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In this paper, we give elementary proofs of Zagier's formula for multiplezeta values involving Hoffman element and its odd variant due to Murakami.Zagier's formula was a key ingredient in the proof of Hoffman's conjecture.Moreover, using the same approach, we prove Murakami's formula for multiple$t$-values. This formula is essential in proving a Brown type result whichasserts that each multiple zeta value is a $\mathbb{Q}$-linear combination ofmultiple $t$-values of the same weight involving $2$'s and $3$'s.

Elementary proofs of Zagier's formula for multiple zeta values and its odd variant