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If there is only one correct answer, I could use radio buttons. If there are multiple correct answers, checkboxes. However, since it's a quiz and not an information gathering form or survey, I could use checkboxes all the time to make it more difficult. Though this is could be a bit deceiving to the user, what say you?

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+1 Great question! – Vitaly Mijiritsky Mar 25 '11 at 20:13

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I think it is only fair to indicate wether you allow multiple answers. You could either state this at the top ("Questions may have multiple correct answers, check all that apply" and then use checkboxes throughout) or state it per question. I would however mention it somewhere, because otherwise you are not making it more difficult, but mostly more random (because interpretations of the UI are more decisive than knowledge, and that makes your quiz less useful. So I wouldn't do that.)

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+1. I think it's fair to give them exactly the same affordances and hints that they would get from a quiz given on paper. On paper they have the physical ability to check multiple answers, and if the question allows it, then it's stated in the question body. I would go with checkboxes throughout. – Vitaly Mijiritsky Mar 25 '11 at 20:12
I wouldn't go with checkboxes if the question only allows one answer. This is also about meeting expectations. Checkboxes for multiple answers and radio buttons for single answers is a standard convention that users expect most of the time. I'm not sure what benefit you derive from using checkboxes when only one answer is allowed. – jameswanless Mar 29 '11 at 18:25
Well, you could have a question of "which technologies were available in 1800, list all that apply: book press, internet, nucleair fission". (Yeah yeah, silly, I know.) The point is: you could have questions where multiple answers are allowed and yet only one is correct. (You could also have the opposite: multiple answers are correct but you have to pick the best match. Then using checkboxes and no indication is misleading.) – Inca Mar 30 '11 at 8:50
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As Inca states, radio buttons and checkboxes attend to different type of answers. Radio buttons will allow for only one choice such as Choose your favorite food in contrast to the checkbox approach which would most likely serve for Choose your favorite foods. Note how the second question implies multiple answers.

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You are quizzing their knowledge, not their interpretation of a web form. Use radios.

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