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Guest Message by DevFuse
 

Perhaps check it again

Member Since 02 Dec 2013
Offline Last Active Yesterday, 07:32 PM
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Topics I've Started

Number of discrete "curves" in two dimensions

14 December 2014 - 08:56 PM

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.

 


Three 8's to make 24 (my modified version)

25 November 2014 - 05:40 AM

In each expression:

 

Use exactly three 8's.

 

Use at most three square root symbols.

 

Use at most two factorial signs.

 

You can use some combination of addition, subtraction/negation, multiplication,

and/or division signs (+, -, *, /).

 

You can use parentheses.  One pair should be sufficient.

 

Concatenation is not allowed.

 

No other numbers, operations, or characters are allowed.

 

To start off, I am giving you this expression for a solution:

 

8 + 8 + 8

 

 

> > > Try to come up with as many as five other expressions for solutions. < < <


Six sixes to make 1,000 (modified version)

25 October 2014 - 06:26 PM

Form as many horizontal-style arithmetical expressions as possible that equal 1,000.

 

In order to limit the total number of possible solutions, the following rules of mine are in place:

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

 

Use exactly six sixes.

 

Use no more than one pair of parentheses.

 

Use no more than one "/" symbol.

 

Use no more than two decimal points.

 

Use no more than one minus (subtraction) sign.

 

Concatenation is allowed.

 

No other symbols/characters/operations are allowed.

 

 


Total number of equilateral triangles

15 August 2014 - 06:08 PM

*             *              *             *    

  

       *              *             *             *

 

               *             *             *            *

 

                      *              *

 

              *              *

 

 

 

The 16 points above lie in a plane on an equilateral triangular lattice.

 

Certain sets of three points of the figure correspond to the vertices of equilateral triangles.

 

Suppose you were to form all of the equilateral triangles possible, such that, for any given

 

equilateral triangle, it must have its three vertices coincide with three of the 16 points.

 

 

How many total equilateral triangles can be formed this way in the figure?

 

 


Minimizing the number of terms of the square of a polynomial

23 July 2014 - 05:42 AM

Suppose a, b, c, and d  belong to the set of nonzero integers.

 

Let  P(x)  =  (x+ ax3 + bx2 + cx + d)2.

 

 

Determine one of the sets of values of a, b, c, and d, such that when P(x) is

expanded into individual terms of an 8th degree polynomial, that polynomial

will have the fewest number of nonzero terms possible.

 

 

Bonus:

Write P(x) in its expanded form.