In traditional mechanics, most interactions are pair-wise; if we omit one of the particles from our description, then the original pair-wise interaction can sometimes only be represented as interaction between triples, etc. It turns out that, vice versa, every possible interaction between N particles can be represented as pair-wise interaction if we represent each of the original N particles as a triple of new ones (and two new ones are not enough for this representation). The resulting three "particles" actually represent a single directly observable particles and in this sense, cannot be separated. So, this representation gives a fundamental reason for the three-quark model of basic barions, and explains quark confinement. The representation is based on a deep mathematical result (namely, a minor modification of Kolmogorov's solution to Hilbert's 13th problem) which has already been used in foundations of neural networks and in the foundations of physics - to explain why fundamental physical equations are of second order, and why all these fundamental equations naturally lead to non-smooth solutions like singularity.