After being spray painted by the kids and driving through the haystack, Arno's face is red, but his hair isn't. Later, in the studio his hair does have some red paint in it.
When Pete and Annie are reviewing the "Applejack" sequence on their reel to reel tape recorder, the reels change speed between the close-ups and medium shots.
In the chase scene, Dexter's friends are seen pouring paint on the road. As the car that A. J. Arno is in drives through the paint, it splatters the side of the car. In the next seen, the paint is gone. All subsequent parts of the chase scene have the same problem.
As Dexter is driving back from Marshfield after picking up the computer part, he is trying to figure out answers to a sample test. There is obviously a window on the driver's side -- there is a reflection in it, and he has also taped several pieces of paper to it. When he arrives at Medfield, he climbs out the side of his dune buggy, indicating that there is no window.
When the doctor, Dean Higgins and Prof. Quigley are looking into Dexter's eyes during the medical examination, they are using an otoscope (a device used to look into ears). The correct instrument should be an ophthalmoscope.
The very last question in the $100,000 knowledge contest asked the name of the Midwest city that is in the area known as the "geographic center of the U.S." Medfield answered Lebanon, KS and won the contest. However, that answer is incorrect because of the way the question was phrased. Lebanon, KS is the geographic center of the 48 states, not the U.S. The official markers there are worded that way. The center of the U.S. is a site in South Dakota about 20 miles north of Belle Fourche. There is a USGS benchmark on the spot which moved from Lebanon, KS in 1959 after Alaska and Hawaii became states, 10 years before this movie came out.
Dexter answers the written test answers incorrectly.
Although the character is known as Dexter "Riley," all onscreen spellings of his name in this film are "Reilly".
When Dexter gets back to the school with the computer part, he gets out of his dune buggy and you can hear a car door slam. The dune buggy does not have car doors.
When asked to name the highest point in South America, Dexter says, "Aconagua" and gets the answer right. In reality the mountain is called, "Aconcagua" (note the added letter "c").
When filling out the multiple choice classification test at breakneck speed with his recently acquired computer smarts, in the closeup the protagonist selects mostly wrong choices, and makes incorrect markings, crossing out two adjacent choices or missing the mark altogether.