An experiment in top-down problem solving

[Jeremy Weissmann]

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

[Dhruv Matani]

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

Regards,
-Dhruv.

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Sent from my not too egotistical device

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

[Jeremy Weissmann]

All can play.

+j

[Dhruv Matani]

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)?

[Jeremy Weissmann]

> 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

[Dhruv Matani]

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 is being asked.

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.

 

[Jeremy Weissmann]

> 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.

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

[Dhruv Matani]

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
> 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.

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...?

Why am I asking this?
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)?

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...)


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

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

[Jeremy Weissmann]

> 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

[Dhruv Matani]

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

> 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.

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.
 

> 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.

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.




 

[Jeremy Weissmann]

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

[Jeremy Weissmann]

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

--

[Dhruv Matani]

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!
 

 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.

okay. will keep it in mind though.
 

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. 

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.

[Jeremy Weissmann]

> 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

[Dhruv Matani]

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.

[Jeremy Weissmann]

> 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

[Dhruv Matani]

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.

[Jeremy Weissmann]

>> • 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

[Henry McLoughlin]

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

[Jeremy Weissmann]

> 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

[Kevin]

Hi,

What is the value of the expression?

Regards,
Kevin.

[Jeremy Weissmann]

Good question.

+j

[Kevin]

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.

[Jeremy Weissmann]

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...@gmail.com> wrote:

> 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:

[Kevin]

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:
> 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
>

[Jeremy Weissmann]

But that question itself is not refined enough.

It is akin to asking me the question: "How can I achieve happiness in life?". That is not a question that I can answer for you at the most refined level because the question itself is not refined. I can help you answer the question, but you have to break the question down first into smaller pieces. This is the process Dhruv has already started.

Abandon the mindset where you think I have the answer written down on a sheet of paper in front of me, and I'm just giving teasing, misleading answers to your questions. A much more apt analogy is to think of me as a service, like Google: when you send a query to Google, it's not like it has the answer waiting for you. It has to work, too, to find the answer -- and accordingly the quality of the answer depends on the quality of the question.

I am an interface with the information, not a seer.

I am a detective who has investigated a crime scene, and you are a newspaper reporter asking for information. If you ask "What happened?", I will give you a similarly broad answer: "A woman was murdered.". If you then ask, "No, I mean tell me down to the ultimate detail what happened.", I'd just shrug my shoulders and say, "I'd have to figure that out, and in fact I'll probably never know everything that happened.".

But if you ask me more specific and relevant questions, perhaps we can come to the answer together.

(The length of my response should hopefully convince you that I was being utterly honest when I said your question was a good one. Indeed, I had hoped that this problem-solving process would have two branches: the problem-solving proper, and discussion about the problem-solving process.)

+j

Sent from my iPhone

[Kevin]

Well, I didn't think you were misleading me or being dishonest! (I
posted my "Apologies..." note before I saw your "Good question"
response but the refresh on my browser/Google groups combination is
lousy.)

So the object of our interest is an interval of time. And, of course,
an interval is defined by a starting point/event and an ending point/
event (details of which you have given). So: do these points/events
coincide?

In one place you say the interval is non-negative and in another place
that it is positive: perhaps I should have sought clarification before
posing my question...

(I have other questions concerning the points/events but I'm asking
them one at a time!)

K.

On Nov 2, 11:39 pm, Jeremy Weissmann <jer...@mathmeth.com> wrote:
> But that question itself is not refined enough.
>
> It is akin to asking me the question: "How can I achieve happiness in life?". That is not a question that I can answer for you at the most refined level because the question itself is not refined. I can help you answer the question, but you have to break the question down first into smaller pieces. This is the process Dhruv has already started.
>
> Abandon the mindset where you think I have the answer written down on a sheet of paper in front of me, and I'm just giving teasing, misleading answers to your questions. A much more apt analogy is to think of me as a service, like Google: when you send a query to Google, it's not like it has the answer waiting for you. It has to work, too, to find the answer -- and accordingly the quality of the answer depends on the quality of the question.
>
> I am an interface with the information, not a seer.
>
> I am a detective who has investigated a crime scene, and you are a newspaper reporter asking for information. If you ask "What happened?", I will give you a similarly broad answer: "A woman was murdered.". If you then ask, "No, I mean tell me down to the ultimate detail what happened.", I'd just shrug my shoulders and say, "I'd have to figure that out, and in fact I'll probably never know everything that happened.".
>
> But if you ask me more specific and relevant questions, perhaps we can come to the answer together.
>
> (The length of my response should hopefully convince you that I was being utterly honest when I said your question was a good one. Indeed, I had hoped that this problem-solving process would have two branches: the problem-solving proper, and discussion about the problem-solving process.)  
>
> +j
>
> Sent from my iPhone
>

[Jeremy Weissmann]

> So the object of our interest is an interval of time. And, of course,
> an interval is defined by a starting point/event and an ending point/
> event (details of which you have given). So: do these points/events
> coincide?

I have no idea if this is relevant or not.

I described the start and end points in an earlier email. Do they coincide? Can they? Must they? Who knows?

> In one place you say the interval is non-negative and in another place
> that it is positive: perhaps I should have sought clarification before
> posing my question...

Both statements were true. One was more sober, one was more informative, but both were absolutely true.

+j
>

[Kevin]

> Both statements were true. One was more sober, one was more informative, but both were absolutely true.

Ok. So the number, the duration, is positive.

> I described the start and end points in an earlier email.

You did indeed. You said "the startpoint is the point in time when, on

a particular day, a school bus passes a teacher".

If I ask "At what time did the startpoint occur?", you will answer "At
the startpoint" or "When, on a particular day...", right?

So I ask: at the startpoint, was the bus headed for the school?

K.

[Jeremy Weissmann]

> If I ask "At what time did the startpoint occur?", you will answer "At
> the startpoint" or "When, on a particular day...", right?

I would not answer "at the startpoint", because that would be to revert back to a coarser (= less refined) grain of detail. That would be pedantic.

Similarly, I would answer "When, on a particular day...", because that is the grain of detail we have reached thus far.

This is a game of information: if your question does not provide any new information, neither will the answer. Your questions have to provoke refinement; the question "What is the value of <expression>?" will never yield a refinement of <expression>.

> So I ask: at the startpoint, was the bus headed for the school?

I hope you can see how utterly different this question is. Given the descriptions of the start and endpoints in terms of a person and a bus, the (dare I say it?) simplest next step is to clarify the relation of these to each other.

Before now, you have had a poverty of data to work with. But now there are times and buses and teachers and the notion of "passing"! The floodgates have burst open, and now it is time to gather the disperate data, to homogenize.

To answer your question: Indeed, at the startpoint, the bus is headed for the school.

* * *

I debated for a long while whether or not to give you lots of other "standard", clarifying information about the teacher and the bus. But I have decided to let you ask these questions on your own, to allow you to make unwarranted assumptions. Cruel though it may seem, it will be a good part of this experience.

+j

[Jeremy Weissmann]

When I was in Eindhoven, studying with the likes of Wim Feijen and Tom
Verhoeff, and collaborating with the likes of Dan Grundy and Apurva
Mehta, my favorite problem solving efforts were the collaborative ones
-- whether formalized in the Eindhoven Tuesday Afternoon Club, or more
casually, in our office, over a nice cup of coffee.

In any case, one of the hallmarks of these collaborations was our
spirited debate over which direction to pursue in an investigation.
There would often be long stretches of quiet thought, followed by a
suggestion, followed by discussion over the merits and demerits of the
suggestion. All this was independent of whether the suggestion was
ultimately pursued! A little taste of the flavor of these discussions
can be found here:

http://www.mathmeth.com/jaw/main/jaw0xx/jaw51.pdf .

Now that there are at least three people involved in this
investigation, I invite you to prod and critique and spur each other
on. Enjoy the game... just be thoughtful about it!

+j

[Kevin]

> I would not answer "at the startpoint", because that would be to revert back to a coarser (= less refined) grain of detail. That would be pedantic.
>
> Similarly, I would answer "When, on a particular day...", because that is the grain of detail we have reached thus far.

Yes.

> This is a game of information: if your question does not provide any new information, neither will the answer. Your questions have to provoke refinement; the question "What is the value of <expression>?" will never yield a refinement of <expression>.

> I hope you can see how utterly different this question is.

I can indeed. I remembered that to refine an expression means to
establish more information about it. The refinements thus far have
established two descriptions, each containing some data. It is sweetly
reasonable to focus on them so...

>Given the descriptions of the start and endpoints in terms of a person and a bus, the (dare I say it?) simplest next step is to clarify the relation of these to each other.

Yes. I forgot again to include my reasons.

> Before now, you have had a poverty of data to work with. But now there are times and buses and teachers and the notion of "passing"! The floodgates have burst open, and now it is time to gather the disperate data, to homogenize.

Yes indeed.

Did the bus arrive at the school at any time between the startpoint
and the endpoint?

Obviously, to establish if the endpoint was the +first+ arrival,
following the startpoint, of the bus at the school.

> I debated for a long while whether or not to give you lots of other "standard", clarifying information about the teacher and the bus. But I have decided to let you ask these questions on your own, to allow you to make unwarranted assumptions. Cruel though it may seem, it will be a good part of this experience.

Not cruel at all. This is the way to do it!

K.

[Srinivas Nayak]

Dear Jeremy,

 

I was very much anxious whn I saw your invitation to solve a problem. But when I was reading line by line your invitation till end, I felt that my knowledge level was gradually falling down.  And finally, I felt, as if my knowledge was really inadequate at the first place, the mercury level was at 0 when I reached the last line in your mail.

 

Tried to guess something about your infinitesimally small problem (only a . ).

But lo, I thought as if it was really a full-stop.

 

Now, going through other posts. Trying to educate myself.

 

Sincerely,

Srinivas Nayak

[Jeremy Weissmann]

Don't be bashful. We're all learning here. :)

+j

[Jeremy Weissmann]

> Did the bus arrive at the school at any time between the startpoint
> and the endpoint?

I thought this was a fair question, and an easy one to answer.  So I thought about the question, and thought and thought.  And finally I thought I knew the answer.  After struggling to write a response for about 10 minutes, I realized that if I have to think this hard, we probably need a better question.

Perhaps if I look again at your reasoning, I can see why I struggled so greatly:

Obviously, to establish if the endpoint was the +first+ arrival,

following the startpoint, of the bus at the school.

I guess I'm not sure why it should be relevant to establish this.  It seems much more reasonable to try to understand the relationship between man and bus.  But if you can help me understand the relevance of your question, I may see how better to answer it.

Once again, I promise I'm not trying to be vague or cryptic.

+j

[Kevin]

Apologies for the delay. (Work, etc.)

> Perhaps if I look again at your reasoning, I can see why I struggled so greatly:
>
> Obviously, to establish if the endpoint was the +first+ arrival, following the startpoint, of the bus at the school.
>

> I guess I'm not sure why it should be relevant to establish this.  But if you can help me understand the relevance of

> your question, I may see how better to answer it.

Ok. Well, we're concerned with a duration, that between the time when
the bus passes a teacher and the time of an (not necessarily "the")
arrival of the bus at school. The bus might have made a number of
journeys to/from the school between startpoint and endpoint, so the
intention of the question is to see where the endpoint might occur in
such a range of arrival times.

> It seems much more reasonable to try to understand the relationship between man and bus.

Does it? Hmm. I've to think about that, so. Another reason I proferred
my last question was to balance out the questions about startpoint and
endpoint. (Faux symmetry, perhaps?) Meantime, we have established that
the teacher is male (or is "man" generic?)!

> Once again, I promise I'm not trying to be vague or cryptic.

I trust you!

K.

[Jeremy Weissmann]

Ok. Well, we're concerned with a duration, that between the time when
the bus passes a teacher and the time of an (not necessarily "the")
arrival of the bus at school. The bus might have made a number of
journeys to/from the school between startpoint and endpoint, so the
intention of the question is to see where the endpoint might occur in
such a range of arrival times.

Certainly there is only one endpoint of the time interval; the question is whether the description of that endpoint is vague. 

I don't think that sort of vagueness would be fair of me. The expression 'when the bus arrives at the school' is grammatically definite, even if it is mathematically indefinite, and it isn't proper to use it if there are multiple points of arrival -- just like I can't properly say 'the dog' if there are several dogs I could be talking about. 

In any case, you can just say you are trying to clarify the route of the bus, and ask if can safely be assumed that when the bus passes the man (the startpoint), it travels directly to the school (the endpoint). My answer would be yes. 

It seems much more reasonable to try to understand the relationship between man and bus.

Does it? Hmm. I've to think about that, so. Another reason I proferred
my last question was to balance out the questions about startpoint and
endpoint. (Faux symmetry, perhaps?)

I'm not sure there is merit to that sort of even-handedness when asking questions. If you have a lead, you should pursue it as long as you have good reason to. In my humble opinion, that is. :)

Meantime, we have established that
the teacher is male (or is "man" generic?)!

I wish I could say that I was being generic, but 'man' probably came from my sex-biased mind. :( Let's make it a robot teacher, an 'it'. 

I trust you!

Good! :)

Know that in this experiment, I'm playing a double role. On the one hand, I'm a Google-style oracle that gives informative answers to pointed questions; on the other, I'm a player of the game. 

When I object to a question asked, don't think of the oracle as balking, think of it as a fellow problem-solver's objection to the pointedness of the question. 

+j

[Kevin]

> In any case, you ... ask if can safely be assumed that when the bus passes the man (the startpoint),

> it travels directly to the school (the endpoint). My answer would be yes.

Ok.

> I wish I could say that I was being generic, but 'man' probably came from my sex-biased mind. :(
> Let's make it a robot teacher, an 'it'.

Indeed :)

> Know that in this experiment, I'm playing a double role.
> On the one hand, I'm a Google-style oracle that gives informative answers to pointed questions;
> on the other, I'm a player of the game.
>
> When I object to a question asked, don't think of the oracle as balking,
> think of it as a fellow problem-solver's objection to the pointedness of the question.

I hear ya!

Ok, well focussing on the relation between teacher and bus, my next
question is:
at the startpoint, which way was the teacher headed?

Why ask this? At this point the bus is headed somewhere and I'm trying
to establish a similar kind of fact for the teacher, with a view to
relating/confronting these facts with each other...

K.

[Jeremy Weissmann]

The teacher is heading for the school as well.

+j

Sent from my iPhone

[Simon]

After reading for a while, I think I finally have an idea of what is
going on and I came up with a few questions and critiques.

Questions
* What is the average speed of the bus between the endpoints?
* What is the distance between the teacher and the school when the bus
passes by?

Since we don't seen to have numerical values for the both endpoints,
using basic physics for calculating them or, at least calculating more
information about them seem like a good way to go forward.

Critique:
* About the relation between the teacher and the bus or the teacher
and the school: it seems like a new level of indirection with respect
to the end points and we haven't found out that much information about
the previous level yet, that is, the trip of the bus. If we can do
without finding much about the teacher, I would be all for it.

Cheers!
Simon

On Nov 8, 2:42 am, Jeremy Weissmann <jer...@mathmeth.com> wrote:
> The teacher is heading for the school as well.
>
> +j
>
> Sent from my iPhone
>

[Jeremy Weissmann]

> * What is the average speed of the bus between the endpoints?

A positive real number.

> * What is the distance between the teacher and the school when the bus
> passes by?

8 miles.

+j

[Simon]

Thank you!

I guess now the attention to the teacher is warranted. To explore its
relation with the bus, I have the following questions:

* Assuming that the teacher arrives eventually, how much time is
elapsed between his arrival at school and that of the bus?
* Is the speed of the teacher constant with respect to time?
* And what is his average speed?

The first one allows me to introduce a new notion in the problem: the
arrival of the teacher. So far we had
* the encounter of the bus and the teacher
* the arrival of the bus in school
* the distance between the place of encounter and the school (8 miles)

With the expected new information, we can find out information about
the teacher which is similar to what we are looking to find out about
the bus and then link the two of them.

Just to make sure that my idea of the average speed is really a dead
alley:
* Is the speed of the bus constant between the time where it
encounters the teacher?
* If not, can we decompose the section of interest in his journey into
segments of constant speed?
* And if not, can we decompose the section of interest in his journey
into segments where his acceleration is constant?

An affirmative to any of the last two questions would send me on an
inquiry about the average speed of the bus. Otherwise, I would jump
back on the idea of investigating the journey of the teacher.

Cheers!
Simon

[Jeremy Weissmann]

> * Assuming that the teacher arrives eventually, how much time is
> elapsed between his arrival at school and that of the bus?

A real number.

> * Is the speed of the teacher constant with respect to time?
> * And what is his average speed?

Yes, 4 mph.

> * Is the speed of the bus constant between the time where it
> encounters the teacher?

Yes.

+j

[Jeremy Weissmann]

So to clarify where we are in the design/refinement process:

• To be able to ask some specific questions about our expression, we
inquired about its type, and found out that it is a number.

• Since numbers can be used to model radically different phenomena, we
inquired further and found that the number represents a duration in
time.

• Since durations in time can always be identified by their boundary
points, we asked if these were salient, and found that they were.
Both are described in terms of the path of a bus on a particular day:
the startpoint is when the bus passes a teacher on that day; the
endpoint is when the bus reaches the school on that day.

• Now that a physical path had entered the picture, we wondered
whether the simple relationship between distance, rate, and time could
help our refinement.

• We clarified the physical scenario: The bus is moving at a constant
rate on a fixed path from startpoint to endpoint. The distance
covered is 8 miles.

• We were not able to refine the bus's rate, which would have ended
our investigation immediately, so we turned to the only other salient
(moving) object, the teacher.

• We learned that the teacher is moving towards the school as well,
presumably on the same course as the bus, at a constant rate of 4
miles per hour.

So, this is where we are! Discussion of how to proceed is just as
welcome as (and perhaps preferable to) more questions!

+j

[Srinu]

Hi,


My approach may be wrong! Just trying...


(0) .
(1) .0 [after decimal point any number of 0s can be present. In (0),
0
number of 0s present, so restored one 0. Ex. 3.0000... = 3.0 = 3]
(2) 0.0 [before decimal point 0 can be omitted. Ex 0.123 = .123]
(3) 0 [because 0.0 =0]
Now, after (3), we can start deriving anything from 0


Please correct me.


Sincerely,
Srinivas Nayak

[Jeremy Weissmann]

I don't understand what you are saying. I think you need more words?

+j

On Mon, 15 Nov 2010 11:20:11 +0530, Srinivas Nayak wrote:
> Hi,
>

> My approach may be wrong! Hust trying...

[Srinivas Nayak]

I interpreted the dot in the given expression

(0) .

as decimal point.

and so I started deriving from that dot.

 

my next step in the derivation is

(1) .0 [after decimal point any number of 0s can be present. In (0), 0 number of 0s present, so restored one 0. Ex. 3.0000... = 3.0 = 3]

 

convinced of this step? Probably many will not be happy with it since it was usual.

 

I argue (may be a wrong argument) that, if

3.0 can be written as 3.00000000000000..., what is the reason behind it?

after decimal point N number of zeros can be placed.

I placed 0 number of zeros. (N=0)

 

If I am wrong, I must be wrong here in this step...

Please have a look.

 

Then rest of the thing follows easily.

 

Srinivas

[Srinivas Nayak]

to clear more,

 

in the expression

(0) .

 

I believe, after dot 0 number of zeros present,

 

so I can add one zero to dot.

 

So

 

(1) .0

[Jeremy Weissmann]

I think there is some confusion here...

When I said

(0) expression .

the . was just a period, a full stop. In any case, have you not followed the rest of the conversation?

+j

[Jeremy Weissmann]

In any case, I am not really sure what kind of "derivation" this is.

+j

[Srinivas Nayak]

ok, then I am wrong. I thought that period is the decimal point.

[Apurva Mehta]

This has been an amazing investigation, and one worth pursuing to its end. So I will pick up the baton :) 

I think it is worth recalling that this is a game where we are all trying to refine an  '<expression>'  into a value we are collectively happy with. The happiest refinement is a definite value from a familiar type, to use Jeremy's words.

The game has two types of players. Jeremy is 'the oracle' to whom we (the investigators?) direct questions in an effort to refine the expression into a happy form. Permissible questions are those which are motivated by the information on the table and which attempt to introduce new information to play with. Only permissible questions will be answered. 

The purpose of such a game, in my books, is to get a flavor for how to design solutions to problems methodically, in a series of motivated steps, and without pulling any rabbits out of our hats. The solution itself is not as important as the process used to arrive at that solution, and the lessons learned from reflecting on that process. 

With that said, lets resume where we left off!

 * * *

I have included Jeremy's summary of the thread below. Let me summarize that summary by listing everything we have gleaned through our questions so far: 

• We know that the expression is a number representing a duration in time. 
  
• The start point of this duration is when a bus passes a school teacher. 
• The end point of this duration is when the bus reaches a school. 
• The bus is headed toward the school at a constant speed. 
• The bus is 8 miles from the school when it passes the teacher. 
• The teacher is moving at a constant speed of 4 miles/hr, and is also headed toward the school. 

We know that the bus has to cover  8  miles and it is traveling at a constant speed. If we knew this speed our investigation would be over. But we don't know the bus' speed and neither does our 'oracle'. 

One direction we could take toward our goal at this point would be to compute the bus' speed from the information we have to work with. 

Another direction is to focus on the movement of the teacher and fix his/her end position when the bus arrives at the school. This is relevant because both the bus and the teacher begin from the same point, at the same time, and are headed in the same direction. So knowing the position of the teacher relative to the school at the moment when the bus arrives at the school will give us the distance walked by the teacher during the interval of time we are interested in. 

And we know that the teacher is moving at  4 mph .  So if we know the distance the teacher covered from when the bus passed him/her, we will be able to calculate the duration of time for which the teacher was walking. And this duration is exactly the same as the duration the bus took to reach the school from the moment it passed him, by definition. 

So, do we know the distance between the teacher and the school at the moment when the bus arrived at the school? (I am assuming that the teacher is moving more slowly than the bus)

Apurva

[Jeremy Weissmann]

Thank you, Apurva, for revitalizing this thread!  I had the chance to read over some old exchanges and found myself really stimulated!

 

The game has two types of players. Jeremy is 'the oracle' to whom we (the investigators?) direct questions in an effort to refine the expression into a happy form. Permissible questions are those which are motivated by the information on the table and which attempt to introduce new information to play with. Only permissible questions will be answered. 

One way I phrased it earlier in the thread is that I am like a search engine -- the quality of the responses is proportional to the quality of the questions.

Another way I sometimes phrase it with the students I tutor and teach is:  I'm not giving you anything for free.  I have lots of answers for sale, but the cost is a good question or explanation.

 

We know that the bus has to cover  8  miles and it is traveling at a constant speed. If we knew this speed our investigation would be over. But we don't know the bus' speed and neither does our 'oracle'. 

One direction we could take toward our goal at this point would be to compute the bus' speed from the information we have to work with.

So in this approach, you would hope to calculate the bus's speed from other quantities, then use that to answer the question.  (Let's keep that in mind.)

That sounds like a reasonable strategy, in principle.  (But see below.)

 

Another direction is to focus on the movement of the teacher and fix his/her end position when the bus arrives at the school. This is relevant because both the bus and the teacher begin from the same point, at the same time, and are headed in the same direction. So knowing the position of the teacher relative to the school at the moment when the bus arrives at the school will give us the distance walked by the teacher during the interval of time we are interested in. 

Everything you say is true -- but how would this be relevant to the problem at hand?  (I am thinking of what Dijkstra wrote about the mathematician's approach to the queens on the chessboard problem:  He conjectured that if the chessboard were divided in four squares of 4*4 fields, each square should contain two queens, and then he started to prove this conjecture without having convinced himself that he could make good use of it. He still has not solved the problem and, as far as I know, has not yet discovered that his conjecture is false. )

In any case, in this approach, you would hope to calculate distance travelled by the teacher from other quantities.  (Let's keep that in mind.)

 

And we know that the teacher is moving at  4 mph .  So if we know the distance the teacher covered from when the bus passed him/her, we will be able to calculate the duration of time for which the teacher was walking. And this duration is exactly the same as the duration the bus took to reach the school from the moment it passed him, by definition. 

Okay -- now I see the relevance of the above.  Why was this in a separate paragraph?  :)

In any case, all of the approaches you suggest have the same flavor, namely using information about distance, rate, and time, to compute other information about distance, rate, and time.

So then perhaps the time has come to really summarize and categorize what we know in each of these dimensions (distance, rate, and time), so that we may ask the question:  "What additional information, in which of these dimensions, would be most useful to us?" .

+j

[Apurva Mehta]

> So then perhaps the time has come to really summarize and categorize what we
> know in each of these dimensions (distance, rate, and time), so that we may
> ask the question:  "What additional information, in which of these
> dimensions, would be most useful to us?" .

I thought I did summarize all the distance, rate, time, and direction
information already available. Based on that summary I asked the
question 'Do we know the distance of the teacher from the school at
the moment when the bus reaches the school?'.

I do think this is a permissible question by the rules of the game: it
is motivated by the information we have, and also is trying to
introduce new information which is relevant to our exercise.

Further, given what we know, and especially since we don't know the
rate of the bus, this seems like the most 'useful' piece of 'distance'
information we could have because it would allow us to solve the
problem with exactly one additional calculation.

If this information is not available, then I will followup with other
questions trying to probe for other (distance, rate, time) quantities!

Apurva

--
http://about.me/apurva.mehta

[Jeremy Weissmann]

> So then perhaps the time has come to really summarize and categorize what we
> know in each of these dimensions (distance, rate, and time), so that we may
> ask the question:  "What additional information, in which of these
> dimensions, would be most useful to us?" .

I thought I did summarize all the distance, rate, time, and direction
information already available.

   You did list all the known information, but, to be fair, that list was not organized in any appreciable way.  My suggestion is to group our data by distance, rate, and time, and then within each category to explore what we do and do not know.

 

Based on that summary I asked the
question 'Do we know the distance of the teacher from the school at
the moment when the bus reaches the school?'.

I do think this is a permissible question by the rules of the game: it
is motivated by the information we have, and also is trying to
introduce new information which is relevant to our exercise.

   There are many questions that could be asked, and many pieces of information that could help us complete the problem!  We owe it to ourselves to ask the best and most useful question, and to make an arbitrary choice only if necessary.  That's why our first question was:  "What is the average speed of the bus between bus and school?" .   We had the distance between bus and school, and we wanted to know the time between bus and school; there is no doubt that this first question was the most direct, and made the most sense.

   Perhaps one could have made the case at that time that this  _wasn't_  a good question, because it ignores the teacher, and why would the teacher be mentioned if it wasn't necessary?  Another could have argued in response,  "Ah, but how do we know if the teacher is arbitrary or not?  Perhaps it is an arbitrary landmark, like the school.  In any case, since we have the distance of the bus and want the time, the simplest piece of information we should look for is the rate of the bus.  If that information is not available, then we will know that the teacher is not arbitrary, and we can explore new directions -- but we really can't know if the teacher is arbitrary or not until we ask this question!" .

   I don't feel your question can be defended in this way.  I agree that it is  *a*  piece of information that could help, but I don't think that's enough reason to ask the question.

Further, given what we know, and especially since we don't know the
rate of the bus, this seems like the most 'useful' piece of 'distance'
information we could have because it would allow us to solve the
problem with exactly one additional calculation.

   Why does it  'seem'  so?  What makes you feel this is the most useful?  I claim that if we organize our data by dimension, and see what we do and do not know in terms of the basic concepts of the problem (bus, teacher, school), we will find a much more pertinent question!

If this information is not available, then I will followup with other
questions trying to probe for other (distance, rate, time) quantities!

   Aha!  But if you have other questions, why are not all out on the table, for us to pick the best?

+j

[Jeremy Weissmann]

By the way, the previous response was posted by  "Jeremy as player" ,  not  "Jeremy as oracle" .

+j

[Jeremy Weissmann]

Forgive all these posts, but I think I have another explanation of why I don't think we are ready to ask your question.

Imagine that the design of our solution is complete, and it comes time to record the design process for posterity, and to help future generations learn techniques of top-down problem solving.  What do we want that record to look like?

For example, at the beginning of this experiment, there were a smattering of questions, like:  Is it boolean?  A number?  Does it have the form nPr or nCr?  Is it positive?  Negative?  Real?  Complex?

I am not at all criticizing the person who asked those questions.  I think he was asking from a very human place, to try to get more information about our unknown, so he could get a mental grip on it.

But is this something we wish to keep in our record of the design process?  Do we want to tell new problem solvers:  "When confronted with a problem to solve, ask every question you can think of.  Something's bound to come out of it." ?   No, humans are already experts at this form of problem solving -- they don't need any more help or encouragement from us!

Instead we decided that the right question to ask at the beginning was:  What is the type of the expression we're looking for? .   And that the justification for this question was:  "We would like to ask more specific questions about the expression in question, and since different types of expressions (like boolean and number) have totally different sets of questions surrounding them, it makes sense to ask for the type of the expression." .

That justification embodies a technique of top-down problem solving that someone can really take away from this investigation.  And I think we can find similarly good questions and good justifications for every step of our design so far.

Now let's relate that to where we are now.  When you first broached your question:  Do we know the distance of the teacher from the school when the bus reaches the school? ,  the first words of that paragraph were:  "Another direction is..." .   So it is clear at the time that you saw it as a possible direction, but I'm not sure that you were claiming at the time it was best possible.  (In your last post you said that you thought this information would be  'most useful' ,  but I don't feel you explained why it is most useful, and indeed I feel we don't yet have a good enough grip on our data to decide whether or not  *any*  particular question is  'most useful' .)

What does that mean for the record of our problem solving process?  Should we content ourselves to say:  "We asked for the simplest piece of information, the speed of the bus, that would help us solve the problem.  When that wasn't available, we tried to think of ways we could get more information to help us to solve the problem and then asked for that information.  If that information wasn't available, we would follow up with other questions trying to probe for other quantities." ?

I think we can do better.  I, as a player of the game, would like our record to say at this point:  "... When that wasn't available, it was clear that the answer would carve a twistier route through the data.  Many ideas for how to proceed came to mind, but which one should we choose?  We didn't want to explore every possible dead-end, so we decided to classify the range of possibilities open to us at this point.  We grouped our data by type and related the pieces of information to each other as best we could, in order to restore asymmetries lost by English-language phrasings, to find out what we really do and do not know, and ultimately to find the best way to proceed." .

But that is only my suggestion!  If someone can suggest a better way to proceed, I am totally open to it.  That suggestion should ideally be written as if it is already in the record of our design process, explain how we decided to proceed from this point of our investigation.

+j

[Apurva Mehta]

Ok, here is what we have in a structured form:

* Both the Bus and the Teacher are moving toward the school.
* Both start their journey at the same moment.
* The bus ends its journey at the school.

| Rate | Distance | Time
Bus | Br | 8 mi | Bt
Teacher | 4 mph | Dt | Tt

We need to find Bt . We do not know Br .

If we fix the 'end moments' of both journeys (of the bus and the
teacher) to be the same, then we have

0) Bt == Tt.

Then, knowing the distance covered in the teacher's journey would
allow us to compute the duration of the teacher's journey, ie. Tt .
Hence we would know Bt by (0).

We could also ask, 'Did the teacher's journey end? If so, how long was
this journey (in distance or time)?'. An answer to this question give
us values for both Dt and Tt for some journey.Then if we asked the
distance covered by the Bus in this interval, we would find Br.

And if we knew Br , we could compute Bt .

* *

Those are the only two options I can see at this moment. Either we
compute Br , or equate the the journey of the teacher with the
journey of the bus through (0) .

If I am missing something, then please prod!

Thanks,
Apurva

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[Apurva Mehta]

>
>              | Rate   |  Distance |  Time
> Bus        |   Br      |  8 mi       | Bt
> Teacher  |  4 mph |  Dt           | Tt
>

Jeez, I should get a smack on the knuckles for that nomenclature. I
meant to write Rb and Tb for the Rate of the Bus and the Time of
the Bus.

[Jeremy Weissmann]

This is a nice fleshing out of the possibilities from a purely mathematical perspective. But what about a conceptual analysis? What about an effort to homogenize our given information, which is variously described in terms of a bus, a teacher, and a school?

In particular, the 'Teacher' row of your chart is a little hasty. A rate does not imply a distance or a time. (From either, we can use the rate to calculate the other.)

As for the 'Bus' , we know we are looking for the time it takes to cover 8 miles, but we do not have its rate and there is no reason to believe we necessarily will be able to calculate it without first knowing the time. In that case, we need to find another way to use our distance. To do this, we should be recast the description "the distance the bus travels on its way to school" in a less-committal way.

+j

Sent from my iPhone

[Jeremy Weissmann]

Hello all!

   Well, it looks like a little stirring the pot might be necessary here, so I am going to give a little hint.  Consider the following:

• One of our goals was to use the relation:  distance = rate x time .
• We felt that we were unable to use this, because while we have the bus's distance, we don't have its rate.
• We would rather not introduce any new data if we don't have to.
• However, we are overlooking that we actually have a distance and a rate that can be combined.
• We are overlooking it because the distance and the rate are described in English very differently.
• I therefore propose we homogenize the phrasings, and apply the formula to the information we already have.

Enjoy!

+j