When Ms. Cross is grading papers in the library, before Max comes to refill her lemonade you see her grade a paper with a yellow paper under it, and her smiley face stamp to the right of her hand. In the next shot when Max refills her Lemonade there is no yellow paper under the one she is grading, and her smiley face stamp is far off to her left.
The Swiss Army Knife Dirk gives Max says "Rushmore Yankee 1985-1997". Since Max is 15, this would mean he has gone to Rushmore since age three. But we know from his first talk with Dr. Guggenheim that he has only gone to Rushmore since second grade.
In the first close-up shot, the box with the "Perfect Attendance" and "Punctuality Award" pins appear from left to right. But when Max reads out the names of the pins (in the same order as above), he points at them from right to left.
When Max is shown in the Model UN delegation as "Russia", he has an old, red Soviet Union flag on his desk (note the hammer and sickle), although the USSR fell in 1991. A closeup of Max's Swiss Army knife reveals that the story takes place in 1997, by which time the current white, blue, and red Russian flag was adopted.
The supposedly unsolvable math problem appearing in Max' dream sequence is to prove that the area of an ellipse with semimajor axis a, and semiminor axis b equals (pi)ab, which is a routine calculus problem, and not ridiculously hard.
In the final party, Mr. Littlejeans is in the background of two consecutive conversations that ostensibly take place in different parts of the party hall - the conversation between Max's Dad and his Grover Cleveland teacher, and between Ms. Blume and Margaret Yang.
When Max is riding the replacement bicycle to Blume's work to sabotage his car, we see from the side that the rack on the back of the bike has nothing in it; when he gets to the gate and we follow him in, there is now a small pack attached to the rack (from which he pulls his tools in the next scene under the car).
In the elevator scene at the hospital with Max and Herman, Herman lights a second cigarette while still finishing the first. In the shots that follow the old (almost finished) and new cigarette (full) switch locations and then switch back.
When Dirk confronts Herman by standing in front of his car, we can see Herman's son in the front seat with his head in his hand, gesturing impatiently. Then he is suddenly sitting upright with his hand on the side visor.
During the math class, the board is shown. y = f(x) = x^0.5. The question is to compute f(x+ deltax) - f(x). The first statement is accurate. The second statement is obtained by multiplying by 1 in the form of [(x+ deltax)^ 0.5 + x^ 0.5]/[(x+ deltax)^ 0.5 + x^ 0.5]. Thus the denominator of the second statement should be [(x+ deltax)^ 0.5 + x^ 0.5], NOT [(x+ deltax) + x^ 0.5]^0.5 as shown on the board. Interestingly, the third statement on the board has the denominator correct.
When one the Blume twins get into their father's Bentley after school, one of them turns up the air conditioning. The automatic heater controls shown are not the same as those in the Bentley, which has manual controls. The controls are from a Mercedes-Benz C-Class 1994 or 1995 model year.
The movie continually updates what time of year we're in. But in the portion of the year labeled "September", the leaves on the trees are all red and gold and falling off. In "October", however, the trees are completely covered with green leaves.
At one point a reporter is interviewing Max Fischer with a tape recorder. As he's holding it up to Max to speak into, it is very obvious that the tape recorder is not moving, the wheels inside are at a standstill.