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Research Article
JPD-Coloring of the Monohedral Tiling for the Plane
S. A. El-Shehawy1and M. Basher2
1Department of Mathematics, Faculty of Science, Menoua University, Shebin El-Kom 32511, Egypt
2Department of Mathematics and Computer Science, Faculty of Science, Suez University, Suez 43518, Egypt
Correspondence should be addressed to S. A. El-Shehawy; shshehawy@yahoo.com
Received November ; Revised January ; Accepted January
Academic Editor: Gaston Mandata N’gu´
er´
ekata
Copyright © S. A. El-Shehawy and M. Basher. is is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We introduce a denition of coloring by using joint probability distribution “JPD-coloring” for the plane which is equipped by
tiling I.WeinvestigatetheJPD-coloringofther-monohedral tiling for the plane by mutually congruent regular convex polygons
which are equilateral triangles at r=orsquaresatr= or regular hexagons at r= . Moreover we present some computations for
determining the corresponding probability values which are used to color in the three studied cases by MAPLE-Package.
1. Introduction
A tiling of the plane is a family of sets—called tiles—that
cover the plane without gaps or overlaps. Tilings are known
as tessellations or pavings; they have appeared in human
activities since prehistoric times. eir mathematical theory
is mostly elementary, but nevertheless it contains a rich
supply of interesting problems at various levels. e same
is true for the special class of tiling called tiling by regular
polygons []. e notions of tiling by regular polygons in the
plane are introduced by Gr¨
unbaum and Shephard in []. For
more details see [–].
Denition 1 (see [,]). A tiling of the plane is a collection
I={
𝑠: = 1,2,3,...}of closed topological discs (tiles)
which covers the Euclidean plane 2and is such that the
interiors of its tiles are disjoint.
More explicitly, the union of the sets 1,
2,
3,...,tiles,
is to be the whole plane, and the interiors of the sets 𝑠are
pairwise disjoint. We will restrict our interest to the case
where each tile is a topological disc; that is, it has a boundary
that is a single simple closed curve. Two tiles are called
adjacent if they have an edge in common, and then each is
called an adjacent of the other. Two distinct edges are adjacent
if they have a common endpoint. e word incident is used to
denote the relation of a tile to each of its edges or vertices and
also of an edge to each of its endpoints. Two tilings I1and
I2are congruent if I1may be made to coincide with I2by
a rigid motion of the plane, possibly including reection [].
Denition 2 (see [,]). A tiling is called edge-to-edge if
therelationofanytwotilesisoneofthefollowingthree
possibilities:
(a) they are disjoint,
(b) they have precisely one common point which is a
vertex of each of the polygons,
(c) they share a segment that is an edge of each of the two
polygons.
Denition 3 (see []). A regular tiling Iwill be called
-monohedral tiling if every tile in Iis congruent to one
xed set .esetis called the prototile of I,whereis
the number of vertices for each tile.
Henceapointoftheplanethatisavertexofoneofthe
polygons in an edge-to-edge tiling is also a vertex of every
other polygon to which it belongs and it is called a vertex
of the tiling. Similarly, each edge of one of the polygons,
regular tiling, is an edge of precisely one other polygon and
it is called an edge of the tiling. It should be noted that the
only possible edge-to-edge tilings of the plane by mutually
congruent regular convex polygons are the three regular
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2015, Article ID 258436, 8 pages
http://dx.doi.org/10.1155/2015/258436
Abstract and Applied Analysis
tilings by equilateral triangles, by squares, or by regular
hexagons.
e notions of the coloring of the monohedral tiling for
the plane have been introduced by Gr¨
unbaum and Shephard
[]. e -coloring and the perfect -coloring for the plane
equipped by the r-monohedral tiling Ihave been introduced
by Basher [].
In this paper we redene the coloring of the r-
monohedral tiling for the plane by using joint probability
distribution (JPD). We aim to investigate the three regular
tilings by equilateral triangles, squares, and regular hexagons
using JPD. ese three tilings are shown graphically and
computationally. Some computations by MAPLE-Package
to determine the probability values (vertices) for the three
studied tilings are presented. We introduce this alternative
techniquetoexpandandupdatethecoloringtechniqueto
implement tiling according to a probabilistic approach. e
probability values refer to percentages in the coloring process
and this contributes to convert the coloring process into a
computational process in the future.
roughout this paper we consider two discrete random
variables and with a joint probability mass function
𝑋,𝑌(,)=(=,=)which satises
() 𝑋,𝑌(𝑖,𝑗)≥0, for all points (𝑖,𝑗)in the range of
(,),
() ∑𝑥𝑖∑𝑦𝑗𝑋,𝑌(𝑖,𝑗)=1.
e value 𝑋,𝑌(𝑖,𝑗)is usually written as 𝑖𝑗 for each point
(𝑖,𝑗)in the range of (,);see[,]. In this paper we
consider 𝑖𝑗 having equal denominators (the large common
multiplication of the denominators of the probabilities) “”.
2. JPD-Coloring of the Regular Tilings
In this section we will investigate the coloring of -
monohedral tiling.
Let 2be equipped by -monohedral tiling I,andlet
(I)be the set of all vertices of the tiling. Here, we consider
the probability values 𝑖𝑗 to represent the coloring of the set
(I)as in the following denition where
𝑖𝑗 ==
𝑖,=𝑗=
𝑋,𝑌 𝑖,𝑗
=
nonzero value with equal denominators ,
if 𝑖,𝑗∈(I),
zero value,if 𝑖,𝑗∉(I).
()
For each , a family of a corresponding JPD is denoted by
“(JPD)”.
Denition 4. AcoloringofthetilingIis a partition of (I)
into color-classes such that
(i) each color represents a probability value 𝑖𝑗,
(ii) the dierent colors appear on adjacent vertices,
(iii) for each prototile 𝑠∈Ithere exists a corresponding
JPD ∈(JPD).
p31
p22
p11
F : e used equilateral triangle with the corresponding JPD
values.
Denition 5. e set of tiles colored by (JPD)is called the
mesh of tiling.
From the above denition the tiling Ican be colored by
horizontal or vertical translation of the mesh.
Denition 6. e order ((JPD)) of (JPD)is the number
of JPDs which construct the mesh.
2.1. JPD-Coloring of the 3-Monohedral Tiling. Here, we con-
sider the JPD 𝑋,𝑌(𝑖,𝑗)=
𝑖𝑗, = 1,2,3;=1,2,with
dierent nonzero values of 11,31,22 and zero values of
21,12,32,where𝑖𝑗 is with equal denominators .eused
equilateral triangle is illustrated in Figure .
eorem 7. If the plane is equipped by 3-monohedral tiling,
then the number of colors “”equals3where ≥6.If<6,
then the tiling cannot be colored.
Proof. Let 2be equipped by equilateral triangle tiling. We
give the proof in two cases.
Case 1. If <6, then the tiling cannot be colored (i.e., the
number of colors equals)becausewecannotndthree
dierent probability values (JPD) to color the three vertices
of the mentioned equilateral triangle tiling (say, at =5the
probability values are {1/5,2/5,2/5},at=4the probability
values are {1/4,1/4,2/4},andat=3the probability values
are {1/3,1/3,1/3}).
Case 2. If ≥6,thenforeachthe number of colors equals
and we can nd three dierent probability values (JPD),
which satised the condition (ii) in Denition ,tocolorthe
three vertices of the mentioned equilateral triangle tiling. We
can nd the following:
(i) at =6the dierent probability values are only
{1/6,2/6,3/6} (see Figure (a)),
(ii) at =7the dierent probability values are only
{1/7,2/7,4/7},
(iii) at ≥8there are more than one JPD with three
dierent probability values (Figures (b) and (c)).
Abstract and Applied Analysis
1/6
2/6 3/6
(a)
5/8
1/8
2/8
(b)
4/8
1/8
3/8
(c)
F : Some JPD values at =6and =8to color the three vertices of the equilateral triangle tiling.
p12 p22
p11 p21
F : e used square with the corresponding JPD values.
Remark 8. ((JPD)) of the -monohedral tiling equals .
2.2. JPD-Coloring of the 4-Monohedral Tiling. Here,wecon-
sider the following JPD: 𝑋,𝑌(𝑖,𝑗)=
𝑖𝑗,=1,2;=1,2,
with the assumptions 11 =
21,11 =
12,22 =
21,
22 =
12 and where 𝑖𝑗 is with equal denominators .e
used square is illustrated in Figure .
eorem 9. If the plane is equipped by the 4-monohedral
tiling, then the greatest number of colors “”isgivenasfollows:
=
− 4, ≥6 ,
3, = 7,
− 5, ≥9 ,
()
and if <6, then the tiling cannot be colored.
Proof. Let 2be equipped by square tiling. We give the proof
in four cases.
Case 1.If<6, then the tiling cannot be colored (i.e., the
greatest number of colors equals ) because we cannot nd
four probability values (JPD) to color the four vertices of the
mentioned square tiling which satised (ii) in Denition
1/7
1/7
2/7
3/7
F : e JPD values at =7to color the four vertices of the
square triangle tiling.
for coloring (say, at =5the probability values are
{1/5,1/5,2/5,1/5} and at the minimum value =4the
probability values are {1/4,1/4,1/4,1/4}).
Case 2. For =7,takethecorrespondingprobability
values of two adjacent vertices 1/7 and 2/7.en,therest
corresponding probability values must be 1/7and 3/7 which
satised the condition (ii) in Denition .So,thegreatest
number of colors “” equals and the dierent probability
values are only {1/7,2/7,3/7},seeFigure .
Case 3.For≥6and is even, take the corresponding
probability values of two adjacent vertices 1/ and 2/.
en, the rest corresponding probability value is ( − 3)/.
is probability value must be distributed on the other two
vertices as follows: {(−4)/,1/},{(−5)/,2/},...,{(−
)/,(−3)/},whereis an integer number and 4≤≤−1.
is implies that the available total probability values to
obtain the mesh are {1/,2/,(−4)/,(−5)/,...,(−)/},
where 4≤≤−1. To avoid the repetition, the last two
probability values ( − 3)/,( − 1)/ are excluded. In this
case we obtain the following.
(i) For =6, the available total probability values
{1/6,2/6,(6 − 4)/6,(6 − 5)/6} are equivalent to
{1/6,2/6},andthegreatestnumberofcolorsequals
(i.e.,−4).
Abstract and Applied Analysis
T : e relation between ,,((JPD)), and the (JPD) of the square tiling at =13.
(F(JPD)) Example of the corresponding (JPD)
{{5/13,2/13,1/13,5/13}}
{{2/13,1/13,1/13,9/13},{1/13,2/13,9/13,1/13},{2/13,8/13,1/13, 2/13}}
{{2/13,1/13,1/13,9/13},{1/13,2/13,9/13,1/13},{2/13,3/13,1/13, 7/13}}
{{2/13,1/13,1/13,9/13},{1/13,2/13,9/13,1/13},{2/13,8/13,1/13, 2/13},
{8/13,2/13,2/13,1/13},{2/13,3/13,1/13,7/13}}
{{2/13,1/13,1/13,9/13},{1/13,2/13,9/13,1/13},{2/13,3/13,1/13, 7/13},
{3/13,2/13,7/13,1/13},{2/13,6/13,1/13,4/13}}
{{2/13,1/13,1/13,9/13},{1/13,2/13,9/13,1/13},{2/13,8/13,1/13, 2/13},
{8/13,2/13,2/13,1/13},{2/13,3/13,1/13,7/13},{3/13,2/13,7/13, 1/13},{2/13,4/13, 1/13,6/13}}
6/14
3/14
8/14
10/145/144/14
7/142/14 2/14 9/14
2/14
2/14
1/14
2/14
1/141/14
1/142/14
1/141/14
(a)
2/13
2/13
2/13 6/13
4/13
8/131/13
1/13 1/139/13
2/13
1/13
2/13
1/13
7/13
3/13
(b)
F : Some JPD values at =13and =14to color the four vertices of the square tiling.
(ii) For ≥8and is even, the available total probability
values (without repetition) are {1/,2/,(−4)/,(−
5)/,...,( − )/},where4≤≤−3.e
greatest number of colors equals (−4); for example,
at =8, the available total probability values
are {1/8,2/8,3/8,4/8},andsoequals (i.e., −
4); at =14, the available total probability val-
ues are {1/14,2/14,3/14,4/14,5/14,6/14,7/14,8/14,
9/14,10/14},andsothegreatestnumberofcolors
equals (i.e., −4); see Figure (a).
Case 4. For ≥9and is odd, the proof is similar to
Case . Since is odd, then in this case the rest corresponding
probability value ( − 3)/ has even value of its numerator.
en, this probability value can be distributed on the other
two vertices by two equal probability values “((− 3)/2)/”.
As in Case , the last two probability values (−3)/,(−
1)/ and (( − 3)/2)/ are excluded. In this case we obtain
that, for ≥9and is odd, the available total probability
values are {1/,2/,(−4)/,(−5)/,...,(((−3)/2)+1)/,
((( − 3)/2) − 1)/,...,(−)/},where4≤≤−3.e
greatest number of colors equals (− 5): for example,
(i) at =9,theavailabletotaldistinctprobabilityvalues
are {1/9,2/9,4/9,5/9} and the greatest number of
colors equals (i.e., −5);
(ii) at =13,thetotaldistinctprobabilityvaluesare
{1/13,2/13,3/13,4/13,6/13,7/13,8/13,9/13} and
thegreatestnumberofcolorsequals (i.e., −5);
see Figure (b).
p23
p12
p21 p31
p42
p33
F : e used regular hexagon with the corresponding JPD
values.
ere are a relation between ,,((JPD)) and the
corresponding (JPD). Tables and show this relation:
(i) for =13,seeTab l e ;
(ii) for =14,seeTab l e .
Corollary 10. e smallest number of colors for square tiling
is 2 if is even and 3 if is odd.
2.3. JPD-Coloring of the 6-Monohedral Tiling. Here, we con-
sider the JPD: 𝑋,𝑌(𝑖,𝑗)=
𝑖𝑗, = 1,2,3,4; = 1, 2,3,with
the assumptions 21 =
31,31 =
42,42 =
33,33 =
23,
23 =
12,12 =
21 (where 𝑖𝑗 is with equal denominators )
and zero values of 11,41,22 32,13,43.eusedregular
hexagon is illustrated in Figure .
Abstract and Applied Analysis
T : e relation between ,,((JPD)), and the (JPD) of the square tiling at =14.
((JPD)) Example of the corresponding (JPD)
{{4/14,3/14,3/14,4/14}}
{{8/14,1/14,1/14,4/14}}
{{2/14,7/14,1/14,4/14}}
{{3/14,1/14,1/14,9/14},{1/14,3/14, 9/14,1/14},{3/14, 4/14,1/14,6/14}}
{{6/14,1/14,1/14,6/14},{1/14,5/14,6/14,2/14},{5/14,1/14,2/14, 6/14},
{1/14,4/14,6/14,3/14}}
{{2/14,7/14,1/14,4/14},{7/14,2/14,4/14,1/14},{2/14,6/14,1/14, 5/14},{6/14,2/14, 5/14,1/14},
{2/14,1/14,1/14,10/14}}
{{2/14,7/14,1/14,4/14},{7/14,2/14,4/14,1/14},{2/14,6/14,1/14, 5/14},{6/14,2/14, 5/14,1/14},
{2/14,8/14,1/14,3/14}}
{{2/14,7/14,1/14,4/14},{7/14,2/14,4/14,1/14},{2/14,6/14,1/14, 5/14},{6/14,2/14, 5/14,1/14},
{2/14,1/14,1/14,10/14},{1/14,2/14,1/14,10/14},{2/14,8/14,1/14, 3/14}}
{{2/14,7/14,1/14,4/14},{7/14,2/14,4/14,1/14},{2/14,6/14,1/14, 5/14},{6/14,2/14, 5/14,1/14},
{2/14,1/14,1/14,10/14},{1/14,2/14,10/14,1/14},{2/14,9/14,1/14, 2/14},{9/14,2/14, 2/14,1/14},
{2/14,8/14,1/14,3/14}}
1/18
1/18
1/18
1/18
1/18
1/18
1/188/18
1/18
1/18
1/18 1/18 1/18
1/18
1/18
1/18
1/18
1/18
1/181/184/18
4/18 4/18
4/18
4/18
4/18
4/185/18
5/185/18
7/18 7/18
7/18
7/18
6/18
5/18
1/18 5/18
6/18
6/186/18
6/18
6/18
4/18
1/18 1/18
1/18
1/18
8/18
1/18
3/183/18
3/18
3/18
3/183/183/18
3/18
3/18 3/18 3/18
3/18 9/18
3/18
1/18
6/18
4/18
2/18
2/18
2/18 2/18
2/18
2/18
2/18 2/18
2/18
2/18
2/18
2/18
2/18
2/18
2/18 2/18
2/18 2/18
2/18
2/18
2/18
2/18
2/18
2/18
8/18
2/18
2/18
10/18
11/18
3/18
3/18
F : e JPD values at =18to color the six vertices of the regular hexagon tiling.
eorem 11. If the plane is equipped by 6-monohedral tiling,
then the greatest number of colors “”isgivenas=−7where
≥9.If<9, then the tiling cannot be colored.
Proof. Let 2be equipped by hexagon tiling. e proof can
be given as follows.
Case 1. If <9, then the tiling cannot be colored (i.e., the
greatest number of colors equals ) because we cannot nd
six probability values (JPD), which satised the condition (ii)
in Denition , to color the six vertices of the mentioned
hexagon tiling (say, at =8the available probability
values are {1/8,2/8,1/8,2/8,1/8,1/8},at=7the available
probability values are {1/7,2/7,1/7,1/7,1/7,1/7},andatthe
smallest value =6the available probability values are
{1/6,1/6,1/6,1/6,1/6,1/6}).
Case 2. For ≥9, take the corresponding probability value
of a vertex ( − )/ where ≥7becauseitisimpossibleto
take the value of less than . en, the rest corresponding
probability value is /.isprobabilityvaluemustbe
distributed on the other ve vertices under consideration of
Abstract and Applied Analysis
1/25
1/25
4/25
3/25
3/25
2/25
6/25
2/25
4/25
4/25
3/24 3/25 2/25
2/252/25
2/25
2/25
7/25 3/25 3/25
2/25
6/25
2/25
1/25 8/25
5/25
5/25
5/25
7/25
2/25
3/254/25
4/256/25
6/25
9/25
5/25 3/25 8/25
7/25
4/25
8/25
3/252/25 6/25 4/25
3/25
2/25
3/25 3/25
1/25
1/251/25
1/251/25 1/25
3/25
3/25
3/25
3/25
3/25
5/25
5/25
5/25
6/25
3/25
1/25
1/25
1/25
1/25
1/25
1/25
1/251/25
1/251/25
1/25
1/25
1/25
1/25
1/25
1/253/25
3/25
6/25
7/25
1/25
1/25 1/25
1/25
1/25
1/25
1/25
1/25
1/251/25
1/25
1/25
1/25
2/25
2/25 2/25
2/25 2/25
2/25
2/25
2/25
2/25
2/252/25
2/25
2/252/255/25
4/25
2/25
2/252/253/25
3/25
4/25
4/25
4/25
1/25
1/25
1/25
10/25
13/25
16/25
12/24
14/25
10/25
10/25
10/25 17/25
12/25
13/25
12/25
11/25
12/25
15/25
18/25
13/25 5/25
F : e JPD values at =25to color the six vertices of the regular hexagon tiling.
the conditions in Denition for an integer number ≥
7. is implies that the available total probability values to
obtain the mesh under consideration of the conditions in
Denition are {1/,2/,3/,...,(−−1)/},where7≤
≤−2. Avoiding the repetition, we obtain the following:
(i) for =9,thetotalprobabilityvaluesare{/, /}and
equals ,
(ii) for =18,thetotalprobabilityvaluesare{/, /,
/, /, /, /, /, /, /, /, /}and
equals (see Figure ),
(iii) for =25,thetotalprobabilityvaluesare{/, /,
/, /, /, /, /, /, /, /, /,
/, /, /, /, /, /, /}and
equals (see Figure ).
In this case, the greatest number of colors equals −7.
Corollary 12. e relation between and can be shown in
Tab l e 3.
T : e relation between and of the regular hexagon tiling.
6 +3, ≥ 1
6 +3, 6+ 4,6 +5, ≥ 1
6,6 + 1,6 +2, ≥ 2
6,6 + 1,6 +2,6 +3, 6+ 4,6 +5, ≥ 2
6 +5, ≥ 2
6,6 + 1,6 +2,6 + 3,6 +4, ≥ 3
6 +3, 6+ 4,6 +5, ≥ 3
6,6 + 1,6 +2, ≥ 4
Appendix
As a MAPLE programming guide see [].
Determination of the Probability Values by Using the MAPLE
Program
Case A.1 (JPD-coloring of the -monohedral tiling). See
Box .
Case A.2 (JPD-coloring of the -monohedral tiling). See
Box .
Abstract and Applied Analysis
>restart: with(linalg):
Probability Values:=proc(n::posint,i::posint,j::posint)
local A,b,T,r,v,C; global PV;
A[T]:= matrix([[1,1,1]]): b:= vector([1]): C:=sum(V[w],w=1..2):
linsolve(A[T],b,‘r): r: linsolve(A[T],b,‘r,v):
PV[T[n,i,j]]:=subs(v[1]=i/n,v[2]=j/n,v[3]=1-C,%):
end proc:
n:=3:i:=1:j:=1: PV[T[n,i,j]]:=Probability Values(n,i,j);
n:=4:i:=1:j:=2: PV[T[n,i,j]]:=Probability Values(n,i,j);
n:=5:i:=1:j:=2: PV[T[n,i,j]]:=Probability Values(n,i,j);
n:=6:i:=1:j:=2: PV[T[n,i,j]]:=Probability Values(n,i,j);
n:=7:i:=1:j:=2: PV[T[n,i,j]]:=Probability Values(n,i,j);
n:=8:i:=1:j:=2: PV[T[n,i,j]]:=Probability Values(n,i,j);
n:=8:i:=1:j:=3: PV[T[n,i,j]]:=Probability Values(n,i,j);
PV𝑇3,1,1 := 1
3,1
3,1
3,PV
𝑇4,1,2 := 1
2,1
4,1
4,PV
𝑇5,1,2 := 2
5,1
5,2
5,PV
𝑇6,1,2 := 1
2,1
6,1
3,
PV𝑇7,1,2 := 2
7,4
7,1
7,PV
𝑇8,1,2 := 5
8,1
8,1
4,PV
𝑇8,1,3 := 1
2,1
8,3
8
B
>restart: with(linalg):
Probability Values:=proc(n::posint,i::posint,j::posint,k::posint)
local A,b,F,r,v,C; global PV;
A[F]:=matrix([[1,1,1,1]]): b:=vector([1]): C:=sum(V[w],w=1..3):
linsolve(A[F],b,‘r): r: linsolve(A[F],b,‘r,v):
PV[S[n,i,j,k]]:=subs(v[1]=i/n,v[2]=j/n,v[3]=k/n,v[4]=1-C,%):
end proc:
n:=4: i:=1:j:=1:k:=1: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=5: i:=1:j:=2:k:=1: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=6: i:=1:j:=2:k:=1: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=7: i:=1:j:=2:k:=1: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=8: i:=1:j:=2:k:=1: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=8: i:=1:j:=2:k:=3: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=8: i:=1:j:=3:k:=1: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=9: i:=1:j:=2:k:=1: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=9: i:=1:j:=4:k:=3: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=13:i:=1:j:=2:k:=1: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=13:i:=1:j:=3:k:=4: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=13:i:=1:j:=1:k:=6: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=13:i:=1:j:=1:k:=7: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=13:i:=1:j:=1:k:=8: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=14:i:=1:j:=1:k:=2: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=14:i:=2:j:=3:k:=4: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=14:i:=1:j:=5:k:=6: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=14:i:=1:j:=2:k:=7: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=14:i:=1:j:=2:k:=8: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
n:=14:i:=2:j:=2:k:=9: PV[F[n,i,j,k]]:=Probability Values(n,i,j,k);
PV𝐹4,1,1,1 := 1
4,1
4,1
4,1
4,PV
𝐹5,1,2,1 := 1
5,1
5,2
5,1
5,PV
𝐹6,1,2,1 := 1
6,1
3,1
6,1
3,PV
𝐹7,1,2,1 := 3
7,1
7,2
7,1
7,
PV𝐹8,1,2,1 := 1
8,1
4,1
2,1
8,PV
𝐹8,1,2,3 := 1
4,1
8,1
4,3
8,PV
𝐹8,1,3,1 := 3
8,1
8,3
8,1
8,PV
𝐹9,1,2,1 := 5
9,1
9,2
9,1
9,
PV𝐹9,1,4,3 := 1
3,4
9,1
9,1
9,PV
𝐹13,1,2,1 := 9
13,1
13,1
13,2
13,PV
𝐹13,1,3,4 := 5
13,4
13,3
13,1
13,PV
𝐹13,1,1,6 := 1
13,6
13,5
13,1
13,
PV𝐹13,1,1,7 := 1
13,7
13,4
13,1
13,PV
𝐹13,1,1,8 := 8
13,1
13,1
13,3
13,PV
𝐹14,1,1,2 := 1
7,5
7,1
14,1
14,PV
𝐹14,2,3,4 := 1
7,3
14,2
7,5
14,
PV𝐹14,1,5,6 := 1
14,5
14,3
7,1
7,PV
𝐹14,1,2,7 := 2
7,1
14,1
7,1
4,PV
𝐹14,1,2,8 := 1
14,1
7,4
7,3
14,PV
𝐹14,2,2,9 := 1
14,1
7,1
7,9
14
B
Abstract and Applied Analysis
>restart: with(linalg):
Probability Values:=proc(n::posint,i::posint,j::posint,k::posint,l::posint,m::posint)
local A,b,S,r,v,C; global PV;
A[S]:=matrix([[1,1,1,1,1,1]]): b:=vector([1]): C:=sum(v[w],w=1..5):
linsolve(A[S],b,‘r): r: linsolve(A[S],b,‘r,v):
PV[S[n,i,j,k,l,m]]:=subs(v[1]=i/n,v[2]=j/n,v[3]=k/n,v[4]=l/n,v[5]=m/n,v[6]=1-C,%):
end proc:
n:=6: i:=1:j:=1:k:=1:l:=1:m:=1: PV[S[n,i,j,k,l,m]]:=Probability Values(n,i,j,k,l,m);
n:=7: i:=1:j:=2:k:=1:l:=1:m:=1: PV[S[n,i,j,k,l,m]]:=Probability Values(n,i,j,k,l,m);
n:=8: i:=1:j:=2:k:=1:l:=2:m:=1: PV[S[n,i,j,k,l,m]]:=Probability Values(n,i,j,k,l,m);
n:=9: i:=1:j:=2:k:=1:l:=2:m:=1: PV[S[n,i,j,k,l,m]]:=Probability Values(n,i,j,k,l,m);
n:=18:i:=1:j:=2:k:=3:l:=4:m:=6: PV[S[n,i,j,k,l,m]]:=Probability Values(n,i,j,k,l,m);
n:=25:i:=1:j:=2:k:=3:l:=4:m:=5: PV[S[n,i,j,k,l,m]]:=Probability Values(n,i,j,k,l,m);
n:=25:i:=1:j:=2:k:=3:l:=4:m:=5: PV[S[n,i,j,k,l,m]]:=Probability Values(n,i,j,k,l,m);
PV𝑆6,1,1,1,1,1 := 1
6,1
6,1
6,1
6,1
6,1
6,PV
𝑆7,1,2,1,1,1 := 1
7,1
7,2
7,1
7,1
7,1
7,PV
𝑆8,1,2,1,2,1 := 1
4,1
8,1
4,1
8,1
8,1
8,
PV𝑆9,1,2,1,2,1 := 2
9,1
9,2
9,1
9,2
9,1
9,PV
𝑆18,1,2,3,4,6 := 1
9,1
18,1
9,1
6,2
9,1
3,PV
𝑆25,1,2,3,4,5 := 1
5,4
25,3
25,2
25,1
25,2
25,
PV𝑆25,1,2,3,4,5 := 1
5,4
25,2
5,3
25,2
25,1
25
B
Case A.3 (JPD-coloring of the -monohedral tiling). See
Box .
Conflict of Interests
e authors declare that there is no conict of interests
regarding the publication of this paper.
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