The foundation of equivariant stable homotopy theory is laid by Lewis-May-Steinberger in the 80's, while people's understanding of the computational aspect of the subject is very limited even until today. The reason is that the equivariant homotopy groups are $RO(G)$-graded, and even the coefficient rings of Eilenberg-Maclane spectra involve complicated combinatorics of cell structures. In this talk I'll illustrate the advantages of Tate squares in doing $RO(G)$-graded computations. Several Eilenberg-Maclane spectra of particular interest will be discussed: the Eilenberg-Maclane spectra associated with the constant Mackey functors $\mathbb{Z}$, $\mathbb{F}_2$, and the Burnside ring. Time permitting, I'll also talk about some structures of the homotopy of $HM$, for $M$ a general $C_{2^n}$-Mackey functor.