Date: 10/28/2002 at 11:22:18 From: Sam Subject: Math problem of the week A history test has three questions on Presidents of the United States. Here are the answers six students give. 1. Polk, Polk, Taylor 2. Taylor, Taylor, Polk 3. Filmore, Filmore, Polk 4. Taylor, Polk, Filmore 5. Filmore, Taylor, Taylor 6. Taylor, Filmore, Filmore Every student has at least one correct answer. What are the answers?
Date: 10/28/2002 at 16:16:21 From: Doctor Ian Subject: Re: Math problem of the week Hi Sam, Interesting question! Let's start by lining things up (and abbreviating the names), so if there are any patterns, they'll be easier to find: P P T T T P F F P T P F F T T T F F Now, let's look at all the possible answer keys: F -> FF -> FFF FFP FFT FP -> FPF FPP FPT FT -> FTF FTP FTT P -> PF PP etc. PT T -> TF TP etc. TT (All I've done here is start with the possible choices for the first answer - F, P, or T - and then extend each of those with the possible choices for second and third answers. I've left the final 2/3 for you to expand.) Now let's check the answer key FFF against the students' answers: F F F Correct - - - ------- P P T 0 T T P 0 F F P 2 T P F 1 F T T 1 T F F 2 So this clearly can't be the key, because the first two students didn't get any right. How about FFP? F F P Correct - - - ------- P P T 0 T T P 1 F F P 3 T P F 0 F T T 1 T F F 1 Nope, it's not that one either. If you generate all the rest of the keys and test them against the answers this way, you should find only one key for which every student gets at least one answer right. Or you could try to be clever. For example, by making T the first answer, you make sure that at least half of the students get one right: T Correct - - - ------- P P T T T P 1 F F P T P F 1 F T T T F F 1 Let's rearrange the students so we can concentrate on the ones who haven't got any right yet: T Correct - - - ------- T T P 1 T P F 1 T F F 1 P P T F F P F T T By making the final answer T, we can get two more students. And then it's just a matter of getting the remaining student to be right on the second question. Now, this can lead us to _an_ answer, but is it the _only_ answer? The first method I outlined is tedious, but if you check every possible answer key, you'll find every possible set of answers that will satisfy the problem. The second method is much easier, but we don't have any idea whether the solution it finds is unique. That's one of the trade-offs that you often have to make make when you decide to be clever. I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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