Generalizing Barth and Bauer, denote by N_2n(d) the maximal number of smooth degree d rational curves that can lie on a smooth 2n-polarized K3-surface X⊂Pn. Originally, the question was raised in conjunction with smooth spatial quartics, which are K3-surfaces. The numbers N_2n(1) are well understood, whereas the only known value for d=2 is N_6(2)=285. I will discuss my recent discoveries that support the following conjecture on the conic counts in the remaining interesting degrees. Conjecture. One has N_2(2)=8910, N_4(2)=800, and N_8(2)=176. The approach used does not distinguish (till the very last moment) between reducible and irreducible conics. However, extensive experimental evidence suggests that all conics are irreducible whenever their number is large enough. Conjecture. There exists a bound N∗_2n(2)