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Given the following problem:

Alice, Bob and Carl stand on a straight line.
Alice is one rod away from Bob.
Dana stands one rod away from both Bob and Carl.
Carl is as far from Alice as Alice is from Dana.

How far can Alice be from Carl?

Answer using a well-formed sentence, with any distance expressed (in rod) with three significant digits (calculators allowed).

Question: At what education level could we expect that more than one in ten students give a mathematically correct answer, in exam conditions?

Note: I authored this problem, and know its solution.

Update: Spoiler with reference answer to the original problem follows; hover mouse to see it.

Alice is either 1.62… or 0.618… rod away from Carl.

Update: Some have trouble with the problem's wording (which is an important aspect of the question), so here is a restatement in geometry terms; hover mouse to see it.

$A$, $B$, $C$, $D$ are distinct points, with $A$, $B$, $C$ collinear,
such that $|AB|=|BD|=|CD|=1$ and $|AC|=|AD|$.

Give the set of possible $|AC|$ rounded to 3 significant digits.

Update: link to spoiler illustration; ctrl-click for new window or tab.

fgrieu
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  • Spare the rod and spoil the math. – vadim123 May 03 '13 at 14:09
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    Call me cynical, but the "well-formed sentence" requirement alone might establish graduate school as a lower bound :). – Erick Wong May 03 '13 at 14:24
  • Stands Dana on the straight line? If it is,the problem should be simple during the middle school. I can't give hints! Sorry –  May 03 '13 at 14:17
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    Is the question about the maximum distance between A and C? or all possibilities? – long tom May 03 '13 at 14:48
  • @long tom: my intend with can and any was to subtly imply "all possibilities"; I realize now that this is ambiguous (full disclosure: I'm French). – fgrieu May 03 '13 at 15:07
  • @vadim123: Change rod to decameter, or similar unit. Yard could make Alice embarrassingly close to Carl. Or simply use "unit", but I purposely make the problem concrete; that seems to make it harder. Oh I just gave a hint... – fgrieu May 03 '13 at 15:08
  • @Greg ba: it is not implied that Dana is on the line, and can be disproved (assuming two persons can't be at the same place). – fgrieu May 04 '13 at 08:17

2 Answers2

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To be able to solve the problem I think a student would need to know

  • basic algebra
  • some trigonometry
  • some geometry, including the concept of circles, triangles, and equidistance

At what education level...

Education level is a strange concept, but I think a bright student in an advanced secondary mathematics class should be able to solve this. It's somewhat tricky but it's not exceedingly difficult.

It's definitely well beyond the primary / elementary school student, and below the university / college level mathematics student. Probably above the level of the general high school mathematics student but within grasp of the advanced high school student.

Just my opinion, of course.

Kirk Broadhurst
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I might be being silly but it appears to me that the question is not written very well and it is mathematically inaccurate.

My main issue is that Dana might not be able to stand anywhere. How do we know that there is a place that is one rod away from both Bob and Carl? They might be at a distance of more than two rods away from each other, in which case there is no possible place for Dana to stand at all!

fretty
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    This type of question forces us to infer. We know there is a place that is one rod away from Bob and Carl because Dana is standing there! This also tells us that Bob and Carl are not at a distance of more than two rods away from each other. – Kirk Broadhurst May 03 '13 at 20:59
  • Ok fair enough, however it is too easy to imagine there being a problem...the question should be written clearer. – fretty May 03 '13 at 21:04
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    I think it's quite well written. Just like any question, there might be zero, one or many answers. Part of being able to solve the question is being able to identify or prove the existence of solutions... – Kirk Broadhurst May 03 '13 at 21:09
  • Sorry to be pedantic but "Dana stands" is ambiguous...if it said "Dana is stood" then it would mean that there must be a place for her to stand but the use of "stands" could mean either "is stood" or "is in the action of going to stand after A,B,C were already stood". In the second case it might be impossible for this action to be carried out. – fretty May 03 '13 at 21:27
  • ...but yeah assuming there is a place for her to stand then the question works. – fretty May 03 '13 at 21:31
  • That is pedantic! 'Dana stood' is talking about where Dana was yesterday. Again, if the action is impossible to be carried out then the student can explain the impossibility and get full marks. – Kirk Broadhurst May 03 '13 at 21:31
  • "Dana is stood" describes the present... – fretty May 03 '13 at 22:04
  • Anyway I accept the question...just it is easy to misinterpret. When I first read it I wondered how it was always possible for Dana to be positioned so. – fretty May 03 '13 at 22:06
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    In the problem's wording I purposely use present, and stand once for each person to further repel any notion of movement. Some readers, thinking (wrongly) that there is no static solution, introduce movement nevertheless. That's an aspect of the question. – fgrieu May 04 '13 at 06:55
  • But either way students may think in the same way... – fretty May 04 '13 at 14:21