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### Table of Contents 1/2

### Table of Contents 2/2

THE START

Generalized

Checker

Boards

325 Goldstein AC PVL

&

416A WP GC

Dr. Ronald I. Frank

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

☺ Header1[1]

☺ Table of Contents2[2-3]

☺ Definitions: Generalized & Array Shape List1[4]

☺ Definition of Index List of a Regular Array 1[5]

☺ Parity List of Index List of a Regular Array1[6]

☺ Parity of Parity (P of P) List of a Regular Array 1[7]

☺ Boot Strapping the N-D Checker Board 1[8]

☺ Some Definitions & Observations 2[9-10]

☺ Coloration of Diagonals 1[11]

☺ Effects of Changes on the SoI2[12-13]

☺ Effects of Changes on the SoISUMMARY 1[14]

14

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

☺ Example Problems 1 - 3 3[15-17]

☺ 1-D Checker Board w/Parity 1[18]

☺ 1-D Checker Board w/Parity & Parity of Parity 1[19]

☺ 2-D Checker Board w/Parity & Parity of Parity 1[20]

☺ 3-D Checker Board w/Parity & Parity of Parity 1[21]

☺ 3-D Checker Board w/Parity & Parity of Parity 1[22]

☺ Proof of the Structure of an N-D Checker Board1[23]

☺ Long Algorithm and Example1[24]

☺ Short Algorithm and Example1[25]

☺ End Slide. 1[26]

12

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Generalized in the sense of N-D & Non-Equilateral Arrays

Array Shape List

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Index List of A Regular Array, A

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Parity List of the Index List of A Regular Array, A

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Parity of the Parity Listof the Index List of A Regular Array, A

NOTE: All index list entries ODD means

NO (0) evens. (0) is an EVEN number.

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Boot Strapping anN-D Checker Board

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Some Definitions & Observations 1/3

Orthogonal Move (1-D):

Any one index change.

A change can be by an Even or Odd amount.

Even change amount, P of P unchanged

(O+E=O, O->O) (E+E=E, E-> E)

Odd change amount, P of P changed

(O+O = E, O->E) (E+O=O, E->O)

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Some Definitions & Observations 2/3

2. Diagonal Move (dims changed = 2 to N):

Any multiple index change by the same amount.

# dimensions changed can be Even or Odd

Changes Limited to 1.

A change by 1 changes an axis parity.

A change by 1 changes P of P.

An even # of changes by 1: P of P constant.

An odd # of changes by 1: P of P changes.

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Some Definitions & Observations 3/3

2. Diagonal Move (dims changed = 2 to N):

Even Dimensional Diagonals Have Constant Color

Odd Dimensional Diagonals Have Alternating Color [change by 1 => change P of P]

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Effects of Changes on the Sum of Indexes (SoI) 1/3

Orthogonal Change

Any orthogonal change changes Sum of Indices (SoI) by the amount of the change.

An even change does not change the parity of the SoI. An even change does not change P of P.

An odd change changes the SoI parity and the

P of P.

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Effects of Changes on the Sum of Indexes (SoI) 2/3

Diagonal Changes

An even # of dimensional changes by 1:

The SoI Parity is constant. P of P constant.

An odd # of dimensional changesby 1:

The SoI Parity Changes. P of P changes.

An even change to SoI: SoI Parity Constant &

P of P Constant

An odd change to SoI: SoI Parity Changes &

P of P Changes

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Effects of Changes on the Sum of Indexes (SoI) 3/3

SUMMARY

The parity of the SoIChanges => the P of P

=> The cell COLOR

OBSERVATION:

The SoIChange = SoI-N

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Example Problem 1

Example: [1, 1, 1, 1, 1] is Black, what color is [5, 2, 3, 8, 33]?

Short Algorithm:

46 is even => same color = Black.

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Example Problem 2

Example: [1, 1] is Black, what color is

[1, 2]? (Clearly Red)

Short Algorithm:

1 is odd => change color = Red.

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Example Problem 3

Example: [1, 1, 1, 1, 1, 1] is Black,

what color is [1, 1, 1, 1, 1, 1]?

(Clearly Black)

Short Algorithm:

0 is even=> NO change color = Black.

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Parity

{O} is cell index

Parity ODD

{E} is cell index

Parity EVEN

[i] is cell index

{B} is cell color Black

{R} is cell color Red

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Parity

& Parity

Of Parity

(0) or (1) Count of evens

(E) Or (O) Parity of count

Parity of count=Parity of Parity

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

PARITY OF PARITY

Parity of the Index List

( of # of EVEN Indices)

[i, j] is the cell index

{O} is cell index Parity ODD

{E} is cell index Parity EVEN

COLOR ~ PARITY OF PARITY

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

The * cells are the main (3-D) diagonal with alternating color .

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Proof of the Structure of andN-D Checker Board Proof

- Initial cell, all 1s, is E, so Black.
- Any orthogonal move by 1 changes P of P
- so changes color.
- An even # of orthogonal moves does not change color.
- An odd # of orthogonal moves changes color.

- Any even D diagonal move by 1s does not
- change P of P or color.
- Any odd D diagonal move by 1s changes P of P
- so changes color.

COLOR ~ PARITY OF PARITY [E=B, O=R]

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Long Algorithm for N-D Checker Board

Compute P of P of index list of element.

Even P of P = Black.

Odd P of P = Red.

Example: [5, 2, 3, 8, 33]~[O, E, O, E, O]

P of P = E, so it is Black.

Cell is Black. From [1, 1, 1, 1, 1] there are

46 orthogonal changes: [4, 1, 2, 7, 32] so there was an even # of orthogonal changes to an initial E. So it is Black.

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

Short Algorithm for N-D Checker Board

Subtract [1, 1, ..,1] from index list.

Sum result index list = # orthogonal changes

Or just sum original index list & - N.

Even = No changes = Black.

Odd = Changed = Red.

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

THE END

N-D Checker Boards

Mayor Ned McDodd of Whoville (Seuss)

N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

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