[QUOTE=R.D. Silverman;211965]Your recursive formula is correct. The rest is not.

Hint: With N = 1, there are 2 regions.
with N = 2, there are 4 regions
with N = 3, there are 7 regions,
with N = 4, there are 11 regions
etc.

Letting N --> N+1 increases the value of the polynomial [b]linearly[/b] with
N.

What can you say therefore about the degree of the polynomial? Can
you not find a polynomial that fits these values?

Now prove your result by induction using the recursion.[/QUOTE]

Are you going to show us your work on this problem?

On the topic:

[QUOTE=R.D. Silverman;212023] (1) Consider a disc. (circle plus its interior). It is intersected by N lines, each line intersecting at two distinct points (i.e. no tangents) As a function of N, what is the largest number of regions into which the disc can be divided? Prove your answer by induction. [/QUOTE]

There has been considerable correspondence regarding the (missing) functional representation of the answer following the response:

[QUOTE=blob100]Ok so, the function is:
F(N)=F(N-1)+N.[/QUOTE]

However, although that response seems to have been "accepted", I fail to see that anyone has provided the [B]proof[/B] that, even assuming the inclusion of appropriate initial boundary conditions, this recursive definition is correct.

To me, this aspect of the proof is much more difficult to establish than the algebraic task of creating an equivalent polynomial representation of the result.

N.B. However, the inability to perform the algebraic tasks requested (to transform the recursive definition into a polynomial) is indicative that that area of knowledge also deserves additional attention.

[quote=Wacky;212092]
However, although that response seems to have been "accepted", I fail to see that anyone has provided the [B]proof[/B] that, even assuming the inclusion of appropriate initial boundary conditions, this recursive definition is correct.

To me, this aspect of the proof is much more difficult to establish than the algebraic task of creating an equivalent polynomial representation of the result.[/quote]
If you have already drawn N lines and they divide the circle into F(N) areas, then how many of those F(N) old areas a new line can divide into two new areas? (Hint: how many of the N old lines a new line can intersect?)

[QUOTE=Random Poster;212132]If you have already drawn N lines and they divide the circle into F(N) areas, then how many of those F(N) old areas a new line can divide into two new areas? (Hint: how many of the N old lines a new line can intersect?)[/QUOTE]

A [b]truly[/b] rigorous proof would take us into some very deep mathematics.

(1) We need to start by proving that *if* there are N lines present, then
we can always find another one that intersects all of the previous ones.
This is non-trivial. We can use some results from projective geometry to
take the disc and the current set of lines, embed it via a suitable projective
linear transform into E^N. Consider now, when a new line intersects an
existing line. It will do so between two already known points. This gives
us a 2-sided linear inequality. The embedded projective disc then yields a
[b]system[/b] of 2-sided inequalities. We need to show that a basis of full rank exists. This can be done via a modified version of the N-dimensional
Ham Sandwich Theorem. Now, we apply the inverse projective transform to
return us to the orginal disc.

If someone else has a simpler way to do this, I would love to hear it.

(2) We then need to prove that intersecting an existing line with a new line
actually creates a new region. For this, we need some concepts in topology:
compact set, open/closed set, and the Jordan Curve Theorem.

The recursive formula is easily seen to be correct. (Just draw a new line
that intersects the existing lines) But a formal proof is out-of-scope.
way out of scope.

[QUOTE=R.D. Silverman;212147]A [b]truly[/b] rigorous proof would take us into some very deep mathematics.

(1) We need to start by proving that *if* there are N lines present, then
we can always find another one that intersects all of the previous ones.
This is non-trivial. We can use some results from projective geometry to
take the disc and the current set of lines, embed it via a suitable projective
linear transform into E^N. Consider now, when a new line intersects an
existing line. It will do so between two already known points. This gives
us a 2-sided linear inequality. The embedded projective disc then yields a
[b]system[/b] of 2-sided inequalities. We need to show that a basis of full rank exists. This can be done via a modified version of the N-dimensional
Ham Sandwich Theorem. Now, we apply the inverse projective transform to
return us to the orginal disc.

If someone else has a simpler way to do this, I would love to hear it.

(2) We then need to prove that intersecting an existing line with a new line
actually creates a new region. For this, we need some concepts in topology:
compact set, open/closed set, and the Jordan Curve Theorem.

The recursive formula is easily seen to be correct. (Just draw a new line
that intersects the existing lines) But a formal proof is out-of-scope.
way out of scope.[/QUOTE]

Never mind the first part. There is a direct, constructive proof that we can
always add a line. Work in polar coordinates. let r = 1. The first line has
theta_0 = 0.

The next line intersects the circle at (1,theta_1) = (1, theta_0 + epsilon),
and intersects the first line at (epsilon, 0) for arbitary, small, epsilon.

Etiquette Question
 

At the risk of further ridicule, may I add a conjecture attempt for critique in relation to the line/circle problem, or would that be considered a hijack of the thread?

Take Care,
Ed

[QUOTE=EdH;212351]At the risk of further ridicule, may I add a conjecture attempt for critique in relation to the line/circle problem, or would that be considered a hijack of the thread?

Take Care,
Ed[/QUOTE]

Do not make conjectures.

BTW, I am disappointed that blob100 has chosen [b]NOT[/b] to show
his work.

[QUOTE=R.D. Silverman;212390]Do not make conjectures.

BTW, I am disappointed that blob100 has chosen [b]NOT[/b] to show
his work.[/QUOTE]

Am I wasting my time trying to lead him through these problems?
Am I wasting my time trying to point out things he needs to learn
[b]before[/b] he attempts learning Calculus?

While Number Theory is irrelevant to the latter, it does teach how to
put together simple proofs. blob100 is failing in this regard.

Is he just another wannabee who is unwilling to put in the effort needed
to actually learn mathematics? There are already too many of those in this
forum.

If he [b]does[/b] want to continue, then I will help. But I need to see
the work he has put in.

The next set of exercizes will involve polynomial algebra, since it is clear
that he lacks skills in this area.

[QUOTE=blob100;204009]... because 23's factors are 2,11. ...[/QUOTE]

no offense but even i know that 23 is prime hence it's only factors are 1 and itself 2*11 = 22 not 23

[quote=R.D. Silverman;212576]Am I wasting my time trying to lead him through these problems?
Am I wasting my time trying to point out things he needs to learn
[B]before[/B] he attempts learning Calculus?

While Number Theory is irrelevant to the latter, it does teach how to
put together simple proofs. blob100 is failing in this regard.

Is he just another wannabee who is unwilling to put in the effort needed
to actually learn mathematics? There are already too many of those in this
forum.

If he [B]does[/B] want to continue, then I will help. But I need to see
the work he has put in.

The next set of exercizes will involve polynomial algebra, since it is clear
that he lacks skills in this area.[/quote]
One can never know which protégé will realize the Master's dream. The ratio is often very small. That does not mean the others are failures, only disappointments. Perhaps he will return after studying the texts you suggested, maybe even with a renewed drive.

[QUOTE=R.D. Silverman;212576]Am I wasting my time trying to lead him through these problems?
Am I wasting my time trying to point out things he needs to learn
[b]before[/b] he attempts learning Calculus?

While Number Theory is irrelevant to the latter, it does teach how to
put together simple proofs. blob100 is failing in this regard.

Is he just another wannabee who is unwilling to put in the effort needed
to actually learn mathematics? There are already too many of those in this
forum.

If he [b]does[/b] want to continue, then I will help. But I need to see
the work he has put in.

The next set of exercizes will involve polynomial algebra, since it is clear
that he lacks skills in this area.[/QUOTE]

Whether or not you are wasting your time depends on what result you would consider worthwhile. If blob100 is learning but not indicating it here (or indeed acknowledging your help), would that be a waste of time? If he's learning nothing, but others on this forum are benefiting, would that be a waste of time?

My sense is that even if blob100 never comes back, there are enough people on this forum who would profit from your problems/examples that there is value in continuing in this vein. Lots of people need a brush-up or simply enjoy the puzzle aspect.

As for blob100, remember that he's young and needs time to absorb new concepts. Be patient with him; what can it hurt?