= Solution
In molecular <boron trichloride>, <boron> uses three approximately $sp^2$ hybrids to form three equivalent B–Cl <sigma bonds> in a trigonal plane. Its perpendicular empty $2p$ orbital can accept <electron> density from the compatible combination of filled <chlorine> $3p$ <lone pairs>. The resulting <pi donation> is delocalized over the three B–Cl connections.
In the elementary model, one net <pi bond> is distributed over three equivalent <sigma bonds>, giving \b[formal average B–Cl <bond order> $1+1/3=4/3$]. The corresponding <resonance contributors> each put a B=Cl connection in a different position, with formal negative <boron> and positive donating <chlorine>. They do not mean one experimentally distinct double bond and two single bonds. The $4/3$ value is the idealized sigma-plus-delocalized-pi bookkeeping value; a computed bond index need not equal that exact rational number.
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