Piecewise Functions of Chess By Ian Lawson, Ritesh Karsalia, Veronica Behman, Emily Choi, and Shubham Sabnani The King {3, 1≤x≤4}

K(x) = {5, 5≤x≤28}

{8, 29≤x≤64} The Rook R(x) = {14, 1≤x≤64} The Black Bishop {7, x=2, x=4, x=5, x=7, x=9, x=12, x=14, x=16, x=17, x=19, x=21, x=24, x=26, x=28}

B(x) = {9, x=30, x=32, x=34, x=36, x=38, x=40, x=42, x=44, x=46, x=48}

{11, x=50, x=52, x=54, x=56, x=58, x=60}

{13, x=62, x=64} The Knight The Queen {21, 1≤x≤28}

Q(x) = {23, 29≤x≤48}

{25, 49≤x≤60}

{27, 61≤x≤64} {2, 1≤x≤4}

{3, x=5, x=10, x=11, x=16, x=17, x=22, x=23, x=28}

Kn(x) = {4, 6≤x≤9, 12≤x≤15, 18≤x≤21, 24≤x≤27, x=29, x=34, x=39, x=44}

{6, 30≤x≤33, 35≤x≤38, 40≤x≤43, 45≤x≤48}

{8, 49≤x≤64} Relationships Rook+Bishop = Queen

R(x) = {14, 1≤x≤64}

{7, 1≤x≤28

B(x) = {9, 29≤x≤48

{11, 49≤x≤60

{13, 61≤x≤64

{21, 1≤x≤28

Q(x) = {23, 29≤x≤48

{25, 49≤x≤60

{27, 61≤x≤64

R(x)+B(x) = Q(x); Therefore:

{7, 1≤x≤28}

{14, 1≤x≤64} + {9, 29≤x≤48} =

{11, 49≤x≤60}

{13, 61≤x≤64} {21, 1≤x≤28}

{23, 29≤x≤48}

{25, 49≤x≤60}

{27, 61≤x≤64} Relationships Queen - Rook = Bishop

R(x) = {14, 1≤x≤64}

{7, 1≤x≤28

B(x) = {9, 29≤x≤48

{11, 49≤x≤60

{13, 61≤x≤64

{21, 1≤x≤28

Q(x) = {23, 29≤x≤48

{25, 49≤x≤60

{27, 61≤x≤64

Q(x) - R(x) = B(x); Therefore:

{7, 1≤x≤28}

{14, 1≤x≤64} = {9, 29≤x≤48}

{11, 49≤x≤60}

{13, 61≤x≤64} {21, 1≤x≤28

{23, 29≤x≤48 -

{25, 49≤x≤60

{27, 61≤x≤64 Relationships If you were to limit the Queen to move only one space in its normal directions = movement of the King

Since Rook + Bishop = Queen:

Rook and Bishop limited to move one space in its normal directions = movement of King

King and Knight:

In certain areas (i/e-center of board), both Knight and King can move to 8 different spaces to attack

When King is on square range: 29≤x≤64

AND when Knight is on square range: 49≤x≤64

In other areas, both the Knight and King can move to 3 spaces to attack

When King is on square range: 1≤x≤4

AND when Knight is on square range: x=5, x=10, x=11, x=16, x=17, x=22, x=23, x=28 Relationships Queen - Bishop = Rook

R(x) = {14, 1≤x≤64}

{7, 1≤x≤28

B(x) = {9, 29≤x≤48

{11, 49≤x≤60

{13, 61≤x≤64

{21, 1≤x≤28

Q(x) = {23, 29≤x≤48

{25, 49≤x≤60

{27, 61≤x≤64

Q(x) - B(x) = R(x); Therefore:

{7, 1≤x≤28}

{9, 29≤x≤48} = {14, 1≤x≤64}

{11, 49≤x≤60}

{13, 61≤x≤64} {21, 1≤x≤28

{23, 29≤x≤48 -

{25, 49≤x≤60

{27, 61≤x≤64 Thanks for Watching! R(x) = {14, 1≤x≤64} Data Review K(x) = {3, 1≤x≤4} K(x) = {5, 5≤x≤28}

Rook

{14}

Knight

{2,3,4,6,8}

King

{3,5,8}

Queen

{21,23,25,27}

Bishop (Both Black and White)

{7,9,11,13}

Combined

{2,3,4,5,6,7,8,9,11,13,14,21,23,25,27} Domain (For All Pieces Except Bishop): {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,8,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64}

Domain (White Bishop): {1,3,6,8,10,11,13,15,18,20,22,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61, 63}

Domain (Black Bishop): {2,4,5,7,9,12,14,16,17,19,21,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64} K(x) = {8, 29≤x≤64} Q(x) = {21, 1≤x≤28} Q(x) = {23, 29≤x≤48} Q(x) = {25, 49≤x≤60} Data Review Q(x) = {27, 61≤x≤64} Minimum: 2

Maximum: 27

Median: 9

Mean: 10.9

Mode: 3,7,8,9,11,13

Appears 2X each Relationships Black Bishop and White Bishop:

The Black Bishop and White Bishop can both move:

7 spaces

When Black Bishop is on square range: x=2, x=4, x=5, x=7, x=9, x=12, x=14, x=16, x=17, x=19, x=21, x=24, x=26, x=28

AND when White Bishop is on square range: x=1, x=3, x=6, x=8, x=10, x=11, x=13, x=15, x=18, x=20, x=22, x=23, x=25, x=27

9 spaces

When Black Bishop is on square range: x=30, x=32, x=34, x=36, x=38, x=40, x=42, x=44, x=46, x=48

AND when White Bishop is on square range: x=29, x=31, x=33, x=35, x=37, x=39, x=41, x=43, x=45, x=47

CONTINUED ON NEXT SLIDE Relationships Black Bishop and White Bishop(CONTINUED):

11 spaces

When Black Bishop is on square range: x=50, x=52, x=54, x=56, x=58, x=60

AND when White Bishop is on square range: x=49, x=51, x=53, x=55, x=57, x=59

13 spaces

When Black Bishop is on square range: x=62, x=64

AND when White Bishop is on square range: x=61, x=63 Full Data Set:

{2,3,3,4,5,6,7,7,8,8,9,9,11, 11,13,13,14,21,23,25,27} Kn(x) = {2, 1≤x≤4} Kn(x) = {3, x=5, x=10, x=11, x=16, x=17, x=22, x=23, x=28} Kn(x) = {4, 6≤x≤9, 12≤x≤15, 18≤x≤21, 24≤x≤27, x=29, x=34, x=39, x=44} Kn(x) = {6, 30≤x≤33, 35≤x≤38, 40≤x≤43, 45≤x≤48} Kn(x) = {8, 49≤x≤64} The White Bishop WB(x) = {7, x=1, x=3, x=6, x=8, x=10, x=11, x=13, x=15, x=18, x=20, x=22, x=23, x=26, x=28} WB(x) = {9, x=29, x=31, x=33, x=35, x=37, x=39, x=41, x=43, x=45, x=47} WB(x) = {11, x=49, x=51, x=53, x=55, x=57, x=59} WB(x) = {13, x=61, x=63} BB(x) = {7, x=2, x=4, x=5, x=7, x=9, x=12, x=14, x=16, x=17, x=19, x=21, x=24, x=26, x=28} BB(x) = {9, x=30, x=32, x=34, x=36, x=38, x=40, x=42, x=44, x=46, x=48} BB(x) = { 11, x=50, x=52, x=54, x=56, x=58, x=60} BB(x) = {13, x=62, x=64} {7, x=1, x=3, x=6, x=8, x=10, x=11, x=13, x=15, x=18, x=20, x=22, x=23, x=25, x=27

WB(x) = {9, x=29, x=31, x=33, x=35, x=37, x=39, x=41, x=43, x=45, x=47

{11, x=49, x=51, x=53, x=55, x=57, x=59

{13, x=61, x=63 Data Review Range:

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# Math Chessboard Project

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