1. What is Dissertation Research Like? V.59.
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Generalized
Checker
Boards
THE START
325 Goldstein AC PVL
&
416A WP GC
Dr. Ronald I. Frank
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N-D Checker Boards V (C) Ronald I. Frank 2013
(C) Ronald I. Frank

2. What is Dissertation Research Like? V.59.
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Table of Contents 1/2
☺ Header 1[1]
☺ Table of Contents 2[2-3]
☺ Definitions: Generalized & Array Shape List 1[4]
☺ Definition of Index List of a Regular Array 1[5]
☺ Parity List of Index List of a Regular Array 1[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 SoI 2[12-13]
☺ Effects of Changes on the SoI SUMMARY 1[14] 14
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3. What is Dissertation Research Like? V.59.
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Table of Contents 2/2
☺ Example Problems [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 Example 1[24]
☺ Short Algorithm and Example 1[25]
☺ End Slide. 1[26] 12
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(C) Ronald I. Frank

4. Generalized in the sense of N-D & Non-Equilateral Arrays
Array Shape List
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5. Index List of A Regular Array, A
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N-D Checker Boards V (C) Ronald I. Frank 2013

6. Parity List of the Index List of A Regular Array, A
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N-D Checker Boards V (C) Ronald I. Frank 2013

7. Parity of the Parity List of the Index List of A Regular Array, A
NOTE: All index list entries ODD means
NO (0) evens. (0) is an EVEN number.
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8. Boot Strapping an N-D Checker Board
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N-D Checker Boards V (C) Ronald I. Frank 2013

9. 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)
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N-D Checker Boards V (C) Ronald I. Frank 2013

10. 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.
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N-D Checker Boards V (C) Ronald I. Frank 2013

11. 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]
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N-D Checker Boards V (C) Ronald I. Frank 2013

12. 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.
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N-D Checker Boards V (C) Ronald I. Frank 2013

13. 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 changes by 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
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N-D Checker Boards V (C) Ronald I. Frank 2013

14. Effects of Changes on the Sum of Indexes (SoI) 3/3
SUMMARY
The parity of the SoI Changes => the P of P
=> The cell COLOR
OBSERVATION:
The SoI Change = SoI-N
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N-D Checker Boards V (C) Ronald I. Frank 2013

15. 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.
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N-D Checker Boards V (C) Ronald I. Frank 2013

16. 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.
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N-D Checker Boards V (C) Ronald I. Frank 2013

17. 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.
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N-D Checker Boards V (C) Ronald I. Frank 2013

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

19. N-D Checker Boards V. 5. (C) Ronald I. Frank 2013
With
Parity
& Parity
Of Parity
(0) or (1) Count of evens
(E) Or (O) Parity of count
Parity of count=Parity of Parity
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N-D Checker Boards V (C) Ronald I. Frank 2013

20. N-D Checker Boards V. 5. (C) Ronald I. Frank 2013
2-D Checker Board w/P & P2
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
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21. N-D Checker Boards V. 5. (C) Ronald I. Frank 2013
3-D Checker Board w/P & P2
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N-D Checker Boards V (C) Ronald I. Frank 2013

22. N-D Checker Boards V. 5. (C) Ronald I. Frank 2013
3-D Checker Board w/P & P2
The * cells are the main (3-D) diagonal with alternating color .
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N-D Checker Boards V (C) Ronald I. Frank 2013

23. Proof of the Structure of and N-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
COLOR ~ PARITY OF PARITY [E=B, O=R]
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N-D Checker Boards V (C) Ronald I. Frank 2013

24. 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.
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N-D Checker Boards V (C) Ronald I. Frank 2013

25. 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.
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N-D Checker Boards V (C) Ronald I. Frank 2013

26. What is Dissertation Research Like? V.59.
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N-D Checker Boards
Mayor Ned McDodd of Whoville (Seuss)
THE END
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N-D Checker Boards V (C) Ronald I. Frank 2013
(C) Ronald I. Frank