Deliberate mistake: The Batmobile's Bat-turn lever is located at the center upper console, between the driver and passenger seats. Whenever the Bat-turn lever is used while the Dynamic Duo are in the Batmobile, the same closeup always faces the passenger seat and shows Batman's gloved hand pushing the Bat-turn lever, but the problem is Robin vanishes in this closeup - the passenger seat is empty. Episodes S1xE3 "Fine Feathered Finks", S1xE29 "The Bookworm Turns", and S2xE38 "The Joker's Hard Times" are three examples of the Bat-turn lever being used.
The Penguin Declines (3) - S2-E39
Visible crew/equipment: When Venus is at the Batcave, loads of fisticuffs ensue between the Caped Crusaders, Joker, Penguin and all the baddies. While Robin is up on the atomic pile scaffolding he punches one goon off the ladder, then as he falls we can see the edge of the brown crash mat rise up, right before the word "SPLATT!" appears onscreen. (00:20:50)
Batman Stands Pat (2) - S1-E14
Visible crew/equipment: When the Dynamic Duo trace Turkey's bowler to Mad Hatter's lair, they proceed to climb the wall. Then Hatter tells his goons to get into place for the trap, and in the next overhead shot there's a white chalk T-mark on the floor near Mad Hatter, at the right side of the screen. (00:17:45)
The Joker's Hard Times (2) - S2-E38
Continuity mistake: After Joker enters Commissioner Gordon's office, Gordon and Chief O'Hara walk over to the window, and the Batphone table is behind them, beside Gordon's desk. Then Joker makes two "quick-frozen cops" and runs off to snatch Justice, leaving Gordon and O'Hara immobile. When their shock paralysis wears off, Gordon and O'Hara rush over to Gordon's desk, but the Batphone table and the Batphone have vanished (note the rug's pattern). (00:03:00 - 00:04:55)
The Joker's Hard Times (2) - S2-E38
Visible crew/equipment: After Joker and his henchmen steal the statue, it cuts to Commissioner Gordon and Chief O'Hara as they recover from their shock paralysis, and when they walk over to Gordon's desk, the wheel of filming equipment is visible at the right side of the screen. (00:04:55)
When the Rat's Away the Mice Will Play (2) - S1-E12
Visible crew/equipment: While Robin is "fiddling with the atomic pile" up on the scaffolding in the Batcave, Commissioner Gordon calls Batman to ask if he'd been to Gordon's office earlier, and in the overhead shot of Gordon's office there are two white tape marks visible under the desk, with one at Gordon's feet. (00:17:45)
Penguin Is a Girl's Best Friend (1) - S2-E42
Visible crew/equipment: While Batman and Robin are at the studio, Harriet and the others from the Gotham City League of Film Decency confront Penguin about scene 12. When Harriet says, "This scene is bursting with other things as well," there's a shot facing Marsha with two other women, and we can see a curved chalk mark at Marsha's feet and a chalk T-mark at the other woman's feet. (00:13:20)
Penguin Is a Girl's Best Friend (1) - S2-E42
Visible crew/equipment: When Harriet, Alfred, and the others arrive at the studio while Batman and Robin are on set, the actor's curved white chalk mark is visible on the floor where Alfred stands. (00:12:10)
Minerva, Mayhem and Millionaires - S3-E26
Visible crew/equipment: At the start, in the two consecutive scenes with millionaires William Dozier and then Howie Horwitz, while they're being treated to Minerva's renowned scalp massage, the chalk T-mark is visible on the floor at the lower left side of the screen, and in the Horwitz scene Minerva's small mark is also visible on the floor behind the spa chair. (00:00:45)
Minerva, Mayhem and Millionaires - S3-E26
Visible crew/equipment: Minerva tells Freddie that if she'll get the diamonds from the Wayne Foundation vault, he can fence them. When it cuts to Commissioner Gordon, Batgirl, and Chief O'Hara pacing the floor, three small white tape marks are visible on the rug in front of the coffee table, where each of them stop and turn on their marks. (00:05:35)
The Entrancing Dr. Cassandra - S3-E25
Visible crew/equipment: After Batman discovers Cassandra's true identity in the alchemy book, he takes a call from Commissioner Gordon on his cute portable Batphone, then he and Robin head to Spiffany's. When it cuts to Spiffany's, just as narrator William Dozier says, "Gotham City's fashionable 15th Ave," the actor's white marks are visible on the floor under the table. Then we see Cassandra's and Cabala's white T-marks when they walk in. Next, upon the Terrific Trio's entrances, we see their T-marks as well. (00:08:25)
The Entrancing Dr. Cassandra - S3-E25
Visible crew/equipment: After Alfred has placed the two-dimensional Terrific Trio into the Three-Dimensional Bat-Restorer, when they all walk out of the chamber, two chalk T-marks are visible on the floor. First we see Batgirl's T-mark and then Robin's. (00:16:50)
I'll Be a Mummy's Uncle - S3-E23
Visible crew/equipment: When King Tut runs back up the shaft and encounters Batgirl, Commissioner Gordon, and Chief O'Hara, he is reveling giddily in the truth he has discovered. Then, when Batman and Robin show up, their chalk T-marks are visible on the ground. (00:21:10)
The Great Train Robbery (2) - S3-E22
Visible crew/equipment: When Shame has finished reading Batman's skywriting message, as he says, "Darned if I won't be there," the shadow of the boom mic dips into view at the top of the green door, at the right side of the screen. (00:17:20)
Nora Clavicle and the Ladies' Crime Club - S3-E19
Visible crew/equipment: When Batman and Robin are seated in the Batmobile with Batgirl standing alongside, while they listen to the ridiculous police dispatcher, the reflection of the moving boom mic is visible on the front windshield's chrome trim. (00:10:00)
Nora Clavicle and the Ladies' Crime Club - S3-E19
Revealing mistake: When Robin calls Chief Linseed to get her to mobilize the entire force, Linseed and the others are useless, and in the two closeups of the mice scurrying on the rug in front of the desk, a few stunt wires are visible. (00:18:50)
Nora Clavicle and the Ladies' Crime Club - S3-E19
Revealing mistake: When the crime fighting Trio are playing the tune on their toy flutes and reach the docks, in the wide shot the stunt apparatus (a bar) is visible, and in the closeups of the mice procession their wires are visible as well. (00:20:50)
Louie's Lethal Lilac Time - S3-E18
Visible crew/equipment: Commissioner Gordon and Chief O'Hara think Batgirl has joined Batman and Robin inside Louie's hideout, so they decide to stay put where they are. In the next shot, when two of Louie's goons report that the Batmobile is outside, the white tape mark is visible on the floor by their feet. (00:06:15)
The Joke's on Catwoman (2) - S3-E17
Visible crew/equipment: At the start, when the crime fighting trio split up Batgirl heads to her motorcycle, and when she attempts to start it two chalk T-marks are visible on the ground, one under the motorcycle in front of the rear tire and another beside the front tire. (00:00:35)
Visible crew/equipment: When Olga shouts, "Da, da, da! Drowning her in caviar!" in the overhead shot, the boom mic repeatedly dips into view over Olga, at the top of the screen. (00:21:55)
Answer: It's wasn't unusual for multiple actors to play one character on this show. The villains on the TV Batman were played by guest stars, not regular cast members. George Sanders, Otto Preminger, and Eli Wallach played the part at various times. They may simply have only wanted to play the part once or twice, or they were later tied up with other projects, making it necessary to cast someone else. Other characters, like Catwoman, were also played by more than one actress.
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