= Solution
\b[The ions are acetylide $\mathrm{C_2^{2-}}$ and linear tricarbide $\mathrm{C_3^{4-}}$.] Charge balance gives $\mathrm{Mg^{2+}[C_2]^{2-}}$ and $\mathrm{(Mg^{2+})_2[C_3]^{4-}}$.
The acetylide has ten valence electrons, two more than neutral $\mathrm C_2$. They fill the p-derived sigma bonding orbital, giving the formal configuration $\pi_u^4\sigma_g(2p)^2$ above the cancelling $2s$ pair. Its bond order is three, with the <Lewis structure> $\mathrm{^-C\equiv C^-}$ and a lone pair at each end; each carbon is approximately $sp$ hybridized.
The tricarbide has $3(4)+4=16$ valence electrons and is isoelectronic with <carbon dioxide>. The cumulene contributor $\mathrm{^{2-}C=C=C^{2-}}$ has a central $sp$ carbon, two sigma bonds and two perpendicular pi bonds. The end carbons each have two lone pairs and can be described with an $sp^2$ localized basis. Its two C–C bonds have formal order two in this model; the central C–C–C angle is $180^\circ$. Other resonance contributors help describe its protonation chemistry.
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