Perhaps check it again Posted December 14, 2014 Report Share Posted December 14, 2014 Let n = a positive integer. Suppose you choose n distinct points on a line. The distance between any pair of points is irrelevant. A "curve" is either: 1) an individual point 2) a set of at least two points 3) an individual line segment 4) a set of at least two line segments 5) some combination of the above I will label the points with A, B, C, ..., Y, Z, AA, AB, ..., and so on as needed, consecutively from left to right, as we see them on the screen. If a line segment contains more than two points (recall that these are discrete), then they will be labeled with only their endpoints. Each "curve" in a list will be separated by a comma. The ordering in the list is irrelevant. The topic in this post involves the unordered listing and number of the total of discrete "curves" in two dimensions for n distinct points, where n is greater than or equal to one. Examples: -------------- n = 1 <----------------- A ----------------> List: {A} Total: 1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - n = 2 <-------------- A ----------- B -------------> List: {A}, {B}, {A, B}, {AB} Total: 4 Note: Here, you cannot have {A, AB}, {B, AB}, or {A, B, AB}, because neither point A nor point B is isolated once you identify AB as a line segment in the set. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - n = 3 <-------- A -------- B -------------- C -----------> List of points only: {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C} List of line segments only: {AB}, {BC}, {AC} List of points and line segments mixed: {A, BC}, {AB, C} Total: 12 Note: Here, you cannot have {B, AC}, because point B is not an isolated point. It is part of line segment AC, and AC has already been listed. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Puzzle: Compute the total number of "curves" for n = 4. *** Optional additional question: If you continue on getting totals for n = 5, 6, 7, 8, ..., you will get a sequence that might suggest a pattern to you. As with all sequences, there are an infinite number of pos- sibilities for a formula. However, there is a formula that I have in mind. Whatever formula you get, do not attempt to prove it. Just type the formula in your reply. Quote Link to comment Share on other sites More sharing options...

0 bonanova Posted December 15, 2014 Report Share Posted December 15, 2014 2^{n} Quote Link to comment Share on other sites More sharing options...

0 Pickett Posted December 15, 2014 Report Share Posted December 15, 2014 (edited) (2^{n}-1) + ((n^{2}-n)/2) + ((3^{n+1} - 2^{n+3} + 6 + (-1)^{n+1})/12) Validated out to n=5 so far...I'm pretty sure on the first two parts of that formula...it's the last/third that I'm a little questionable on...and if that's correct, here are the values up to 9: n=1: 1 n=2: 4 n=3: 12 n=4: 31 n=5: 81 n=6: 218 n=7: 610 n=8: 1753 n=9: 5127 Edited December 15, 2014 by Pickett Quote Link to comment Share on other sites More sharing options...

0 Perhaps check it again Posted December 15, 2014 Author Report Share Posted December 15, 2014 (edited) bonanova, the main puzzle asks for the answer for when n = 4. Maybe you kept it to yourself. Did you have the value for n = 4? Pickett, you didn't get a confirmation either way for the total for n = 4, before going into more calculations for the totals for larger n and then working on a formula. The values for n = 4, 5, ... are different. In the following spoiler, I show the listing and the total for n = 4: n = 4 <---------- A --------- B ------------ C ---------------- D -------------> list of points only: {A}, {B}, {C}, {D}, {A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D} {A, B, C}, {A, B, D}, {A, C, D}, {B, C, D}, {A, B, C, D} list of line segments only: {AB}, {BC}, {CD}, {AB, CD}, {AC}, {BD}, {AD} list of points and line segments mixed: {A, BC}, {A, CD}, {A, BD}, {B, CD}, {AB, C}, {AB, D}, {BC, D}, {AC, D}, {A, B, CD}, {A, BC, D}, {AB, C, D} Total: 33 Edited December 15, 2014 by Perhaps check it again Quote Link to comment Share on other sites More sharing options...

0 Perhaps check it again Posted December 15, 2014 Author Report Share Posted December 15, 2014 (edited) To follow up to my prior post, I will show the listing and the total for n = 5: n = 5: <------ A --------- B ---------- C -------------- D --------- E ------------> points only: {A}, {B}, {C}, {D}, {E}, {A, B}, {A, C}, {A, D}, {A, E}, {B, C}, {B, D}, {B, E}, {C, D}, {C, E}, {D, E}, {A, B, C}, {A, B, D}, {A, B, E}, {A, C, D}, {A, C, E}, {A, D, E}, {B, C, D}, {B, C, E}, {B, D, E}, {C, D, E}, {A, B, C, D}, {A, B, C, E}, {A, B, D, E}, {A, C, D, E}, {B, C, D, E}, {A, B, C, D, E} line segments only: {AB}, {BC}, {CD}, {DE}, {AB, CD}, {AB, DE}, {BC, DE}, {AC}, {BD}, {CE}, {AB, CE}, {AC, DE}, {AD}, {BE}, {AE} points and line segments mixed: {A, BC}, {A, CD}, {A, DE}, {B, CD}, {B, DE}, {AB, C}, {C, DE}, {AB, D}, {BC, D}, {AB, E}, {BC, E}, {CD, E}, {A, BD}, {A, CE}, {A, BC, DE}, {B, CE}, {AB, C, DE}, {AC, D}, {AC, E}, {BD, E}, {AB, CD, E}, {A, BE}, {AD, E}, {A, B, CD}, {A, B, DE}, {A, C, DE}, {A, BC, D}, {A, BC, E}, {A, CD, E}, {B, C, DE}, {B, CD, E}, {AB, C, D}, {AB, C, E}, {AB, D, E}, {BC, D, E}, {A, B, C, DE}, {A, B, CD, E}, {A, BC, D, E}, {AB, C, D, E}, {AC, D, E}, {A, BD, E}, {A, B, CE} Total: 88 Reminder: The total for n = 4 is actually: 33. Edited December 15, 2014 by Perhaps check it again Quote Link to comment Share on other sites More sharing options...

0 witzar Posted December 17, 2014 Report Share Posted December 17, 2014 Fibonacci(2n+1)-1 Quote Link to comment Share on other sites More sharing options...

0 DejMar Posted December 17, 2014 Report Share Posted December 17, 2014 "curve" sequence [Fibonacci (2n+1)-1 : n ∈ ℤ^{+} ] 1, 4, 12, 33, 88, ... n = 4 List of points only: {A}, {B}, {C}, {D}, {A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}, {A,B,C}, {A,B,D}, {A,C,D}, {B,C,D}, {A,B,C,D} List of line segments only: {AB}, {AC}, {AD}, {BC}, {BD}, {CD}, {AB,CD} List of points and line segments mixed: {AB,C}, {AB,D} {A,BC}, {BC,D}, {A,CD}, {B,CD}, {A,BD}, {AC,D), {A,B,CD}, {A,BC,D}, {AB,C,D} Quote Link to comment Share on other sites More sharing options...

0 Perhaps check it again Posted December 17, 2014 Author Report Share Posted December 17, 2014 witzar, I clicked yours as "Mark Solved," because it's posted before the next poster's, and you both show the same formula. Quote Link to comment Share on other sites More sharing options...

## Question

## Perhaps check it again

Let n = a positive integer.

Suppose you choose n distinct points on a line. The distance between

any pair of points is irrelevant.

A "curve" is either:

1) an individual point

2) a set of at least two points

3) an individual line segment

4) a set of at least two line segments

5) some combination of the above

I will label the points with A, B, C, ..., Y, Z, AA, AB, ..., and so on

as needed, consecutively from left to right, as we see them on

the screen.

If a line segment contains more than two points (recall that these are

discrete), then they will be labeled with only their endpoints. Each

"curve" in a list will be separated by a comma. The ordering in the

list is irrelevant.

The topic in this post involves the unordered listing and number ofthe total of discrete "curves" in two dimensions for n distinct points,where n is greater than or equal to one.Examples:

--------------

n = 1

<----------------- A ---------------->

List: {A}

Total: 1

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

n = 2

<-------------- A ----------- B ------------->

List: {A}, {B}, {A, B}, {AB}

Total: 4

Note: Here, you cannot have {A, AB}, {B, AB}, or {A, B, AB},because neither point A nor point B is isolated once youidentify AB as a line segment in the set.- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

n = 3

<-------- A -------- B -------------- C ----------->

List of points only: {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C}

List of line segments only: {AB}, {BC}, {AC}

List of points and line segments mixed: {A, BC}, {AB, C}

Total: 12

Note: Here, you cannot have {B, AC}, because point B is notan isolated point. It is part of line segment AC, and AC hasalready been listed.- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Puzzle: Compute the total number of "curves" for n = 4.*** Optional additional question: If you continue on getting

totals for n = 5, 6, 7, 8, ..., you will get a sequence that might

suggest a pattern to you.

As with all sequences, there are an infinite number of pos-

sibilities for a formula. However, there is a formula that I have

in mind. Whatever formula you get,

do notattempt toprove it. Just type the formula in your reply.

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