Victor and others,

I have always understood trick questions to be questions that either involve
a special method that the solver is expected to discover for himself/herself
or that the solver is likely to assume that what appears to be the way to
solve the problem is a much harder or longer or messier approach than
necessary.  As Victor had said, there is, according to the solver, a 
surprisingly easy or elegant solution.  One possible example could be this
one:

Suppose an urn contains 15 red balls, 25 green ones, and 12 blue ones.
I draw 8 balls without replacement.  What is the probability I draw at
least one red ball?  

Students who haven't seen a similar example, especially one with the
complement rule for probabilities, might try to find the probability
of drawing one red ball, then two, then three, etc. and then adding 
them.

The proof of the binomial theorem using a counting argument is a
surprisingly elegant way to prove the theorem, much more elegant
than using induction.  Many students who haven't seen that solution
before will probably be surprised, even if they have learned some
permutations and combinations before.    

Or a trick question can be one in which the approach that the solver is 
likely to use involves a subtle error in logic or is not obvious up front 
that the approach will not work.  A good number of Calculus I students will
probably find the question "What is the derivative of x^x?" a trick
question, especially if you ask them when they hadn't seen it before,
because many of them will believe that they can use the power rule for
derivatives on this one.    

Trick questions can also be ones in which most people misread or misinterpret
the information.  For instance, "I have two coins in my pocket worth 26 cents
altogether.  One of them is not a quarter.  What coins do I have?"  Many people
misinterpret this question to mean that the person does not have any quarters,
but this actually means that both coins cannot be quarters.   

Some trick questions I have seen are "proofs" of 1=0 that involve a subtle
error in logic.  Another one I have seen is the "proof" by induction that
all horses are the same color.  The error is subtle, at least according
to many students.      

Here's another such example of one I have seen:

Theorem: We are both the same age!  

Proof:
Suppose x is your age, and y is mine.  
Let our average age be M = (x+y)/2.
Then x+y=2M.
Thus, (x+y)(x-y) = 2M(x-y), or x^2-2Mx=y^2-2My.
Thus, x^2-2Mx+M^2 = y^2-2My+M^2, which says that
(x-M)^2 = (y-M)^2.  
Take square roots to get x-M=y-M.  
Therefore, x=y.  So we are the same age!  QED

This one comes from "Guidelines for Teaching Mathematics," 2nd edition
by Johnson and Rising.

Other such trick questions that fit this description are "why?" questions
that ask why something is not true when it appears that the statement
ought to be true, often because many students hold a misconception that
causes them to believe that this statement ought to be true.  An example:
Why is sqrt(x^6) not equal to x^3 for all real numbers x?  How would you
correct this statement so that it is true for all real numbers x?  

I'm sure many algebra students would find this one puzzling because 
textbooks and teachers often assume that x is non-negative and then
quit stating that assumption and never deal with this lack of assumption
from that point on, and then students develop the habit of simplifying
sqrt(x^6) as x^3 in all cases without comment; many of them then end up
believing that we are supposed to do this all the time.


If I do think of any other examples and can remember to do so, I'll
post them.  



Jonathan Groves
      


On 9/10/2010 at 2:23 pm, Victor Steinbok wrote:

I think, the problem is more likely with the definition. It's only a "trick" the first time you see it. Once you've seen it a few times, it's a method or heuristic (depending on the actual nature of the trick). Furthermore, I am not sure this is what I've ever seen identified as a "trick question". "Trick questions" that I am used to usually involve an unexpected answer or a surprisingly easy path to the answer. The "trick" is in tricking, or fooling, the solver and leading him down the garden path away from the solution, not being a solution trick.

For example, a trivialized version of a trick question: 


Imagine that you are a train engineer. The train has [... fill with useless information about the load the train is carrying, heavy on numerics ...] How old is the engineer?


Such trick questions may be contextually dependent. Charles Trigg had a question that followed a number of difficult dissection problems--dissect a square into five congruent parts. The question is trivial when not contextualized.

    VS-)

On 9/10/2010 11:00 AM, Chris Sangwin wrote: 
I am emailing to ask for some help in identifying "trick questions" in mathematics. 

** Definition: 

A mathematics question/exercise/task is said to be a "trick question" if the reasoning required to solve it is applicable only to the solution of that question. 

** 

I am having some difficulty in identifying "trick questions".  For example, the task "Expand (x-a)(x-b)(x-c)...(x-z)"  felt to me originally to require a trick.  But, on reflection, the reasoning is precisely that needed to construct Lagrange polynomials (HINT!). 

Can anyone identify a trick question?   I'm struggling to do so. 

Chris