Saturday, September 19, 2009 2:56 PM,

If you have your hands full with everything you already have to do, please 
feel free to just ignore this email message for now.  Save it for Christmas 
vacation or next summer or whenever you have some free time.  
I am happy to have you solve problem 6b with three 2to1 muxes.

But if you are turned on by challenges like the two 2to1 mux problem as 
originally stated, feel free to read on.  Again, you are NOT expected to read 
this, and nowhere on any exam in 306 will we test you on what follows in this 
email message. 

Having said that, I note that a student has written:




	Dear Professor Patt,

	This is *******************.  Your "Trick Question" is either a 
	little to tricky for me, or impossible. I spent a large portion 
	of my Friday night pondering the idea of how a mux could yield 
	a value of zero when all inputs are 1.  You have successfully 
	made logical operations circuits diagrams a part of my nightmares. 
	Well done. 

	From my understanding a xor gate evaluates to 0 when all inputs are 1 



Actually, to be more precise, a 2-input xor gate evaluates to 0 when BOTH 
inputs are 1.  I think you will find that an n-input xor gate has a little 
different behavior.  Construct one out of n-1 2-input xor gates and see.



	and a mux selects a specific input to go through. If all inputs to 
	the mux are 1 how could a zero come out without the use of an inverter. 

	The rest of the problem set 
	seemed to be pretty straight forward. However, the fact that this 
	specific problem seems impossible to me leads me to believe that if 
	it is possible, my fundamental understanding of both xor gates and 
	mux's is flawed. Is my understanding of these concepts flawed? 



No, your understanding of these concepts is probably not at all flawed.



	If you read this and come to the conclusion that I am wasting your 
	time I am sorry. 



And, no, you are not wasting my time.  In fact, any time a student struggles 
with a problem like you have, you are not wasting my time.  In fact, in this 
particular case, you are providing me with an opportunity to teach you 
something that in the long run is actually more valuable than 306.  The reason 
I told the class that if they have their hands full without considering this, 
they should just put this aside for later is that I don't want those students 
to struggle with this when they have more immediate pressing problems to work 
on.  BUT, since you asked, I will answer.

The problem you have is that you constrained the problem unnecessarily.  
That is, you put a constraint into the problem as stated that isn't there, and 
in so doing eliminated the solution.

We humans do that all the time.  We adopt a mind set that prevents us from 
solving the problem.  I have several examples.  Perhaps you have seen this
one:  "Arrange six equal-length match sticks to form four equilateral 
triangles."  It is trivial to do, but I have seen many, many people throw up 
their hands and say, "It is impossible." 

I think you have put more than enough time into this problem.  I will show you 
the answer on Monday.  ...and the answer to the match stick problem if you 
have not seen that one either.

Meanwhile, if you are so inclined, enjoy the football game tonight.  Or do 
whatever you enjoy doing on Saturday night.



	Please withhold this from the rest of the class if at all 
	possible. Thank you for your time.



Usually, I will honor such a request.  But in this case, it is too important a 
lesson to keep private.  In case there are others who spent the better part of 
their Friday night on this, I wanted them to see this as well.

	<<name withheld to protect the student wrongfully criticizing himself>>

Good luck with the rest of the course.  I hope to see you in class on Monday.

Yale Patt