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In this paper, we study (1,2) and (2,1) random walks in varying environmentson the lattice of positive half line. We assume that the transitionprobabilities at site $n$ are asymptotically constants as $n\rightarrow\infty.$For (1,2) random walk, we get some elaborate asymptotic behaviours of variousescape probabilities and hitting probabilities of the walk. Such observationsand some delicate analysis of continued fractions and the product ofnonnegative matrices enable us to give criteria for finiteness of the number ofcutpoints of both (1,2) and (2,1) random walks, which generalize E. Cs\'aki, A.F\"oldes and P. R\'ev\'esz [J. Theor. Probab. 23: 624-638 (2010)] and H.-M.Wang [Markov Processes Relat. Fields 25: 125-148 (2019)]. For near-recurrentrandom walks, whenever there are infinitely many cutpoints, we also study theasymptotics of the number of cutpoints in $[0,n].$

Cutpoints of (1,2) and (2,1) random walks on the lattice of positive half line