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We consider a twisted massless multiplet on a two-torus, one side with a normal boundary and the other with a twisted boundary. The Casimir energy is calculated and regularized by means of the Epstein-Hurwitz-type zeta-function introduced by Elizalde. The resulting dimensions of spacetime for the twisted case may be integers.The results are compared with those of the untwisted case. Since twisted Casimir energy is lower than untwisted energy, the untwisted case may change into the twisted state in the spacetime.