  The puzzle rules for computing Schubert calculus on $d$-step flag manifolds,
proven in [Knutson Tao 2003] for $1$-step, in [Buch Kresch Purbhoo Tamvakis
2016] for $2$-step, and conjectured in [Coskun Vakil 2009] for $3$-step, lead
to vector configurations (one vector for each puzzle edge label) that we
recognize as the weights of some minuscule representations. The $R$-matrices of
those representations (which, for $2$-step flag manifolds, involve triality of
$D_4$) degenerate to give us puzzle formulae for two previously unsolved
Schubert calculus problems: $K_T(2$-step flag manifolds$)$ and $K(3$-step flag
manifolds$)$. The $K(3$-step flag manifolds$)$ formula, which involves 151 new
puzzle pieces, implies Buch's correction to the first author's 1999 conjecture
for $H^*(3$-step flag manifolds$)$.
