Movie Mistakes- or are they?

Note the following two movies mistakes. Or are they mistakes?

1. In The Wizard of Oz when the Scarecrow gets his diploma (instead of a brain) he says the following:

   The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.

   This is incorrect. I suspect my readers spotted this mistake while watching the movie. If you type

   "Wizard of Oz" isosceles

   into google you get over 2500 hits-- far less than I would have thought. (Though if you replace isosceles with pythagorean you get over 8000 hits.) So this error is somewhat known. But is it an error? Possibilities:
   1. The writers and everyone who proofread the script did not catch this. Realize that this is not hard math. Wouldn't someone have noticed it?
   2. One of the points of the movie is that the Scarecrow is already smart. Hence the writers are trying to show that he was smart (though didn't know math) both before and after getting the diploma, so the diploma didn't change anything except his confidence. And when it comes to math, a misplaced confidence.
   3. Recall that the movie is all Dorothy's dream. The writers were making the point that Dorothy didn't know the proper way to state the theorem.
2. In Miracle on 34th street (1947 version) Kris Kringle, who claims to be Santa Claus, states that he has passed psychological tests, and brags that he knows that

   John Quincy Adams' vice president was Daniel Tompkins.

   But this is incorrect! John Quincy Adams's VP was John Calhoun! (Tompkins was James Monroe's VP.) How well known is this error? If you type

   "Miracle on 34th street" Tompkins

   into google you get over 1000 hits (far more than I would have thought). Is this really an error?
   1. Daniel Tompkins was our 6th VP. John Quincy Adams was our 6th Prez. They just didn't line up (see chart at the end of the answers to my prez quiz). Hence it was an honest mistake on the part of the writers (I find this far more believable than the Scarecrow-math error not being detected.)
   2. The writers were trying to tell us that Kris Kringle wasn't Santa Claus. Or at least put some doubt in our minds. Of course, that would only work if the audience knew their vice presidents. (Easy now with the excellent book Bland Ambition: From Adams to Quayle-- The Cranks, Criminals, Tax Cheats, and Golfers who made it to Vice President but harder in 1947 when the movie was made.
   3. There have been two remakes of the movie (that I know of) but I don't know if they contain the error. If you know, let me know.

The Scarecrow's math being wrong was intentional

In 2009 I had a post about Movie mistakes (see here). One of them was the Scarecrow in The Wizard of Oz after he got a Diploma (AH- but not a brain) he said

The sum of the square roots of any two sides of an isoscles triangle is equal to the square root of the remaining side. Oh joy! Rapture! I have a brain!

I wrote that either this mistake was (1) a mistake, (2) on purpose and shows the Scarecrow really didn't gain any intelligence (or actually he was always smart, just not in math), or (3) It was Dorothy's dream so it Dorothy was not good at math.

Some of the comments claimed it was (2).  One of the comments said it was on the audio commentary.

We now have further proof AND a longer story: In the book Hollywood Science: The Next Generation, From Spaceships to Microchips (see here) they discuss the issue (page 90). The point to our blog as having discussed it (the first book not written by Lance or Lipton-Regan to mention our blog?) and then give evidence that YES it was intentional.

They got a hold of the original script. The Scarecrow originally had a longer even more incoherent speech that was so over the top that of course it was intentional. Here it is:

The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side: H-2-O Plus H-2-S-O-4 equals H-2-S-O-3 using pi-r-squared as a common denominator Oh rapture! What a brain!

While I am sure the point was that the Scarecrow was no smarter, I'm amused at the thought of Dorothy not knowing math or chemistry and jumbling them up in her dream.