  This paper tackles the extension problems for three far-unsatble homotopy
groups $\pi_{39}(S^{6})$, $\pi_{40}(S^{7})$, and $\pi_{41}(S^{8})$ localized at
2, the puzzles having remained unsolved for forty-five years. By a Toda bracket
indexed by 1 included in $\pi_{39}(S^{6}_{(2)})$, which makes better use of the
deuspension property of homotopy classes, we address the problems. As a
corollary, through Thomeier's 8-step backward theorem of the metastable
homotopy theory, together with the results of Oda, Mukai and Miyauchi, we show
a table of the 33-stem homotopy groups $\pi_{33+n}(S^{n}_{(2)})$, ($2\leq n\leq
9$, $n\geq27$).
