-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
The Angel Problem
Eric FritzRamya Vemuganti
University of Wisconsin - Milwaukee
{fritz, vemugan2}@uwm.edu
May 4, 2014
1 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Setting and Players
k
I The game is played on an infinite chessboard
I The Angel has a single discrete position on the board
I During a turn, the Angel may move up to k units on `I During a
turn, the Devil may eat one square
I The Angel may not move to an eaten square
2 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Setting and Players
k
I The game is played on an infinite chessboard
I The Angel has a single discrete position on the board
I During a turn, the Angel may move up to k units on `I During a
turn, the Devil may eat one square
I The Angel may not move to an eaten square
3 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Setting and Players
k
I The game is played on an infinite chessboard
I The Angel has a single discrete position on the board
I During a turn, the Angel may move up to k units on `
I During a turn, the Devil may eat one square
I The Angel may not move to an eaten square
4 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Setting and Players
k
I The game is played on an infinite chessboard
I The Angel has a single discrete position on the board
I During a turn, the Angel may move up to k units on `I During a
turn, the Devil may eat one square
I The Angel may not move to an eaten square
5 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Setting and Players
k
I The game is played on an infinite chessboard
I The Angel has a single discrete position on the board
I During a turn, the Angel may move up to k units on `I During a
turn, the Devil may eat one square
I The Angel may not move to an eaten square
6 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Winning Condition
I The Devil wins if the Angel is trapped
I The Angel does not win (only survive)
7 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Winning Condition
I The Devil wins if the Angel is trapped
I The Angel does not win (only survive)
8 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Trapping (Slide 1)
12
I Is the Angel trapped?
I An Angel of power k = 1 is trappedI An Angel of power k 2 is
not
9 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Trapping (Slide 1)
1
2
I Is the Angel trapped?I An Angel of power k = 1 is trapped
I An Angel of power k 2 is not
10 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Trapping (Slide 1)
1
2
I Is the Angel trapped?I An Angel of power k = 1 is trappedI An
Angel of power k 2 is not
11 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Trapping (Slide 2)
I Is the Angel trapped?
I An Angel of power k = 1 is trapped (with certainty)I An Angel
of power k 2 is not
12 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Trapping (Slide 2)
I Is the Angel trapped?I An Angel of power k = 1 is trapped
(with certainty)
I An Angel of power k 2 is not
13 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Trapping (Slide 2)
I Is the Angel trapped?I An Angel of power k = 1 is trapped
(with certainty)I An Angel of power k 2 is not
14 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Trapping (Slide 3)
k
k
k k
I The Devil must build a k-wide moat on all four sides
15 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Trapping (Slide 3)
k
k
k k
I The Devil must build a k-wide moat on all four sides
16 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
The Question
I Can the Devil trap the k-Angel?
I Can the k-Angel survive indefinitely?
17 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - Partitioning
I The Devil can partition the board into halves
I The Devil must set up three blocks appropriately
18 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - Partitioning
I The Devil can partition the board into halves
I The Devil must set up three blocks appropriately
19 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - Partitioning
I The Devil can partition the board into halves
I The Devil must set up three blocks appropriately
20 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - Block Setup
a b
a
b
ab c
c
d
a
b
a
bc
d
a
b
c
c
c
c
de
a
b
c
de
a b bc d
d
b
b
ab
cd
ab
c
c
d
d
da
bc
d
1. Set a block directly ahead of the Angel on the wall
2. Place block bx when the Angel moves to bx
21 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - Block Setup (Case 1)
a b
a
b
ab c
c
d
a
b
a
bc
d
a
b
c
c
c
c
de
a
b
c
de
a b bc d
d
b
b
ab
cd
ab
c
c
d
d
da
bc
d
1. Set a block directly ahead of the Angel on the wall
2. Place block bx when the Angel moves to bx
22 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - Block Setup (Case 2)
a b
a
b
ab c
c
d
a
b
a
bc
d
a
b
c
c
c
c
de
a
b
c
de
a b bc d
d
b
b
ab
cd
ab
c
c
d
d
da
bc
d
1. Set a block directly ahead of the Angel on the wall
2. Place block bx when the Angel moves to bx
23 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - Block Setup (Case 3)
a b
a
b
ab c
c
d
a
b
a
bc
d
a
b
c
c
c
c
de
a
b
c
de
a b bc d
d
b
b
ab
cd
ab
c
c
d
d
da
bc
d
1. Set a block directly ahead of the Angel on the wall
2. Place block bx when the Angel moves to bx
24 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - Block Setup (Case 4)
a b
a
b
ab c
c
d
a
b
a
bc
d
a
b
c
c
c
c
de
a
b
c
de
a b bc d
d
b
b
ab
cd
ab
c
c
d
d
da
bc
d
1. Set a block directly ahead of the Angel on the wall
2. Place block bx when the Angel moves to bx
25 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - Block Setup (Case 5)
a b
a
b
ab c
c
d
a
b
a
bc
d
a
b
c
c
c
c
de
a
b
c
de
a b bc d
d
b
b
ab
cd
ab
c
c
d
d
da
bc
d
1. Set a block directly ahead of the Angel on the wall
2. Place block bx when the Angel moves to bx
26 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - The Trap
2c + 8
c
c
1. Remove 2c + 1 blocks in each corner
2. The Angel moves at most 8c + 4 blocks
3. Build walls as necessary using partition strategy
4. The Angel is trapped - remove remaining squares
27 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - The Trap
2c + 8c
c
1. Remove 2c + 1 blocks in each corner
2. The Angel moves at most 8c + 4 blocks
3. Build walls as necessary using partition strategy
4. The Angel is trapped - remove remaining squares
28 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - The Trap
2c + 8
c
c
1. Remove 2c + 1 blocks in each corner
2. The Angel moves at most 8c + 4 blocks
3. Build walls as necessary using partition strategy
4. The Angel is trapped - remove remaining squares
29 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching a King - The Trap
2c + 8
c
c
1. Remove 2c + 1 blocks in each corner
2. The Angel moves at most 8c + 4 blocks
3. Build walls as necessary using partition strategy
4. The Angel is trapped - remove remaining squares
30 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - Definition
I Can the Angel of power k 2 simply outrun the Devil?I Let a
Fool be an Angel with a strictly increasing
y -coordinate
31 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Necessary
P
h
k
2hk + 1
I Let P be the position of the Fool
I To win, the Devil must build a k-wide barrierI Barrier cannot
be completely constructed by the time it
is breached (even for k = 1)
32 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Necessary
P
h
k
2hk + 1
I Let P be the position of the FoolI To win, the Devil must
build a k-wide barrier
I Barrier cannot be completely constructed by the time itis
breached (even for k = 1)
33 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Necessary
P
h
k
2hk + 1
I Let P be the position of the FoolI To win, the Devil must
build a k-wide barrierI Barrier cannot be completely constructed by
the time it
is breached (even for k = 1)
34 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Trick
P
Qh
k
h/2
2hk + 1
I As the Angel moves from P to Q, remove h/(2k)squares from the
barrier uniformly
I When Q is reached, the Angel can only reach half ofthe
original barrier (which is h/(4k) squares complete)
I Repeat until the Angel has no where to go
35 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Trick
P
Qh
k
h/2
2hk + 1
I As the Angel moves from P to Q, remove h/(2k)squares from the
barrier uniformly
I When Q is reached, the Angel can only reach half ofthe
original barrier (which is h/(4k) squares complete)
I Repeat until the Angel has no where to go
36 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Trick
P
Qh
k
h/2
2hk + 1
I As the Angel moves from P to Q, remove h/(2k)squares from the
barrier uniformly
I When Q is reached, the Angel can only reach half ofthe
original barrier (which is h/(4k) squares complete)
I Repeat until the Angel has no where to go 37 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Proof
I Let R be the location of the Angel at the barrier
I Let BR be the barrier reachable from R
I |BR | = k(2k + 1)I We iterate n = log2h times
I We remove h/(2nk) squares from BR per iteration
ni=1
h
2nk k(2k + 1) (1)
I Works when h 2k2(2k+1)I k = 1, then h 8I k = 2, then h 1, 048,
576I k = 3, then h 9, 223, 372, 036, 854, 775, 808I Huge, but
finite
38 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Proof
I Let R be the location of the Angel at the barrier
I Let BR be the barrier reachable from R
I |BR | = k(2k + 1)I We iterate n = log2h times
I We remove h/(2nk) squares from BR per iteration
ni=1
h
2nk k(2k + 1) (1)
I Works when h 2k2(2k+1)I k = 1, then h 8I k = 2, then h 1, 048,
576I k = 3, then h 9, 223, 372, 036, 854, 775, 808I Huge, but
finite
39 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Proof
I Let R be the location of the Angel at the barrier
I Let BR be the barrier reachable from R
I |BR | = k(2k + 1)
I We iterate n = log2h times
I We remove h/(2nk) squares from BR per iteration
ni=1
h
2nk k(2k + 1) (1)
I Works when h 2k2(2k+1)I k = 1, then h 8I k = 2, then h 1, 048,
576I k = 3, then h 9, 223, 372, 036, 854, 775, 808I Huge, but
finite
40 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Proof
I Let R be the location of the Angel at the barrier
I Let BR be the barrier reachable from R
I |BR | = k(2k + 1)I We iterate n = log2h times
I We remove h/(2nk) squares from BR per iteration
ni=1
h
2nk k(2k + 1) (1)
I Works when h 2k2(2k+1)I k = 1, then h 8I k = 2, then h 1, 048,
576I k = 3, then h 9, 223, 372, 036, 854, 775, 808I Huge, but
finite
41 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Proof
I Let R be the location of the Angel at the barrier
I Let BR be the barrier reachable from R
I |BR | = k(2k + 1)I We iterate n = log2h times
I We remove h/(2nk) squares from BR per iteration
ni=1
h
2nk k(2k + 1) (1)
I Works when h 2k2(2k+1)I k = 1, then h 8I k = 2, then h 1, 048,
576I k = 3, then h 9, 223, 372, 036, 854, 775, 808I Huge, but
finite
42 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Proof
I Let R be the location of the Angel at the barrier
I Let BR be the barrier reachable from R
I |BR | = k(2k + 1)I We iterate n = log2h times
I We remove h/(2nk) squares from BR per iteration
ni=1
h
2nk k(2k + 1) (1)
I Works when h 2k2(2k+1)I k = 1, then h 8I k = 2, then h 1, 048,
576I k = 3, then h 9, 223, 372, 036, 854, 775, 808I Huge, but
finite
43 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Proof
I Let R be the location of the Angel at the barrier
I Let BR be the barrier reachable from R
I |BR | = k(2k + 1)I We iterate n = log2h times
I We remove h/(2nk) squares from BR per iteration
ni=1
h
2nk k(2k + 1) (1)
I Works when h 2k2(2k+1)
I k = 1, then h 8I k = 2, then h 1, 048, 576I k = 3, then h 9,
223, 372, 036, 854, 775, 808I Huge, but finite
44 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Proof
I Let R be the location of the Angel at the barrier
I Let BR be the barrier reachable from R
I |BR | = k(2k + 1)I We iterate n = log2h times
I We remove h/(2nk) squares from BR per iteration
ni=1
h
2nk k(2k + 1) (1)
I Works when h 2k2(2k+1)I k = 1, then h 8
I k = 2, then h 1, 048, 576I k = 3, then h 9, 223, 372, 036,
854, 775, 808I Huge, but finite
45 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Proof
I Let R be the location of the Angel at the barrier
I Let BR be the barrier reachable from R
I |BR | = k(2k + 1)I We iterate n = log2h times
I We remove h/(2nk) squares from BR per iteration
ni=1
h
2nk k(2k + 1) (1)
I Works when h 2k2(2k+1)I k = 1, then h 8I k = 2, then h 1, 048,
576
I k = 3, then h 9, 223, 372, 036, 854, 775, 808I Huge, but
finite
46 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Proof
I Let R be the location of the Angel at the barrier
I Let BR be the barrier reachable from R
I |BR | = k(2k + 1)I We iterate n = log2h times
I We remove h/(2nk) squares from BR per iteration
ni=1
h
2nk k(2k + 1) (1)
I Works when h 2k2(2k+1)I k = 1, then h 8I k = 2, then h 1, 048,
576I k = 3, then h 9, 223, 372, 036, 854, 775, 808
I Huge, but finite
47 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Catching the Fool - The Proof
I Let R be the location of the Angel at the barrier
I Let BR be the barrier reachable from R
I |BR | = k(2k + 1)I We iterate n = log2h times
I We remove h/(2nk) squares from BR per iteration
ni=1
h
2nk k(2k + 1) (1)
I Works when h 2k2(2k+1)I k = 1, then h 8I k = 2, then h 1, 048,
576I k = 3, then h 9, 223, 372, 036, 854, 775, 808I Huge, but
finite
48 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Curves
DefinitionA segment is an border between two adjacent
squares
DefinitionA curve is a sequence of segments. A curve is
I Continuous
I Infinite
I Directed from past to future
DefinitionA right square sr is a square to the right of a curve
segments when oriented past to future - a left square s` is
analogous
49 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Border Curves
DefinitionA border curve is a curve with some special
properties
I {s, s, s} 6 I if {s, s} , they trace opposite directionsI
partitions the board into a connected left set LS and
(possibly disconnected) right set RS of squares
I if s , s` LS and sr RSI if {s, s} , s`, sr LSI if s 6 , sr and
s` are in the same set
50 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Border Curves - Operations (Slide 1)
DefinitionA contraction operation removes dead ends from a
bordercurve by removing consecutive identical segments
I Contracting yields I is a valid border curveI is strictly
shorter than I and have identical left and right sets
51 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Border Curves - Operations (Slide 2)
DefinitionAn extension operation widens the left set of a border
curveby swallowing a right square rs if rs RS
I Contracting yields I is a valid border curveI is longer than
by two segmentsI has one more square in the left set than I has one
less square in the right set than
52 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
Border Curves - Parentage
DefinitionA border curve is a descendant of if there exists
afinite series of contraction and extension operations
whichtransform into
I The left set of is a superset of the left set of
53 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
The Angels Strategy - Overview
1. Use a border curve to partition the board into aforbidden
zone (left set) and a safe zone (right set)
2. Walk along the curve in the safe zone
3. Swallow traps constructed by the devil usingcontraction and
extension operations
4. . . .
5. Walk forever
54 / 63
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The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
The Angels Strategy - The Perch
DefinitionThe perch is a segment on the border curve, and the
Angelalways sits on the perchs right square. On each turn, theAngel
will update the perch two segments towards the future(from a to
b)
a
b
ab
a b
55 / 63
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The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
The Angels Strategy - Traps
a
b
a
b
ab
I Suppose the Devil is a big jerk and eats a square youwere
going to land on. What should the Angel do?
I Swallowing that jerks move sounds like a pretty goodidea
56 / 63
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The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
The Angels Strategy - Traps
a
b
a
b
ab
I Suppose the Devil is a big jerk and eats a square youwere
going to land on. What should the Angel do?
I Swallowing that jerks move sounds like a pretty goodidea
57 / 63
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The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
The Angels Strategy - The Update
I The Angel will modify the border curve if there is asufficient
threat
I The Angel will choose a new border curve such thatI The perch
existsI The past is unchangedI It is a descendant of the current
border curveI It maximizes swallowed threatsI It minimizes
increased length
a a
58 / 63
-
The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
The Angels Strategy - The Proof (Slide 1)
TheoremThe Angel cannot land on an eaten square in the safe
zone
Proof.Such an eaten square must be a right square of somesegment
s. If rs is eaten, then the Angel would prefer toswallow the square
over any other solution (it strictlyincreases the number of
swallowed threats and does so witha minimum increase of length)
59 / 63
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The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
The Angels Strategy - The Proof (Slide 2)
TheoremThe Angel cannot land in the forbidden zone
Proof.Kloster shows the Angel may be forced to change the
bordercurve to include forbidden squares reachable by the perch[3].
However, if this square exists, it exists one segment afterthe
current perch (and the Angel skips right over it in thenext
step)
60 / 63
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The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
The Angels Strategy - The Proof (Slide 3)
TheoremThe Angel will always land on a free square
Proof.The Angel always updates the perch and never lands on
aneaten square or a square in the forbidden square - nothingbut
free squares are left
Corollary
The Angel of power k 2 can move forever
61 / 63
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The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
The Angels Strategy - The Proof (Slide 3)
TheoremThe Angel will always land on a free square
Proof.The Angel always updates the perch and never lands on
aneaten square or a square in the forbidden square - nothingbut
free squares are left
Corollary
The Angel of power k 2 can move forever
62 / 63
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The AngelProblem
Eric FritzRamya Vemuganti
The Game
The Devil Wins
Catching a King
Catching a Fool
The Angel Wins
Border Curves
The Angels Strategy
John H. Conway, The angel problem, Games of NoChance 29 (1996),
312.
Peter Gacs, The angel wins.
Oddvar Kloster, A solution to the angel problem,Theoretical
Computer Science 389 (2007), no. 1,152161.
Martin Kutz, Conways angel in three dimensions,Theoret. Comp.
Sci. 349 (2005), no. 3, 443451.
Andras Mathe, The angel of power 2 wins,Combinatorics,
Probability and Computing 16 (2007),no. 3, 363374.
Johan Wastlund, A weaker winning angel, 2008.
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The GameThe Devil WinsCatching a KingCatching a Fool
The Angel WinsBorder CurvesThe Angel's Strategy
![A weaker winning angel - Chalmerswastlund/weakerAngel.pdf · The angel problem was introduced by Berlekamp, Conway and Guy [1] in 1982. An angel is a nite range chess piece. On an](https://img.dokumen.tips/doc/110x75/60098ae5ae7b0202c530c0f5/a-weaker-winning-angel-wastlundweakerangelpdf-the-angel-problem-was-introduced.jpg)
