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We present the first model independent search for three-jet hadronicresonances within multijet events in $\sqrt{s} = 1.96$ TeV $p\bar{p}$collisions at the Fermilab Tevatron using the CDF II detector. Pair productionof supersymmetric gluinos and squarks with hadronic R-parity violating decaysis employed as an example of a new physics benchmark for this signature.Selection criteria based on the kinematic properties of an ensemble of jetcombinations within each event help to extract signal from copious QCDbackground. Our background estimates include all-hadronic $t\bar{t}$ decaysthat have a signature similar to the signal. No significant excess outside thetop quark mass window is observed in data with an integrated luminosity of 3.2fb$^{-1}$. We place 95% confidence level limits on the production cross section$\sigma(p \bar{p} \rightarrow XX^{'}) \times\rm{BR}$($\widetilde{g}\widetilde{g}$$ \rightarrow 3 {\rm jet}+3 {\rm jet})$where $X, X' = \widetilde{g}, \widetilde{q}$, or $\widetilde{\bar{q}}$, with$\widetilde{q}$, $\widetilde{\bar{q}} \rightarrow \widetilde{g}$ + jet, as afunction of gluino mass, in the range of 77 GeV/$c^2$ to 240 GeV/$c^2$.

First Search for Multijet Resonances in $\sqrt{s} = 1.96$ TeV $ p\bar{p}$ Collisions