Batman vs. the Joker #3 (D2969)
In "The Challenge of the Joker," Batman and the Joker find themselves caught in a bizarre twist when a chance encounter under mysterious epsilon rays swaps their personalities—leaving the Dark Knight with the Joker’s chaotic mind and the Clown Prince of Crime trapped in Batman’s body. Written by David Vern and illustrated by Sheldon Moldoff, with inks by Charles Paris and letters by Pat Gordon, this 1966 classic features a cover by Dick Sprang that captures the surreal tension of the moment.
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During a tussle with the Joker, the pair step under and are bathed in "epsilon rays", which have the effect of transfering the personality on one person to that of the other.
Plot details indexed by the Grand Comics Database (CC BY-SA).