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THE START

THE START. Generalized Checker Boards. 325 Goldstein AC PVL & 416A WP GC RFrank@pace.edu. Dr. Ronald I. Frank. ☺ 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]

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THE START

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  1. THE START Generalized Checker Boards 325 Goldstein AC PVL & 416A WP GC RFrank@pace.edu Dr. Ronald I. Frank N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

  2. ☺ 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 Table of Contents 1/2 N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

  3. ☺ 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 Table of Contents 2/2 N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

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

  5. Index List of A Regular Array, A N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

  6. Parity List of the Index List of A Regular Array, A N-D Checker Boards V. 5. (C) Ronald I. Frank 2013

  7. 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

  8. Boot Strapping anN-D Checker Board N-D Checker Boards V. 5. (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) N-D Checker Boards V. 5. (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. N-D Checker Boards V. 5. (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] N-D Checker Boards V. 5. (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. N-D Checker Boards V. 5. (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 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

  14. 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

  15. 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

  16. 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

  17. 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

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

  19. With 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

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

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

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

  23. 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

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

  26. 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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