Continuity mistake: When Kyle is taking the test, he's doing it with precision and accuracy, but he marks the wrong answers for #31-33. Even though it's meant to be a continuous shot, when he's answering #35, his answers to #31-33 are now changed to the correct answer. Kyle is able to answer the questions right the first time and didn't go back to change his answers, plus there's no sign of the previous marks if they were erased.
Character mistake: When Kyle is re-taking the test at the end of the school day, you see him bubble in 4 answer and turn the page and continue to answer questions. However there were 7 questions on the first page so he ended up putting the answer to #8 on #5 of the scantron. Kyle is too smart to do this.
Continuity mistake: This is actually both continuity and character mistake error. In the scene where Kyle helps Josh with his math test, the first time we look at the test, Josh has stuck in problem number 3, and in the problem above, number 2 has already given the answer "A". The second time we see the test, they jumped in problem number 13, which ironically in the previous shot was half covered with the kid's hand, so obviously he couldn't look at it. Also he has now answered question 12 above as well. Moreover Kyle says the correct answer is X=49. It's a complete random answer that has nothing to do with either question 3 or 13. In question 3, "X" isn't even mentioned and is just a multiple choice between A and E. In problem 13, if we do the math, we have X<4.5. In simple words, X must be smaller than 4.5, but Kyle says 49. Considering that Kyle is a math genius, it's impossible to have given such a stupid answer.
Plot hole: About three and a half minutes into the episode, Kyle looks through the photo album and sees a picture of his fake mother (the woman paid to pretend to be his mother) giving him a dog for Christmas. Two minutes later in a different scene, the paid parents claim in front of everyone that they're both allergic and had no pets. So by now it's obvious they're lying. Another two minutes later in another scene, Kyle's foster parents are looking through that same album and nobody ever notices the inconsistency.
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