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An experiment in top-down problem solving
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More options Oct 30 2010, 7:42 pm
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Sat, 30 Oct 2010 19:42:04 -0400
Local: Sat, Oct 30 2010 7:42 pm
Subject: An experiment in top-down problem solving
Hey all!

I recently came across a wonderful problem that I'd like to share with y'all.  However, I don't want to do this in the standard way, where I give you a lot of information followed by a question, and then you sit around  "thinking"  about the information, and try to answer the question.  I want top-down problem solving only!

To enforce this, I am not going to give you any information:  You have to ask for it, and you have to give a good reason for it.  (Just like if you were working in a company and needed more money for your department, you'd have to prove that you really needed it!)

Accordingly, I'm not going to give you a  "question"  either, because I don't want you to think about  "answers" ;  I want you to think about refining what you've been given, in sweetly reasonable ways.

So, without further ado, here is your refinandum  ("that which is to be refined") :

(0)   expression     .

That's it.  You are given as little about  (0)  as I could possibly manage:  it is an expression.  You don't even know its type!  Your goal is to refine  (0)  until you are  "happy" .   (The happiest possible refinement is probably a definite value from a familiar type.)   As stated above, you are welcome to ask me for any information you like, but I will be very eager to ask why you want it, and I won't hesitate to deny your request.

Enjoy!

+j

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More options Oct 31 2010, 3:23 am
From: Dhruv Matani <dhruvb...@gmail.com>
Date: Sun, 31 Oct 2010 12:53:40 +0530
Local: Sun, Oct 31 2010 3:23 am
Subject: Re: [mathmeth] An experiment in top-down problem solving

Hello Jeremy,
Is this specific to calculational math, or can people w/o knowledge of it
also participate?

Regards,
-Dhruv.

On Sun, Oct 31, 2010 at 5:12 AM, Jeremy Weissmann <jer...@mathmeth.com>wrote:

--
-Dhruv Matani.
http://duckduckgo.com/?t=chucknorris

Sent from my not too egotistical device

"What's the simplest thing that could possibly work?"
-- Ward Cunningham

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More options Oct 31 2010, 3:31 am
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Sun, 31 Oct 2010 03:31:51 -0400
Local: Sun, Oct 31 2010 3:31 am
Subject: Re: [mathmeth] An experiment in top-down problem solving

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More options Oct 31 2010, 3:36 am
From: Dhruv Matani <dhruvb...@gmail.com>
Date: Sun, 31 Oct 2010 13:06:19 +0530
Local: Sun, Oct 31 2010 3:36 am
Subject: Re: [mathmeth] An experiment in top-down problem solving

On Sun, Oct 31, 2010 at 1:01 PM, Jeremy Weissmann <jer...@mathmeth.com>wrote:

> All can play.

ha! nice!

I'm not familiar with the terminology, so please feel free to correct me or
indicate any ambiguity in the questions that I may have.
I can define my happiness level once I have asked a few questions.

q1. Is this expression a math expression? like (23 + 45.8)?

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More options Oct 31 2010, 3:38 am
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Sun, 31 Oct 2010 03:38:38 -0400
Local: Sun, Oct 31 2010 3:38 am
Subject: Re: [mathmeth] An experiment in top-down problem solving

> q1. Is this expression a math expression? like (23 + 45.8)?

All questions should be submitted with an explanation of why the question is being asked.

+j

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More options Oct 31 2010, 3:51 am
From: Dhruv Matani <dhruvb...@gmail.com>
Date: Sun, 31 Oct 2010 13:21:06 +0530
Local: Sun, Oct 31 2010 3:51 am
Subject: Re: [mathmeth] An experiment in top-down problem solving

On Sun, Oct 31, 2010 at 1:08 PM, Jeremy Weissmann <jer...@mathmeth.com>wrote:

> > q1. Is this expression a math expression? like (23 + 45.8)?

> All questions should be submitted with an explanation of why the question

okay. Well, multiple reasons:

1. I want to be able to narrow down my domain to be able to reason about it
more concretely (like choose between cross-roads)

2. When you mentioned "expression", the first thing that came to mind was
either a math expression (math) or a boolean expression (computers), or a
parsable expression (compiler) and the first one seemed most common. The
literal "facial expression" seemed unlikely, but if some of the more
plausible ones fail, I'll try that as well.

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More options Oct 31 2010, 4:06 am
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Sun, 31 Oct 2010 04:06:49 -0400
Local: Sun, Oct 31 2010 4:06 am
Subject: Re: [mathmeth] An experiment in top-down problem solving

> okay. Well, multiple reasons:

> 1. I want to be able to narrow down my domain to be able to reason

> 2. When you mentioned "expression", the first thing that came to mind
> was either a math expression (math) or a boolean expression
> (computers), or a parsable expression (compiler) and the first one
> seemed most common. The literal "facial expression" seemed unlikely,
> but if some of the more plausible ones fail, I'll try that as well.

It's not a trick question, so no on  "facial expression" .

Why is a boolean expression not a math expression?  By  'math expression' ,  did you maybe mean  'numerical'  expression?

In essence, your question is exactly the right one to be asking:  "What is the type of the expression?" .

I don't know how specific of a type you need, so I'll just say  "number" .   Thus:

(1)  number

is the next refinement.

+j

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More options Oct 31 2010, 4:43 am
From: Dhruv Matani <dhruvb...@gmail.com>
Date: Sun, 31 Oct 2010 14:13:31 +0530
Local: Sun, Oct 31 2010 4:43 am
Subject: Re: [mathmeth] An experiment in top-down problem solving

On Sun, Oct 31, 2010 at 1:36 PM, Jeremy Weissmann <jer...@mathmeth.com>wrote:

> > okay. Well, multiple reasons:

> > 1. I want to be able to narrow down my domain to be able to reason

> > 2. When you mentioned "expression", the first thing that came to mind
> > was either a math expression (math) or a boolean expression
> > (computers), or a parsable expression (compiler) and the first one
> > seemed most common. The literal "facial expression" seemed unlikely,
> > but if some of the more plausible ones fail, I'll try that as well.

> It's not a trick question, so no on  "facial expression" .

> Why is a boolean expression not a math expression?

I don't know the exact difference, but what would AND and OR mean in (43 AND
26)?
If it's a boolean one then I only have True & False to play with (rather
than numbers) and the boolean operators can be used (not mathematical ones
like * & +).

However, looking at your next question, it seems the real question I wanted
to ask was between numerical and boolean. I think I misclassified numerical
as math.

> By  'math expression' ,  did you maybe mean  'numerical'  expression?

Yes.

> In essence, your question is exactly the right one to be asking:  "What is
> the type of the expression?" .

> I don't know how specific of a type you need, so I'll just say  "number" .
>   Thus:

> (1)  number

> is the next refinement.

Assumption: Since it is a numeric expression, it would contain standard math
operators and functions as well.

q2: Does it include stuff like summation, product, factorial, nCr, nPr,
etc...?

Just trying to get more of an insight into the symbols and manipulations
involved and the complexity of the expression.

q3.
a. Does it include imaginary numbers?
b. If not (a), then does it include reals?
c. If not (b), then it must include intgers
d. Either ways, (+ve/-ve/includes zero)?

This is the next question that came to mind. Though in hind-sight, it seems
it doesn't add too much value, but it would be nice to know the more
specific type of quantities I am working with. It might help me later on.
For example, if it only includes non -ve integers, there is a chance that
the quantities being referred to are real objects (3 apples, 5 oranges,
etc...)

q4. Does in include variables? (x,y,z, etc...)
If so, how many?

If it doesn't, I can try to land up at a final value. Othewise, I'll be
happy leaving it as-is.

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More options Oct 31 2010, 10:34 am
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Sun, 31 Oct 2010 10:34:36 -0400
Local: Sun, Oct 31 2010 10:34 am
Subject: Re: [mathmeth] An experiment in top-down problem solving

> I don't know the exact difference, but what would AND and OR mean in
> (43 AND 26)?

For me, this expression would not be a number, or a boolean.  It has the same status for me as  1 / 0 ,  which is even composed of numbers and numerical operators, and yet is not a number.

But I think Eric Hehner has a system where he unifies the numerical and boolean domains.  You might want to look at that; I'll try to find a link.

> However, looking at your next question, it seems the real question I
> wanted to ask was between numerical and boolean. I think I
> misclassified numerical as math.

Yes.  What counts as  "math"  is quite hazy, and certainly includes numerical, boolean, and much else.  But now you know that I'm looking for a number.

> Assumption: Since it is a numeric expression, it would contain
> standard math operators and functions as well.

> q2: Does it include stuff like summation, product, factorial, nCr,
> nPr, etc...?

This is not a real question.  Note that  2+3  and  5  are both math expressions, and both equal, but one includes  '+'  and one doesn't.  You are certainly welcome to use mathematical operators to describe  (0)  as we go along, because that is why mathematical operators were invented:  to describe numbers.  But whether those expressions belong in the most refined version of  (0)  is not really a question worth asking.

> q4. Does in include variables? (x,y,z, etc...)
> If so, how many?

This is a question in the same vein.  A variable is just a funny symbol used to denote a number, just like  2  and  17  and  8+4  are funny symbols used to denote numbers.  The only difference is that a symbol we'd use for a variable doesn't give any indication of the number it represents; it's just a placeholder, a name.  You could give  (0)  a name right now, you could call it  'x' .   That would be a pleasant shorthand, even though all you'd know about  x  is that it is of type number.  But then the answer to your question would be yes, because clearly the expression contains variables.  In the process of refinement, however, we may be able to replace the name  x  by some more descriptive name, like  23 .   So this is a question of refinement and I will not answer it.

> q3.
> a. Does it include imaginary numbers?
> b. If not (a), then does it include reals?
> c. If not (b), then it must include intgers
> d. Either ways, (+ve/-ve/includes zero)?

> Why am I asking this?
> This is the next question that came to mind. Though in hind-sight, it
> seems it doesn't add too much value, but it would be nice to know the
> more specific type of quantities I am working with. It might help me
> later on. For example, if it only includes non -ve integers, there is
> a chance that the quantities being referred to are real objects (3
> apples, 5 oranges, etc...)

I think this sort of question adds a lot of value.  Numbers are used to model many different things, and you are trying to get a sense of what my number is being used to model.  So rather than answer each of your questions, I'll get at the question you're really asking, and tell you that the number is positive and real, because in fact it represents a duration in time.

+j

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More options Oct 31 2010, 12:17 pm
From: Dhruv Matani <dhruvb...@gmail.com>
Date: Sun, 31 Oct 2010 21:47:06 +0530
Local: Sun, Oct 31 2010 12:17 pm
Subject: Re: [mathmeth] An experiment in top-down problem solving

On Sun, Oct 31, 2010 at 8:04 PM, Jeremy Weissmann <jer...@mathmeth.com>wrote:

okay. I didn't know what "refined" meant, but now I do!!
Just to make it more explicit, it is that version of the formula which can
not be reduced any further and yields a definite value.

q4. At this point in time, I would like to guess to check if is something to
do with speed-distance-and-time since that's the first thing that came to my
mind.

q5. Does it fall under any of these:
a. Time required to do something
b. Time for which something/someone is waiting
c. Time since an event
d. Time to an event

q6. What are the units of this time? sec/min/hr?

Either ways, there were other questions I wanted to ask:

q7. It is a recurrence relation or a closed form of a recurrence relation?
The reason I ask is again to get more of a feel for the structure of the
expression.

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More options Oct 31 2010, 1:44 pm
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Sun, 31 Oct 2010 13:44:55 -0400
Local: Sun, Oct 31 2010 1:44 pm
Subject: Re: [mathmeth] An experiment in top-down problem solving

> okay. I didn't know what "refined" meant, but now I do!!
> Just to make it more explicit, it is that version of the formula which can not be reduced any further and yields a definite value.

That is an extreme sort of refinement, only possible when that which you wish to refine is a sort of mathematical object with a definite value.

Refinement can happen in lots of ways. For example, your first aim at a solution to a problem might be "some restriction on the value of x", an intermediate refinement might be "C < x < 2 for some C", then "C < x < 2 for some C < -4", and finally "-5 < x < 2". All solutions are correct, but the solutions get more and more refined (ie useful, implementable, descriptive).

Another example of (multiple levels of) refinement: A program which starts as "compute the following value", then turns into pseudocode, then C code, then machine language. The programs are equivalent in meaning, but they become increasingly low-level and implementable.

All the above examples should make it clear that there is not always a notion of "completely refined", and also that refinement can easily go beyond the point of usefulness.

>  At this point in time, I would like to guess to check if is something to do with speed-distance-and-time since that's the first thing that came to my mind.

You're asking whether the time is related to distance and/or rate. That's a reasonable thought to keep on the backburner, but I can't answer the question yet because there is nothing in the picture which is capable of having a distance or rate.

> q7. It is a recurrence relation or a closed form of a recurrence relation?
> The reason I ask is again to get more of a feel for the structure of the expression.

That's too specific a question. Let it come out in the wash. In top-down problem solving, questions need to be intimately tied to what you have in front of you, not what might someday be.

> q6. What are the units of this time? sec/min/hr?

That's not really relevant, is it?

> q5. Does it fall under any of these:
>   a. Time required to do something
>   b. Time for which something/someone is waiting
>   c. Time since an event
>   d. Time to an event

Here is a very reasonable sort of question. Just articulate why you should be asking for this sort of information, to separate it from all the other question you asked.

Also, it would be better if you phrase it not as a smorgasbord of "which of these 30 scenarios is it", but instead ask a focused yet general question about the duration, directly tied to the problem-solving process, ie what you need to know about the duration to help you compute it.

+j

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More options Oct 31 2010, 1:47 pm
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Sun, 31 Oct 2010 13:47:29 -0400
Local: Sun, Oct 31 2010 1:47 pm
Subject: Re: [mathmeth] An experiment in top-down problem solving

Regarding the word 'refined', think too of the physical notion of refining rocks into dust, for example. It's breaking something down into smaller parts. But the process can sometimes go too far!

+j

Sent from my iPhone

On Oct 31, 2010, at 12:17, Dhruv Matani <dhruvb...@gmail.com> wrote:

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More options Oct 31 2010, 2:02 pm
From: Dhruv Matani <dhruvb...@gmail.com>
Date: Sun, 31 Oct 2010 23:32:18 +0530
Local: Sun, Oct 31 2010 2:02 pm
Subject: Re: [mathmeth] An experiment in top-down problem solving

On Sun, Oct 31, 2010 at 11:14 PM, Jeremy Weissmann <jer...@mathmeth.com>wrote:

> All the above examples should make it clear that there is not always a
> notion of "completely refined", and also that refinement can easily go
> beyond the point of usefulness.

got it!

okay. will keep it in mind though.

okay, this is another shat at the same:

q5. Is it a time interval like (time to do XYZ) or an absolute value of time
like Jan 10, 1970.
(I ask beause the latter is represented in machines as a timestamp).

The reason for asking this question is that it will give me some more
insight into the original question that was asked for this expression to
have been generated.

Regards,
-Dhruv.

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More options Oct 31 2010, 8:38 pm
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Sun, 31 Oct 2010 20:38:19 -0400
Local: Sun, Oct 31 2010 8:38 pm
Subject: Re: [mathmeth] An experiment in top-down problem solving

> q5. Is it a time interval like (time to do XYZ) or an absolute value of time like Jan 10, 1970.
> (I ask beause the latter is represented in machines as a timestamp).

I believe "duration" rules out something like "Jan 10, 1970" or "7:30 PM".

+j

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More options Nov 1 2010, 3:16 am
From: Dhruv Matani <dhruvb...@gmail.com>
Date: Mon, 1 Nov 2010 12:46:38 +0530
Local: Mon, Nov 1 2010 3:16 am
Subject: Re: [mathmeth] An experiment in top-down problem solving

On Mon, Nov 1, 2010 at 6:08 AM, Jeremy Weissmann <jer...@mathmeth.com>wrote:

> > q5. Is it a time interval like (time to do XYZ) or an absolute value of
> time like Jan 10, 1970.
> > (I ask beause the latter is represented in machines as a timestamp).

> I believe "duration" rules out something like "Jan 10, 1970" or "7:30 PM".

If it is a duration of time, there is a high probability of it being a
difference of 2 quantities (end - start) or a speed-distance-time kind of
deltaT. The latter would also include stuff like time required to fill a
tank with water, etc...
q8. Which one is it (if it is any of the above 2)?

Regards,
-Dhruv.

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More options Nov 1 2010, 9:06 am
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Mon, 1 Nov 2010 09:06:56 -0400
Local: Mon, Nov 1 2010 9:06 am
Subject: Re: [mathmeth] An experiment in top-down problem solving

> If it is a duration of time, there is a high probability of it being
> a difference of 2 quantities (end - start) or a speed-distance-time
> kind of deltaT. The latter would also include stuff like time
> required to fill a tank with water, etc...

I confess I don't know what a  "speed-distance-time kind of deltaT" .   I think you mean something like  "time required to complete a task" ,  but I think this is just a special case of your first description, since it could be described as  "end of task - start of task" .

Certainly in most cases, a duration of time is characterized by its start and endpoints in time, and thus it makes sense to ask if these points are salient in any way.

Indeed they are, and I can tell you about them:

•  The startpoint is the point in time when, on a particular day, a school bus passes a teacher.

•  The endpoint is the point in time when, on that same day, the school bus arrives at school.

+j

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More options Nov 1 2010, 9:54 am
From: Dhruv Matani <dhruvb...@gmail.com>
Date: Mon, 1 Nov 2010 19:24:40 +0530
Local: Mon, Nov 1 2010 9:54 am
Subject: Re: [mathmeth] An experiment in top-down problem solving

On Mon, Nov 1, 2010 at 6:36 PM, Jeremy Weissmann <jer...@mathmeth.com>wrote:

> > If it is a duration of time, there is a high probability of it being
> > a difference of 2 quantities (end - start) or a speed-distance-time
> > kind of deltaT. The latter would also include stuff like time
> > required to fill a tank with water, etc...

> I confess I don't know what a  "speed-distance-time kind of deltaT" .   I
> think you mean something like  "time required to complete a task" ,  but I
> think this is just a special case of your first description, since it could
> be described as  "end of task - start of task" .

Yes, I know what you are trying to say, but it seemed important for me to
know if in this case it was expressed as a difference between 2 absolute
quantities or as a free-standing delta.

> Certainly in most cases, a duration of time is characterized by its start
> and endpoints in time, and thus it makes sense to ask if these points are
> salient in any way.

> Indeed they are, and I can tell you about them:

>  •  The startpoint is the point in time when, on a particular day, a school
> bus passes a teacher.

>  •  The endpoint is the point in time when, on that same day, the school
> bus arrives at school.

Nice! This reveals a lot.
However, I would like to know if this is the time taken by the bus to drop
all students home and return or pick them all up and return?
The reason being that I would like to know if this represents the pick-up
time or the drop time.

Regards,
-Dhruv.

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More options Nov 1 2010, 10:01 am
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Mon, 1 Nov 2010 10:01:55 -0400
Local: Mon, Nov 1 2010 10:01 am
Subject: Re: [mathmeth] An experiment in top-down problem solving

>>  •  The startpoint is the point in time when, on a particular day, a
>> school bus passes a teacher.

>>  •  The endpoint is the point in time when, on that same day, the
>> school bus arrives at school.

> Nice! This reveals a lot.
> However, I would like to know if this is the time taken by the bus to
> drop all students home and return or pick them all up and return?
> The reason being that I would like to know if this represents the
> pick-up time or the drop time.

?  What students are you talking about?  Nothing has been said about picking up or dropping off, either.

I would think some other people might want to chime in to debate which way the questions should be taken at this point.  I feel a lot more thinking than questioning is in order now.

+j

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More options Nov 1 2010, 10:17 am
From: Henry McLoughlin <Henry.McLough...@ucd.ie>
Date: Mon, 01 Nov 2010 14:17:58 +0000
Local: Mon, Nov 1 2010 10:17 am
Subject: Re: [mathmeth] An experiment in top-down problem solving
Hi Jeremy,

Can you tell me something about the shape of the expression? In
particular, what operators occur in it? Also is there any bracketing?

I ask these because I usually find the shape of an expression can hint
at how it might be refined.

Best wishes,

Henry

On 1 Nov 2010, at 13:06, Jeremy Weissmann wrote:

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More options Nov 1 2010, 8:44 pm
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Mon, 1 Nov 2010 20:44:04 -0400
Local: Mon, Nov 1 2010 8:44 pm
Subject: Re: [mathmeth] An experiment in top-down problem solving

> Can you tell me something about the shape of the expression? In particular, what operators occur in it? Also is there any bracketing?

Expressions do not by nature have operators or particular symbols in them, viz 2+3 and 5, which are equal, yet have completely different set of symbols in them.

It has been established in the meantime that the expression in question represents a duration in time. Therefore it is a nonnegative real scalar.

+j

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More options Nov 2 2010, 4:00 pm
From: Kevin <kevin.h...@gmail.com>
Date: Tue, 2 Nov 2010 13:00:51 -0700 (PDT)
Local: Tues, Nov 2 2010 4:00 pm
Subject: Re: An experiment in top-down problem solving
Hi,

What is the value of the expression?

Regards,
Kevin.

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More options Nov 2 2010, 4:01 pm
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Tue, 2 Nov 2010 16:01:56 -0400
Local: Tues, Nov 2 2010 4:01 pm
Subject: Re: [mathmeth] Re: An experiment in top-down problem solving

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More options Nov 2 2010, 4:37 pm
From: Kevin <kevin.h...@gmail.com>
Date: Tue, 2 Nov 2010 13:37:37 -0700 (PDT)
Local: Tues, Nov 2 2010 4:37 pm
Subject: Re: An experiment in top-down problem solving
Apologies, I forgot to include my justification: characteristic of an
expression is that it has a value, i.e. that it can be evaluated (even
if the "value" is "undefined").

K.

On Nov 2, 8:00 pm, Kevin <kevin.h...@gmail.com> wrote:

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More options Nov 2 2010, 5:59 pm
From: Jeremy Weissmann <jer...@mathmeth.com>
Date: Tue, 2 Nov 2010 17:59:44 -0400
Local: Tues, Nov 2 2010 5:59 pm
Subject: Re: [mathmeth] Re: An experiment in top-down problem solving
Your question is a good one, but hasn't it already been answered?  The value in question was first called "expression", then "number", then "duration", then "end point minus start point".

All of these are correct names; they vary in their degree of refinement. The refinement is your job, with my help of course.

If I gave you the expression "x", you could ask what its value is. But my answer would just be "Its value is x!".

Several questions later, you may determine that x equals 5+7. If you ask me what the value of this expression is, I would say "5+7". If you want the answer "12", you are not asking the right question.

+j

Sent from my iPhone

On Nov 2, 2010, at 16:37, Kevin <kevin.h...@gmail.com> wrote:

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More options Nov 2 2010, 6:55 pm
From: Kevin <kevin.h...@gmail.com>
Date: Tue, 2 Nov 2010 15:55:06 -0700 (PDT)
Local: Tues, Nov 2 2010 6:55 pm
Subject: Re: An experiment in top-down problem solving
Yes. Nice.

How about "what is the most-refined value of the expression?" ?

K.

On Nov 2, 9:59 pm, Jeremy Weissmann <jer...@mathmeth.com> wrote: