Answer to January 17, 2005 Problem |
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| The Nine Students Problem |
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Nine students from three local high schools met at the Apple Blossom Mall. Three students attended Sherando High School, three went to Millbrook, and three were Handley students.
Handley students always tell the truth. Each of the nine students has made three statements, each individual statement being either entirely true or entirely false.
Al made the following statements:
Bonnie made the following statements:
Cal made the following statements:
Dana made the following statements:
Ellie made the following statements:
Fran made the following statements:
Gary made the following statements:
Hank made the following statements:
Isabel made the following statements: Using these statements, can you determine which of these students attend Handley, which attend Sherando, and which attend Millbrook? |
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Solution to the Problem: students at Handley: Cal, Dana, Hank.students at Sherando: Al, Ellie, Gary. students at Millbrook: Bonnie, Fran, Isabel. This puzzle involves following hypothetical paths of truth to find those that hold up. One helpful fact is that all nine students say they're students at Handley, so six of them must be lying, and the other three must be truthful. You can start by figuring out which students can't be students at Handley - that is, can't be telling the whole truth.
One path to the truth is: 2) If Isabel is a student at Handley, then Cal is a student at Sherando (i3). But she lies that Ellie is not a student at Handley (c2). We know Ellie is not a student at Handley (above). So Isabel can't be a student at Handley. 3) Since Isabel can't be a student at Handley, Al can't truthfully say Isabel is (a3). So Al can't be a student at Handley. 4) If Fran is a student at Handley, then so is Gary (f2) and so is Hank (g3). Hank truthfully says that Cal is not a student at Sherando (h2), so since all at the Handley slots are full, Cal must be a student at Millbrook. Bonnie's right that Cal isn't a student at Sherando (b2), so Bonnie is a student at Millbrook. Dana's right that Bonnie is a student at Millbrook (d2), so Dana is too. Ellie's right that Dana isn't a student at Handley (e2), so Ellie is a student at Millbrook. But that's four students at Millbrook, and that's impossible. So Fran can't be a student at Handley. 5) If Gary is a student at Handley, then so is Hank (g3). Gary says that Fran isn't a student at Millbrook (g2), and Hank says Fran isn't a student at Sherando (h2), so since they're both truth-tellers, Fran must also be a student at Handley. But we know from paragraph 4 above that she isn't. So Gary can't be a student at Handley. 6) All four remaining students (Bonnie, Cal, Dana, and Hank) seem to be telling the truth, except that Dana says Bonnie is a student at Millbrook (d2). One at them must be lying, meaning that Cal and Hank must be students at Handley. Hank says Fran isn't a student at Sherando (h2), and we know Fran isn't a student at Handley, so she goes to Millbrook. Since by paragraph 5 above she is lying about Gary being a student at Handley, she must be telling the truth about something, namely that Bonnie is not a student at Handley. Now we know that Cal, Dana, and Hank are students at Handley, and Fran is a student at Millbrook. Dana confirms Gary is a student at Sherando (d3). Isabel's first and third statements are false, and her second is true, so she is a student at Millbrook. All three of both Ellie's and Al's statements have now been proven false, so they are students at Sherando.
In summary: |
| 1. David & Judy Dixon | Bennettsville, South Carolina |
| 2. Cameron Burkholder | Winchester, Virginia |
| 3. Emily Auerbach | Columbus, Georgia |
| 4. Tristan Collins | Winchester, Virginia |
| 5. Jeffrey Gaither | Winchester, Virginia |
| 6. Chris Rogers | Winchester, Virginia |