a family of balls touching it, then another family of balls touching the first family, and so on until the nth family (layer). We find the maximal possible number of balls in this arrangement provided that no two balls intersect by interiors, and all balls are congruent. We show that the answer for the plane is asymptotically equalto 2пn2/√3.