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In the opening credits, the S.S. Minnow that is shown before the cruise, and after beaching on the island is different of the one shown during the storm. (No upper deck above the windows where the helm is, and the front of the cabin is straight, rather than pointed.)
In the first season, the Minnow shown in the opening credits is beached, leaning on its left (port) side. In the closing credits, the Minnow is beached perfectly upright.
In the opening credits during the black and white and color episodes, when the Minnow hits the storm its top deck where the helm is located is missing.
The boat is called the SS Minnow, but "SS" stands for "steam ship". The Minnow would be powered by a diesel or gas engine and would be named MV (Motor Vessel) Minnow.
Incorrectly regarded as goofs
The opening lyrics are incorrect. They landed on an uncharted jungle island, not a desert island. Furthermore, it's very obvious that Gilligan was not a "mighty sailing man". However, a "desert island" means a "remote uninhabited tropical island" according to Oxford English Dictionary. The fact that Gilligan is not a "mighty sailing man" can be attributed to comedic irony.
During the closing credits of Season 3, a large building is visible in the upper left hand corner of the screen behind the foliage on the "deserted island". (This may be true of other seasons also.)
In Feed the Kitty, it's established that Mrs Howell is allergic to cats. Yet, several fans count the fact that there's a cat in The Friendly Physician, and she doesn't sneeze near it. However, it should be added she doesn't share any scenes with it, and her brief contact with it is obviously not enough to trigger her allergy.
Errors in geography
Occasionally the radio announcer would announce the call sign of the radio station the castaways were listening to, and it would always start with W. With few exceptions (and even those don't go any further west than Texas), radio stations with call signs that start with W are east of the Mississippi River. The castaways were somewhere in the South Pacific and had only an AM radio, and even in the most perfect conditions, it would be highly unlikely the strongest AM radio station east of the Mississippi could have reached them.
On a number of occasions, the castaways encounter people, whether other castaways or fugitives in hiding, who have been on the island for 10 years or more, prior to the arrival of the Minnow (Wrongway Feldman, Dubov the painter, and the jungle boy). All these characters regard themselves each as being the only person on the island. How big is this island that all these people could exist there so long and never know about each other?
It has often been observed that although they were on "a three hour tour," the castaways have an awful lot of supplies. Mary Ann and Ginger have seemingly not ever worn the same clothes twice, the Professor has an extensive library, the Howells have myriad luxury items, etcetera.
The boat shows as the Minnow was a 1964 Wheeler, which had a top cruising speed of 12 knots. Which means that in three hours, it couldn't have traveled more than about 40 miles.