:.'"
ON THE CONSTRUCTION OF SETS OF PAIRWISE ORTHOGONAL LATIN
SQUARES AND THE FALSITY OF A CONJECTURE OF EULER
by
••
R. C. Bose and S. S. Shrikhande
University of North Carolina
This research was supported by the United
states Air Force through the Air Office of
Scientific Research of the Air Research
and Development Command, under Contract No.
AF 49(638)-213. Reproduction in whole or
in part is permitted for any purpose of the
United States Government.
Institute of Statistics
Mimeograph Series No. 222
April 1959
•
l
ON THE CONSTRUCTION OF SETS OF PAIRWISE ORTHOGONAL LATIN SQUARES
AND THE FALSITY OF A CONJECTURE OF EULERl
by
R. C. Bose and S. S. Shrikhande
University of North Carolina
n
1.
••• Puu is the prime power decomposition
Introduction.
of an integer v, and we define the arithmetic function n(v) by
••• I
then it is known, MacNeish
["16J and
Mann ["17_7 that there exists a set
of at least n(v) pairwise orthogonal Latin squares (p.o.l.s.) of order v.
It seemed plausible that n(v) is also the maximum possible number of p.o.l.s.
of order v.
This would have implied the correctness of Euler's ["13_7
conjecture about the non-existence of two orthogonal Latin squares of order
v when v
=2
(mod
4), since n(v) = 1 in this case.
If we denote by N(v)
the maximum possible number of Latin squares of order v, then Parker 1:18_7
showed that in certain cases N(v) > n(v) by proving that if there exists a
balanced incomplete block (BIB) design with v treatments,
"II.
= 1,
and block
size k which is a prime power then N(v)
~
on the validity of Eulerts conjecture.
In this paper we generalize Parker's
k-2.
This result cast a doubt
method of constructing p.o.los., by showing that a very general class of
designs, which we have called pairwise balanced designs of index unity,
can be used for the construction of sets of p.o.l.s..
of this method have been made.
Various applications
In particular we show that Euler t s
1 This research was supported by the United states Air Force through the
Air Force Office of Scientific Research of the Air Research and Development Command, under Contract No. P:F 49(638)-213. Reproduction in whole
or in part is permitted for any purpose of the United States Government.
- 2 conjecture is false for an infinity of values of v, including all values
of the form 36m + 22.
We also provide a table for all values of v
for which our methods enable us to show that N(v)
5
150,
> n(v). The smallest
number for which we have been able to demonstrate the falsity of Euler's
conjecture is 22.
Two orthogonal squares of order 22 constructed by our
method are given in the Appendix.
Several attempts, necessarily erroneous,
have been made in the past to prove Euler's conjecture, e.g., Peterson
L19_7,
Wernicke
L20J and MacNeish Ll6J.
Levi £:14, p.
l4J points
out the inadmissibility of the argument used by Peterson and MacNeish.
Wernicke's argument was shown to be fallacious by MacNeish
2.
Orthogonal Latin squares and orthogonal arrays.
L15_7.
A Latin square
of order v may be defined as an arrangement of v symbols, say 1,2, .... ,v
in a vX'! square such that each symbol occurs once in every row and every
column.
Two Latin squares are said to be orthogonal, when if they are
superposed each symbol of the first square occurs just once with each
symbol of the second square.
A set of pairwise orthogonal Latin squares
(p.o.l.s.) is a set of Latin squares any two of which are orthogonal.
Defining N(v) and n(v) as in the introduction, we have already referred to the MacNeish-Mann theorem viz. N(v) ~ n(v).
We shall now give
a useful Lemma which generalizes this result.
».
u then N(v) .~ min(N(vl ),N(v2 ), ••• ,N(Vu
We shall derive this Lemma from a general result on orthogonal arrays
Lemma 1.
If v
= vl
v2 '"
V
due to Bush and one of the authors L5J
A
= (aij ),
m
= A.
vt
where each a
ij
1o_7.
Let us consider a matrix
represents one of the symbols 1,2, ••• ,v, with
columns and r rows.
Consider all trowed submatrices which can
- 3 •
be formed from A by choosing trows, t
can be regarded as an orderedt-pl· t.
~
r.
Each column of the submatrix
The matrix A is called an orthogonal
array (m,r,v,t) of size m, r constraints, v levels, strength t and index
A; if each trowed submatrix which can be formed from A contains every
one of the v t possible ordered t-pJets each repeated A times.
If
f{m,v,t) denotes the maximum number of constraints for given m, v and t,
then the following theorem is known
Theorem.
L5, 10J .
If m is divisible by
i
f{m l m2 ... mu '
v~ for i
= 1,2, ••• ,u,
then
t) ~ miner]! r 2, ...,ru ) where r i =f(mi,vi,t).
that an orthogonal array (v2 ,q,v,2) of strength
v l v2 ... vu '
Now it is known
LllJ
2 and index unity is equivalent to the existence of a set of q-2 p.o.los.
of order v, and may be obtained from this array by using two of the rows
for coordinatization.
Conversely the array can be obtained from the
squares.
Hence f(v2 ,v,2)
taking mi
2
= Vi'
t
=2
= N{v)
+ 2.
The required Lemma follows by
in the theorem.
n
The MacNeish-Mann theorem follows from the Lemma by taking Vi
i
= 1,2" ••• ,u.
nothing new.
If N(v) has always been equal to n(v) the Lemma would add
However this is not the case.
in Section 5, that N{2l) ~ 4.
N(105) ~ min(N(2l),
3.
= Pi1,
N(5»
~
For example
it will be shown
It follows from the Lemma that
4 whereas n(l05)
= min(3,5,7)
Pairwise balanced designs of index unitl-
- 1
= 2.
An arrangement of v
objects (called treatments) in b set (called blocks) will be called a
pairwise balanced design of index unity and type (v; k "k , ••• ,k ) if each
m
l 2
block contains either k , k , ••• or
l
2
(ki ~ v, k
i
~
treatments which are all distinct
~ k ), and every pair of distinct treatments occurs in exactly
j
- 4one block of the design.
If the number of blocks containing ki treatments
is b , then clearly
i
(3.1)
b
=
m
E bi ,
i=l
Consider a pairwise balanced design (D) of index unity and type
(v; kl'~, ... ,km).
The subdesign (D ) formed by the blocks of size k i
i
will be called the i-th equiblock component of (D), i
= 1,2,. •• ,m.
A subset of blocks belonging to any equiblock component (Di ) will be
said to be of type I if every treatment occurs in the subset exactly k
times.
The number of blocks in such a subset if clearly v.
i
As noted by
Levi using Konig's theorem on the decomposition of even regular graphs
("14, pp.
4-6_7,
we can rearrange the treatments within the blocks of the
subset in such a way that every treatment comes in each position exactly
once.
If the v blocks of the subset are written out as columns, each
treatment occurs exactly once in every row.
'When so written out the blocks
will be said to be in the standard form.
A subset of blocks belonging to any equiblock component (D ) will be
i
said to be of type II if every treatment occurs in the subset exactly
once.
The number of blocks in such a subset is v/k •
i
The component (D ) will be defined to be separable if the blocks can
i
be divided into subsets of type I or type II (both types may occur in (D )
i
at the same time).
Let r
be the number of blocks of type I and si the
i
number of blocks of type II in (D ), then clearly
i
v(ri k1 + s1)
(3.2)
= kib i
•
The design (D) is defined to be separable 1f each equiblock component
(Di ) is separable.
It follows from (3.1) that
- 5 -
4.
Main theorem.
Lemma 20
We shall first prove the folloWing:
Suppose there exists a set E of q-l p.o.l.s. of order k,
then we can construct a qxk(k-l) matrix P, whose elements are the symbols
1,2, ••• ,k and such that (i) any ordered pair (~), i ~ j occurs as a
column exactly once in any two rowed submatrix of P, (ii) P can be subdivided into k-l submatrices Pl'P2,,, •• ,Pk- l of order qxk such that in
each row of P , 1 < c < k-l, each of the symbols 1,2, ••• ,k occurs exactly
c
-
-
once.
Proof.
It has been shown in
1:9_7 that matrix P with
(1) can be obtained in the following manner.
the property
Without loss of generality
we can take the set E in a form in which the first row of each Latin square
contains the symbols 1,2, ••• ,k in that order and prefix to the set E a
square containing the symbol i in each position in the i-th column.
If
we then write the elements of each square in a single row such that the
symbol in the i-th row and j-th column occupies the n-th position in the
row where n = k(i-l) + j, and from the resulting qxk2 matriX we delete
the first k columns, we get the matrix P.
formed by the columns k(c-l) + j, j
If Pc is the submatrix of P
= 1,2, ••• ,k,
then the property (ii)
follows from the fact that each of the symbols 1,2, ••• ,k appears exactly
once in any given row of a Latin square belonging to E.
Let 1 be a column of k distinct treatments chosen from the set
tl't2, ... ,tv then following the notation used in
1:9_7,
we shall denote
by P(1), the qxk(k-l) matrix obtained from P on replacing the symbol i
by the treatment occurring in the i-th position in 1.
A similar meaning
- 6 will be .assigned to Pie')') and
Jf
Cj (')') where
Jf
cj denotes the j-th column
of Pc'
Clearly every treatment of ')' occurs once in every row of pc(')'),
c = l,2, ••• ,k-l, and if t
t
a
and
~
are any two treatments of ')' then the
ordered pair (~) occurs as a column exactly once in any two rowed submatrix of pc')').
Theorem 1.
Let there exist a pairwise balanced design (D) of index
unity and type (v; kl ,k , ... ,km) and suppose there exist qi-l p.o.l.s. of
2
order k • If
i
then there exist at least q-2 p.o.l.s. of order v.
If
the design (D) is
separable then the number of p.o.l.s. of order v is at least q-l.
Proof.
The first part of the theorem was proved in
L9_7.
However
in order to prove the second part it will be necessary to reproduce this
proof with slightly altered notation.
Let the treatments of the design
be t l ,t2 " .. ,tv ' and let the blocks of the design (written out as columns)
belonging to the equiblock component (Di ) be 8il,8i2, ... ,8ib (i =1,2, ... ,m)<
i
We now define the k.xb matrix D. by
J.
i
J.
Di = L8il,8i2,···,8ibi-l •
Let Pi be the matrix of order ~ x ki(ki-l) defined in Lemma 2, the
elements of Pi being the symbols 1,2, ••• ,ki •
Let Pie' c
= l,2, •• o,ki -l
be the submatrices of Pi' such that each row of PiC contains the symbols
iCj be the j-th c01umnof'\c J = 1,2, ... ,ki
Let Pi (8iu ), u = 1,2, ••• ,b i , be the matrix obtained from Pi and 8iu' and
let
1,2, ... ,ki exactly once and let
ft
o
- 7 Pi():)1)
Then PieD!)
= LP i (8il ),
.u,
Pi (812 ),
1s1~c1s'·q1.biki(ki-l).
If t
Pi (81b )_7 •
1
a and t b are any
tw~
treatments
occurring in the same block of (D ), then the ordered pair (~) occurs
i
exactly once as a column in any two rowed submatrix of Pi (D ).
i
Let 6 i
be the matrix obtained from Pi(D ) by retaining only the first q rows, and
i
let
Then from (3.1), 6 is of order qxv(v-l), and it follows from the properties
of the design that any two rowed submatrix of 6 contains as a column each
ordered pair of two distinct treatments chosen from t ,t , ••• ,t exactly
v
l 2
once. This property of 6 will be referred to as the property 1"2.
Let 6
tion (i
0
be a qxv matrix whose i-th column contains t
= 1,2, ... ,v).
orthogonal array
strength 2.
Then from
L v2 , q, v, 2J
LllJ the
i
in every posi-
matrixL6o ' 6_7 is an
of or ;er v2 , q constraints, v levels and
Using two rows to coordinatize we get a set of q-2 p.o.l.s.
which proves the first part of the theorem.
To prove the second part of our theorem we shall introduce the concept 'column equivalence." Two matrices A and B will 'be called column
equivalent if they can be obtained from each other by a rearrangement of
columns.
This relation will be denoted by A'" B.
Suppose now that the design (D) is separable.
Let (Dil), ••• ,(D.
J.r
)
i
be the subsets of type I, and (Dtl), ••• ,(Dts ) be the subsets of type II
i
into which the blocks of (D.) are divisible, the blocks of subsets of
1
.
type I being written in the standard form.
We shall show that after a
rearrangement of columns, 6 can be divided into submatrices of order qxv,
- 8 -
possessing the property
that each row contains each of the treatments
T
l
tl' t 2 , • • • , tv exactly once.
Now
= 1,2, ••• ,ri ,
Since fiiCj(Dif)' f
clearly possesses the property T l ,
Pi(Dif ) is divisible (up to a rearrangement of columns) into ki(ki-l)
submatrices each having the property T •
l
Again
Pi(D!g) - 1:... , PiC(D!g)' ..._7, c = 1,2,... ,ki -1.
g ~ 1, 2, ••• ,si' possesses the property Tl , Pi(D!g)
is divisible (up to a rearrangement of columns) into ki-l submatrices
Since Pic (D;g"
each with the property T 1.
Pi(Di )
Since
= I:Pi(Dil),···,Pi(Diri)'
and since the property
Pi(D!l),·.·,Pi(Dfs ) _7
i
remains unchanged by deletion of rows it follows
l
from- (3.3) that after a rearrangement of columns 6 can be divided into
T
v-l submatrices 6(1), 6(2), ••• , 6(v-l) each with the property
should also be noted that the property
ordering the columns.
T
2
It follows that if
T
l
•
It
of 6 remains unchanged by rea~
J.
denotes a lxv row vector
each element of which is t., the matrix
,[ a~, a;:l ··., ~.l l
J
J.
6
is an orthogonal array
0
,
Lv2 ,
.
6
, ••• ,
6(V-l)
q+l, v, 2_7 of order v 2 , q+l constraints, v
levels and strength 2, and can therefore be used to construct a set of q-l
p.o.l.s. of order v.
- 95'0
Use of balanced incomplete block designs for obtaining sets of
pairwise orthogonal Latin squares.
A balanced incomplete block (BIB) design with parameters v,b,r,k,A.
is an arrangement of v objects or treatments into b sets or blocks such
that (i) each block contains k < v different treatments, (ii) each treatment occurs in r different blocks, (iii) each pair of treatments occurs
together in exactly A. blocks.
The parameters satisfy the relations
>v •
=
These conditions are necessary but not sufficient for the eXistence of a
b
BIB design.
by Yates
BIB designs were first introduced into statistical studies
['"21_7, but occur in earlier literature in connection With various
combinatorial problems.
SUbsequent to Yates many authors have dealt with
the problem of constructing these designs.
bibliography we shall only refer to
symmetrical if v
be resolvable
= b,
£2J.
and in consequence k
L3J if the
Without attempting a complete
A BIB design is said to be
= r.
A BIB design is said to
blocks can be divided into sets, such that the
blocks of a given set contain each treatment exactly once.
A BIB design With A.
=1
dex unity and type (v; k).
is clearly a pairwise balanced design of in-
We shall denote such a design by BIB(v; k).
It is evident that a symmetric or a resolvable BIB design is separable,
since in the first case the blocks constitute a single set of type I, and
in the second case the blocks can be divided into r subsets of type II.
It follows from Theorem 1.
'!heorem 2.
size k and A.
N(v) ~ N(k).
-
= 1,
If there eXists a BIB design With v treatments, block
then N(v) ~ N(k)-l.
If the BIB design is separable
- 10 When k is a prime or prime power N(k)
L18J mentioned
theorem
= k-l, hence we have Parker's
in the introduction which asserts that N(v) ~ k-2,
when k is a prime power.
This result can be improved by unity when the
design is separable.
Theorem 3.
N(v)
~
N(k) in the following cases:
(i)
v
= s2
(ii)
v
(iii)
v
= s3 + 1, k = s + 1, where
= s(s-1)/2, k = s/2 = ~-l,
Proof.
+ s + 1,
k
=s
+ 1,
where s is a prime power,
s is a prime power,
m > 2.
Part (i) follows from Theorem 2, since there exists a sym-
metrical BIB design with parameters v
when s is a prime or prime power
L2,
= b = s2+s +1,
p. 364J.
=k
r
= s+l, A. = 1
The existence of the
design is equivalent to the existence of a projective plane PG(2,s) of
order os.
Example (1).
= 4,
If s
4 p.o.l.s. of order 21.
N(k)
= N(5) = 4.
Hence there exist at least
The MacNeish-Mann result only assures the ex-
istence of 2 orthogonal Latin squares.
Example (2).
N(k)
= ~-l.
take s
If s is a Mersenne prime
> 3 then k
Hence N(s2+s +1) > s, whereas n(s2+s +1)
~ 31, n(993)
= 31 then N(993)
Example (3).
=
s +1 = 21' and
= 2, e.g., if we
= 2.
= 49, k = 50. Parker I s theorem is inapplicable.
If s
It is shown in section 6, example (7), that N(50) ~ 5.
2
N(49 +49+1) ~ 5 whereas n(492+49+1) = 2.
Hence
Part (11) follows from the existence of the series of resolvable BIB
designs
v
= s3+1 ,
b
which has been proved in
= s2(s2_ s +1 ),
L6_7.
k
= s+l,
A.
=1
- 11 Let v = 313 +1
Example (4).
N(v) ~ 31 whereas n(v)
= ~.72.19 = 29792,
k
= 32
=
25 •
Hence
= 18.
Te prove part (iii) we shall establish the existence of a series of
resolvable BIB designs with parameters
(5.1)
v
= s(s-1)/2 1
b = s2_ 11
r = s+ll
k
= s / 2 = 2m-l ,
Consider the finite projective plane PG(2 1 ~), and take a non-degenerate
conic C on this plane, e.g., the conic
xz
=
y
2
•
["4,
p. 158_7 that there are s+l points Pl ,P2 , ••• ,Ps +l on this
conic, where s = 2m• Through any point Pi = (x',y' ,z') on the conic
We know
there will pass s+l lines s of which meet the conic in the other s points
on the conic 1 and the remaining line z'x + x'z = 0 1 meets the conic in
the single point P..
~
This may be called a tangent to the conic at P.•
~
The s+l tangents to the conic all pass through the Point Po
which may be called the pole of the conic.
may be divided into three classes:
= (°11,0)
The s2+s +1 lines of the plane
(a) The s(s+1)/2 intersectors each of
which meets the conic in 2 points but does not pass through Po; (b) The
s+l tangents each of which meets the conic in one point and passes through
Po; (c) The s(s-1)/2 non-intereectors which do not meet the conic and
hence do not pass through P.
o
Let the s2_1 points of the plane other than
Po and the points of the conic be called retained points.
Consider the
configuration of the retained points and non-intersectors.
If the re-
tained points are called blocks, and the non-intersectors are called
treatments 1 and if treatment is considered to belong to a block if the
corresponding line and point are incident then we get the design (5.1).
This follows from the fact that on each non-intersector there lie s+l
- 12 points.
This gives r
= s+1.
passes one tangent" viz. P Po.
Also through each retained point P there
If the point of contact of this tangent is
Pi then the other s points on the conic lie two by two on s/2 intersectors
through P.
k = s/2.
Hence there pass s/2 non-intersectors through P.
This gives
That A = 1 follows from the fact that any two intersectors meet
in one and only one point which is a retained point.
Next we have to prove that the design is resolvable" i.e., the treatments can be separated in groups.
We have to find sets of s-l blocks
which contain each treatment exactly once" i.e." sets of s-l retained
points such that each non-intersector passes through one and only one point
of the set.
The s-l retained points on any intersector or tangent pro-
vide such a set.
Finally we have to choose a set of s+l intersectors or
tangents which contain between them all the retained points.
Such a set
is provided by the s+l tangents, or the s+l lines through any point on the
conic.
This proves the resolvability of the design.
Example (5).
Parker leaves undecided the question whether there are
an infinity of values of v for which N(v) > n(v).
This question is de-
cided in the affirmative by part (iii) of our theorem" since i f we take
_m-l) = 22n-l-1. Also v = 22n-l( 2n +1 )( 2n -1 ) •
m = 2n, then N()
v ~ N( ~.
n
Hence n(v) ~ 2 -2 < N(v).
-
N(120 ~ N(k)
= 7,
n(120)
For example, let m = 4, then v
= 120,
~
= 8,
= n(8.5.3) = 2.
If from a BIB design with v treatments, block size k and
omit x
k
r..
=
1, we
k treatments belonging to any block, we obViously get a pairwise
balanced design of index unity and t;ype (v-x; k, k-l; k-x).
Theorem 1, we get
Applying
- 13 Theorem 4.
and A
= 1,
Existence of a BIB design with v treatments, block size k
implies that
N(v-x) ~ min(N(k), r;(k-l), m(k-x» - 1
if x
-< k.
Example (6).
r
= k = s+l,
Consider the symmetrical BIB design v
A = 1, when s
= 16.
If we take x
= 6,
= b = s2+s +1,
8 and 9 respectively,
we obtain N(267) ~ 9, N(265) ~ 7, N(264) ~ 6, Whereas n(267)
n(265)
6.
= 4,
-
n(264)
-
-
= 2.
= 2,
The falsity of Euler's conjecture regarding the non-existence of
two orthogonal Latin squares of order v :: 2 (mod 4), and the method of
adjunction of treatments.
I.emma--.l.
There exists a resolvable series of BIB designs with
parameters
(6.1)
v* = 24m+15,
Proof.
b*
= (8m+5)(12m+7),
r*
= 12m+7,
k*
= 3,
A*
= 1.
The existence of the design is equivalent to a solution of
the Kirkman school girl problem for triplets with n
= 24m+15
girls.
There
is an extensive literature on this problem and a complete bibliography of
all work up to 1911 is given in
L12J,
and an excellent summary of the
known results is contained in Ll, Chapter X, p. 267-298_7.
is solvable for all values of n of the form 6t+3.
qUired by us is t
= 4m+2.
The problem
The special case re-
The girls can be considered as the treatments
of (6.1), triplet as a block, and each day' s walk gives a complete replication.
In the special case when 12m+7 is a prime power, an analytical solu-
tion can be given as follows.
We first prove
- 14 Lemma 4.
If' x is a primitive element of the Galois field GF(12m+7)"
where l2m+7 is a prime power, then there exist odd integers q and
Q
such
that
Proof.
'!he Lemma is proved by considering the (1,1) correspondence
y
=
(Z+l)/(z-l)
between the elements of GF(12m+7), augmented by infinity.
'!he correspond-
ence is an involution and it is easily shown that there must exist a pair
of corresponding add powers of x.
To each element u of the field GF(l2m+7) let there correspond two
treatments u
treatment
l
00.
and u • To the 24m+14 treatments thus obtained adjoin the
2
'!he existence and resolvability of the design can now be
proved by using the second fundamental theorem of the method of differences developed in
2i
(Xl '
1:2_7.
2i+4m+2
Xl
Q+j
2j
( ~, x2
Consider the set of &+5 initial blocks
2i+&+4)
'
Xl
'
x2
i
= 0,1, ••• ,2m;
j
= 0,1, ••• ,6m+2j
'
6m+3+2j)
'
(01' 02' 00) •
Using methods on pages 374 and 384 of
1:2_7 it
is easy to verify that all
the non-zero pure differences and all the mixed differences are symmetrically repeated once.
the initial blocks.
Hence the design (6.1) can be obtained by developing
Since the set of initial blocks gives a complete
replication" the design is resolvable.
We can now proceed to prove the main theorem of this section.
Theorem 5.
order v
= 36m+22"
'!here exist at least two orthogonal Latin squares of
where m is any non-negative integer.
- 15 Proof.
by (6.1).
Let us start with the BIB design whose parameters are given
Take l2m+7 new treatments Ql' Q2'
of the i-th replication add the treatment Q.•
J.
Sisting of the new treatments.
.ee,
Ql2m+7 and to each block
Also take a new block con-
The augmented design is now a pairwise
balanced design of index unity of the type (v; k ,k ) where v
l 2
kl
q
= 4,
k2
= l2m+7.
Since ql-l
= N(kl ) = 3,
~-l
= N{k2 )
= 36m+22,
~ n(l2m+7) ~
4,
= min(ql'~) = 4, it follows from Theorem 1, that there exist at least
two orthogonal Latin squares of order v
= 36m+22.
Two superposed orthogonal Latin squares of order 22 constructed by
this method are exhibited in the Appendix, the capital letters corresponding to the first square and the small letters to the second.
We shall now show that there exist numbers v
=2
(mod 4) for which
there exist many more than two orthogonal Latin squares of order v.
Theorem 6..
i~teger
If sand s+l are both prime or prime powers, then for any
m~ 2
N(sm+ l ) ~ s-2
Proof.. Consider the Euclidean geometry EG (m,s). Taking the points
and lines of this geometry as the treatments and blocks respectively, we
get a BIB design with parameters v*=sm, b*=r*sm-l , r*=l+s+ •••+sm-l ,
A;H!~l.
Furt~er
the design is resolvable into r* replications
corresponding to the lines of the Euclidean geometry passing thEough a
point of the (m-l)-'lat at infinity of the Projective geometry PG{m,s)
in which the Euclidean geometry can be embedded. Add a new treatment
Q
to the blocks of the first replication. This is equivalent to the adjunction of a point at infinity to the lines of a parallel pencil.
The
-16augmented design is a pairwise balanced design of index unity and type
(stn+1; s+l" s). Sinee s aDd s+l are prime or prime powers Theorem 1
gives N(sm+ l ) ~ 6-2.
We may note that sand s+1 will be prime or prime powers if (t)
numbe~
s is a Mersenne
Remark:
if
6
> 3"
(ii) s+l is a Fermat prime, (iii) s=8.
If s is a Mersenne number, then s2m+l ::2 (mod.
4).
Hence
this Theorem provides another infinite set of values v of the
form 4t+2 for which at least s-2 p.O.l.s. of order v can be constructed,
thu6 again disproving Euler's conjecture.
Instead of one we may add x new treatments,
x~*,
one to each
block of a complete replication, and add a new block containing the new
treatments. We then get a pairwise balanced of index unity and type
m
(8 +Xj 8+1, S,x).. Hence the following
Corollary 1.
Exe:Dl.ple (7)
0
If 6 and s+l are prime o:r prime powers
Taking m=l,p s='7 and s:::3l, we get H(,O)
~
5,
N(962) ~ 29.
If we do not insist that sand 8+1 are both prime or priIlle powers, then
taking m = 1, in the above theorem, we get the following cOl"ollary.
Corollary 2. If s is a prime or prime power, then
N(s2+l ) ~min(6-l, N(s+l»
Example (8).
q-l ~ 3.
Let s = 19, then N(s+l)
Therefore N(362) ~ 2.
- 1 •
= N(20)
~
n(20)
Taking s = 8, N(s+l) = 8.
= 3.
Hence
Hence
N(65) ~ 6, whereas n(65) = 4.
Theorem 7.
If there exist at least two orthogonal Latin squares of
order 4t+2, there exist a.t least two orthogonal Latin squa.res of any order
v which is an odd multiple of 4t+2.
- 17 Proof.
Let v
l
= 4t+2,
v
= 2u+l,
2
and v
= v l v2 •
N(v) ~ min(N(vl ), N(V2 »
Now N(V )
2
=2
Then from Lemma 1
•
by hypothesis, and sincev is odd, N(V~:) ~ n(v2 ) ~ 2.
2
This
proves the theorem.
Let v
Example ( 9) •
= 110 = 22.5.
Now from Theorem 5, N(22)
> 2.
=
Hence N(llO) ~ 2.
The method of adjunction considered in Theorem 6 ·canbe generalized.
Let us take x new treatments Ql'Q2' ••• ,Q x (1
< x < s+l), and adjoin the
treatment Q. to each block of i-th replication of the BIB design with
J.
parameters (6.2).
treatments.
We also take an additional block consisting of the new
We then get a pairwise balanced design of index unity and
type (s2+x; s, s+l, x).
Theorem 8.
Applying Theorem 1, we get
If s is a prime power and 1
> min(s-l, N(s+l), N(x»
Example (10).
< x < s+l, then N(s2+x )
- 1.
Let s
= 7,
x = 5.
We get N(54)
2: 3.
Similar results may be obtained by adjunction to the blocks of the
resolvable BIB designs ill (U) &:Illd (i11)of Theorem 3.
Example (11).
Take s
= 16
in (5.1).
Take 9 new treatments and adjoin
the i-th treatments to the blocks of the i-th complete replication.
take a new block consisting of the new treatments.
balanced design of index unity and type (129; 8, 9).
N(129) ~ 6, whereas n(129)
= 2.
Also
We get a pairwise
Hence from Theorem 1
- 18 7.
Certain pairWise balanced designs of index unity associated with
the finite planes PG(2, ~) ~ EG(2, 3 2 ), and the corresponding sets '!.f
pairwise orthogonal Latin squares.
We shall now consider certain designs
based on finite geometry which take advantage of the fact that 7, 8 and 9
are three consecutive integers which are prime powers.
Let us consider the finite projective plane PG(2, 23 ).
It has been
shown in proving part (iii) of Theorem 3, that there exists on this plane
a set
1-1 of 10 points,
viz., the set of points Pl ,P2 "",P on a non9
degenerate conic 0, and its pole Po' such that every line on the plane
intersects
~1.
{-1 in either 2 points
or no points.
Then the number of points in
1-1* is
Let
x ~ 10.
1-1* be
a subset of
Consider the set of
retained points obtained by deleting from the plane the points of
If
'We
1-1*.
take the set of retained points as treatments, and lines as blocks,
then clearly each block contains either 7, 8 or 9 points.
of retained points occurs exactly in one block.
Hence
'We
Also any pair
obtain a pair-
Wise balanced design of index unity and type (73-x; 7,,8,9).
Applying
Theorem 1, we have
Theorem 9.
N(73-x) ~ 5
Example (12).
if x ~ 10.
Taking x = 3 or x = 7 we find that there exist at
least 5 p.o.l.s. of orders 70 or 66.
Again consider the Euclidean plane EG(2,3 2 ).
Ifax2 + 2hxy + by2
is irreducible in GF(3 2 ), where a, b, c belong to this field" and if
f, g" h are so chosen that
2 - Ch2 ~ 0
A = abc + 2fgh - af2 - bg
then the conic
ax2 + 2bxy + by2 + 2gx + 2fy + c = 0 is non-degenerate
and does not meet the line at infinity.
If
1-1 is
the set of 10 points
- 19 on this conic" then no line meets .r1in more than two points.
1-1* to be a subset of
.1-1,
If we take
then the set of retained points obtained by
deleting the points of 1-1* from the plane" provide a pairwise balanced
design of the type (8l-x; 7,8,9) where x < 10.
=
Hence using Theorem 1 we
have
N(8l-x) > 6, if x < 10.
=
Example (13). Taking x = 3 or 7 we find that there exist at least 6
Theorem 10.
p.o.l.s. of orders 78 and 74.
8.
Use of group divisible designs in constructing sets of pairwise
orthogonal Latin squares.
An
arrangement of v objects (treatments) in b sets (blocks) each
containing k distinct treatments is said to be a grouP divisible (OD) desig~
if the treatments can be divided into
t
groups of m treatments each"
so that any two treatments belonging to the same group occur together in
"'1 blocks" and any two treatments from different groups occur together in
We will denote such a design by the notation GD(v; k"m; A.l "A. )
2
The combinatorial properties of these designs have been studied in ~7-7
A.
2
blocks.
where it has been shown that
(8.1)
r being the number of replications, i.e., the number of times each treatment occurs in the design.
(8,,2)
P
= rk
It has also been show that
- "'2v ~ 0,
Q
=r
- "'1 ~ O.
The GD designs can be divided into three classes.
(i)
(ii)
(iii)
Regular (R) characterized by P > 0" Q > O.
8emiregula.r (SR) characterized by P
Singular (S) characterized by Q
= o.
= 0"
Q
> O.
Q
- 20 Methods of constructing these designs have been given in ~8_7o
So far
as the construction of poo.l.s. is concerned a special role is played by
GD designs with "'1
GD(v; k,mj 0,1).
= 0,
"'2
= 1,
which in our notation can be denoted by
It further this design is regular
'We
shall denote it by
RGD(vj k,m,O,l) and if it is semiregular we shall denote it by
SRGD(vj k,m; 0,1).
Lemma
5~
Existence of a GD(v; k,m; 0,1) implies the existence of
pairwise balanced design (D) of index unity and type (v; k,m).
If to the b block of the GD design, we add
Knew blocks
corresponding
to the groups we get a pairwise balanced design of index unity and type
It follows from Theorem 1:
(v; k,m).
Theorem 11.
If there exists a GD(v; k,mj 0,1), then
N( v) > min(N(k), N(m»
• 1.
Further if the GD design is separable
N(v) ~ min(N(k), N(m».
Corollary.
Proof.
from
L8,
If s is a prime or a prime power N(s2. l )
Since s is a prime or a prime power N(s)
~ N(s-l).
= s-l,
and we know
p. l73J that there exists a symmetric and hence separable
RGD(s2_ lj s, s-l; 0,1).
Example (14).
Taking s
= 9,
N(80)
~
N(8)
= 7,
whereas n(Bo)
= 4.
We note that Parker's result is inapplicable in this case since there
does not exist a BIB design With v
we get N(1023) ~ N(3l)
Theorem 12.
N(v-l)
~
= 30
= 80,
and k > 2, ;..
whereas n(1023)
= 1.
= 2.
Suppose there exists a GD(v; k,mj 0,1).
q-2 where
q-l
= min(N(k),
Taking s
N(k-l), N{m), N(m-l»
•
Then
= 32,
- 21 If further the GD design is resolvable then N(v-l) ~ q-l
Proof.
From Lemma 5, we get a pairwise balanced design (D) of index
unity and t~e (v; k,m).
Omitting a particular treatment say a from the
blocks in which it occurs, we get a pairwise balanced design (Da ) of the
t~e (v-l; k" k-l, m" m-l).
Theorem 1 now gives N(v-l) ~ q-2.
We shall now show that we can get q-l p.o.l.s. when the GD design is
resolvable.
There are
i
blocks in (D)" each of which contains m treat-
ments, Viz." the blocks corresponding to the groups.
responding set of blocks of (D ).
a
Consider the cor-
Since only one group contains the treat-
this set of blocks consists of one block 6
1 containing m-l
g"mtreatments". and a set (Sg,m ) of i-l blocks each of which contains m treat-
ment
(X"
ments.
Since the initial GD design is resolvable" the other blocks of (D)
are divisible into r sets, such that each set of blocks contains every
treatment exactly once.
(X
From the i-th of these sets" we obtain by dropping
the corresponding set of blocks of (D ), i
a
= 1"2, ••• ,,r.
This set consists
-
of v/k blocks, one of which 6. k 1 contains k-l tr€8;1:;,,-c,ents and a set (S. k)
~,
~,
of (v/k)-lblOOka each of which contains k treatuer.ts.
Put q3=N(m';;1}-+1 and Q4cN (m)+L Let -P and l?3c'
c::;1,,2, ... ,m..2 be
3
the matrices of o::.,der Cl x(m-l) (m-2) and q,x(m-l) respectively of Lemma
3
2. Similarly let P4 arid., P4d l <1=1,2" ••• ,m-l be me:.trices of order Q4xm(m-l)
and Cl4xmreapectively.
Let .6.g be the matrix of order q x
(m-l)(m-2) + m(m-l)(f-l)
ob-
tained from
by retaining the first q rows.
Defining column equivalence as in 'the proof
- 22 of Theorem 1 and remembering that (i) the number of submatrices P4d is one
greater than the number of submatrices P3c " (11) the block 8g "m_l and the
set of blocks (S
) contain between them all the v-l treatments in the
g"m
.
design (D ) we can write
a
6
g
- 16(1) 6(2)
6(m-2) 6* 7
L' g , g " ••• " g
, g_
where each row of 6(j) contains each of the v-l retained treatments
g
exactly once (j
= 1,,2, ••• ,m-2).
Thus
tive to the v-l retained treatments.
6~j)
possesses the property T relal
Each row of 6*
contains all the
g
v-m retained treatments other than those contained in 8g
,m- 1.
Also any
pair of treatments occurring in the block 8g,m- 1 or blocks (Sg ,,~
m) of {D,J
occurs as a column exactly once in any two rowed submatrix of 6 g •
Likewise ·trom 81 k-l and (Si k..) we get.a qx( (k~1.){k-2)+(k-1Hv..k»
matrix 6
'
~
i
r (1)
(2)
(k-2)
* 7
6 i - L 6 i ,,6i ' ••• , 6 i
' 6 i _, i = 1,2, ••• ,r
possessing similar properties in relation to the blocks 8 i ,k-l and (Si,k).
Let 6 be a q x (v-l) matrix whose j-th column contains the j-th
0
treatment in (D ) in every position.
a
It follows from the property of
pairwise balance possessed by (D), that the set of blocks 8g,m- 1" 81, k- l'
••• , 8 k 1 taken together contain each of the v-l retained treatments
r" -
exactly once.
Let 6** be a q x (m-l) matrix each of whose rows is 8
g
I
g,m-
Similarly let 6~* be a q x (k-l) matrix each of whose rows is 8~,k_l
(i
= 1,2, ••• ,r).
clearly
Then
1.
- 23 has the property
T
in relation to the v-l treatments of (D ). It follows
a
2
from this or can be directly verified using (801) that A is of order
q x (v-l){v-2).
Hence LAo, A_7 is an orthogonal array L(V-l)2, q" v-l" 2_7
of order (V_l)2" q constraints" v-l levels and strength 2.
We now note that the matrices LA;, A*;'"_7 and LA;, A;*_7" i= 1"2" ••• ,,r
have the property
exactly once.
, that each row contains every treatment of (Da )
l
Hence by a reordering of columns, L4
rA,
A_ 7 can be divided
0
T
into (v-l) submatrices of order q x (v-l) such that each submatrix has the
property T l • After this reordering has been mad: LAo, A_7 can be converted into an orthogonal array L(V-l)2, q+l, v-l, 2_7 of q+l constraints,
by adding another row such that the part of this row corresponding to the
c-th submatrix With the property
peated v-l times.
T
l
consists of the c-th treatment re-
It follows that there exist q-l p.o.l.s. of order v-l.
This proves the second part of Theorem 12.
In the same manner we can prove:
Corollary.
If
there exists a resolvable BIB (v; k) then
N(v-l) ~ min(N(k), N(k-l».
Let there exist a GD(v; k,m; 0,1).
and adding it to each of the
t
Then by taking a new treatment
Q,
blocks of the design (D) of Lemma 5, we
get a pairwise balanced design of index unity and type (v+l; k, m+l).
Using Theorem 1, we have
Theorem 13.
If there exists a GD(v; k,m; 0,1) then
N(v+l) ~ min(N(k), N(m+l»
- 1.
Suppose the design GD(v; k,m; 0,1) is resolvable and r is the number
of replications.
We can take new treatments Ql,Q2" ••• "Qx' x
Qi to each block of the i-th replication.
~
r" and add
Also we take blocks corresponding
- 24 to the groups of treatments and a new block consisting of the treatments
Ql"Q2, ••• "Qx when x > 1.
We then get a pairwise balanced design of index
unity and type (v+x; k+l, k, m) if x
= 1,
type (v+x; k+l" k" m" x) if
1 < x < r and type (v+x; k+l, m" x) if x = r.
Theorem 14.
Using Theorem 1, we get
Suppose there exists a resolvable GD(v; k, m; 0, 1)" then
N(v+x) > min{N(k+l)" N(k), N(m» - 1 if x = 1, N(v+x) > min(N(k+l), N(k)"
-
N(m), N(x»
x
-
- 1 if 1 < x < rand N(v+x) ~min(N(k+l)" N(m), N(x»
- 1, if
= r.
Theorems 12, 13 and 14 can be applied to semiregular GD designs,
which provide a large class of resolvable GD designs.
sult is taken from
Lemma 6.
["8,
p.
The following re-
181_7.
The existence of an SRGD ({m;
existence of an orthogonal array A
= (m2, f,
f,
m; 0, 1) implies the
m" 2) of strength 2 (or
equivalently the existence of a set of [-2 p.ool.s. of order m), and
conversely.
Suppose there exists an SRGD {f.m; {, m; 0, 1).
there exists an orthogonal array A
= (m2 ,f,m,2).
Then from Lemma 6,
Without loss of gen-
erality we can take the first row of A as (€i,,€2"."€~) where €iiS a
lxm row vector each of whose elements is i.
If we
now delete the first
row, there remains an orthogonal array A = (m2 ,{-1,m,,2) of f-l conl
straints which can be subdiVided into m submatrices each of order([-l)xm
possessing the property
that every row of each submatrix contains the
l
symbols 1,2, ••• "m exactly once. This property remains unchanged if we
2
get the A2 = (m ,ll,m,2) by deleting some rows of A , \~ [-1. Replace
l
the symbol j in the i-th row of A2 by the treatment Qij (1 = 1,,2"'.'[1;
j =
T
1,2" ... ,m), and take the columns as blocks.
We then get an
- 25 SRGD({lm; ll,m; 0,1) for which the groups are given by the m treatments
occurring in any given row.
This design is resolvable, the blocks cor-
responding to the submatrices with the property T give a complete replil
cation.
Clearly r
Lemma 7.
= m for
this design.
We thus have
The existence of an SRGD(lm; I,m; 0,1) implies the exist-
ence of a resolvable SRGD(llm; ll,m; 0,1), and an orthogonal array
A2
= (m2,11,m,2)
11
< 1.
Corollary.
if
11 ~
divisible into submatrices with property Tl , for any
There exists a resolvable a resolvable SRGD({lm; Ipm; 0,1)
N(m)+l.
Proof.
(m2 ,!,m,2).
Let
1 = N(m)+2.
Then there exists an orthogonal array
From Lemma 6, there exists an SRGD(!m; I,m; 0,1).
The re-
quired result now follows from Lemma 7.
Example (15).
From SRGD(40; 5,8; 0,1) which is resolvable, using
Theorem 12, we get N(39) > 3, whereas n(39) = 2. Similarly from
=
SRGD(136; 8,17; 0,1) we get N(135) > 6, whereas n(135) = 4.
Example (16).
From SRGD(144; 9,16; 0,1) and Theorem 13, we get
N(145) ~ 7, whereas n(145)
Example (17).
= 4.
From SRGD(77; 7,11; 0,1) and Theorem 14 with x
and 9 respectively, we get N(82) > 3, N(86) > 5, whereas n(82)
=
9.
Table of lower bounds of N( v)
=
!2!:
v
5
=5
= n(86) = lo
150.
We provide below a table of lower bounds (lob.) of N(v) for those
values of v ~ 150, where the lower bound exceeds n(v).
In the last
column under remarks we indicate how the corresponding lower bound was
obtained.
The numbers for which v :: 2 (mod 4) are starred.
- 26 Table I
v
n(v)
1.b.
for
Remarks
N(v)
21
22*
24
39
50*
54*
57
58*
65
66*
68
69
70*
74*
75
76
78*
80
82*
84
85
86*
92
94*
95
2
1
2
2
1
1
2
1
4
1
3
2
1
1
2
3
1
4
1
2
4
1
3
1
4
4
2
3
3
5
3
7
2
6
5
5
5
5
5
5
5
5
7
3
5
5
5
5
2
5
Th. 3, Ex. (1)
Th. 5, m =
Th. 11, Cor. 6=5
Th. 12, Ex. (15)
Th. 6, Ex. (7)
SRGD(49; 7,7; 0,1), Th. 14, x=5
Th. 2, BIB (57; 8)
Th. 5, m=l
Th. 6, 6=8, m=2
Th. 9, x=7
Th. 9, x=5
Th. 9, x=4
Th.• 9, x=3
Th. 10, x=7
Th. 10, x=6
Th. 10, x=5
Th. 10, x=3
Th. 11, Cor. s=9
Th. 14, Ex. (17)
SRGD(77; 7,11; 0,1), Th. 14, x=7
SRGD(77; 7,11; 0,1), Th. 14, x=8
Th. 14, Ex. (17)
SRGD(91; 7,13; 0,1), Th. 14, x=1
Th. 5, m=2
SRGD(88; 8,11; 0,1), Th. 14, x=7
°
- 27 Table I (Cont'd.)
v
n(v)
96
98*
100
102*
105
110*
111
115
120
123
126*
129
130*
132
134*
135
138*
140
141
142*
145
146*
147
150*
2
1
3
1
2
1
2
4
2
2
1
2
1
2
1
4
1
3
2
1
4
1
2
1
1.b.-for
N(v)
6
5
5
5
6
2
5
6
7
5
5
6
5
5
5
6
3
5
6
5
7
5
6
5
Remarks
SRGD(88; 8,11; 0,1), Th. 14, x=8
SRGD(91; 7,13; 0,1), Th. 14, x=7
SRGD(91; 7,13; 0,1), Th. 14, .x=9
SRGD(91; 7,13; 0,1), Th. 14, x=ll
SRGD(104; 8,13; 0,1), Th. 14, x=l
Th. 7, Ex. (9)
SRGD(104j 8,13 j 0,1), Th. 14, x=7
SRGD(104; 8,13; 0,1), Th. 14, x=11
Th. 3, pt. (111), 8=16
SRGD(l12; 7,16; 0,1), Th. 14, x=ll
SRGD(119; 7,17; 0,1), Th. 14, x=7
SRGD(128; 8,16; 0,1), Th. 13
SRGD(119; 7,17; 0,1), Th. 14, x=ll
SRGD(U9; 7,17; 0,1), Th. 14, x=13
SRGD(133; 7,19; 0,1), Th. 14, x=l
Th. 12, Ex. (15)
SRGD(133; 7,19; 0,1), Th. 14, x=5
SRGD(133; 7,19; 0,1), Th. 14, x=7
SRGD(128; 8,16; 0,1), Th. 14, x=13
SRGD(133; 7,19; 0,1), Th. 14, x=9
Th. 13, Ex. (16)
SRGD(133; 7,19; 0,1), Th. 14, x=13
SRGD(136; 8,17; 0,1), Th. 14, x=ll
SRGD(133 1 7,19; 0,1), Th. 14, x=17
- 2"8 -
10.
Concluding remarks.
We are conscious of the fact that the table
given in the last section may be incomplete.
There may be other values
of v for which N(v) > n(v), as the pairwise balanced designs of index
unity have not been exhaustively studied.
Indeed they were improvised
for the purposes of the present paper, though we hope that they may be
useful in other combinatorial problems.
Only a lowr bound for N(v) has
been given and it may be possible to improve it in many cases.
Many new interesting problems arise as a result of the present work.
A positive integer n may be called an Eulerian number if there do not
exist more than one orthogonal Latin square of order n.
The density and
distribution of Eulerian numbers is a question of great interest.
The
table of the last section shows that there exists a run of 24 consecutive
non-Eulerian numbers from 63 to 86.
We shall show in a later communica-
tion by extending the methods of section 7, that there exist arbitrarily
large runs of consecutive non-Eulerian numbers.
In other words there
exist arbitrarily large gaps betwen consecutive Eulerian numbers.
It may also be observed that analogous methods for constructing
orthogonal arrays of strength two and index A can be made to depend on
pairwise balanced designs of index A, which are defined as in section 3,
except that every pair of distinct treatments occurs in A different
blocks.
It is hoped to consider these and allied problems in a subse-
quent communication.
- 29 Bibliography
1.
W. W. R. Ball, Mathematical recreations and essays, revised by R.S.M.
Coxeter, Macmillan and Co. Ltd." London (1942).
2.
R. C. Bose, "On the construction of balanced incomplete block designs,1I
Ann. of Eugen.London" Vol. 9 (1939), pp. 353-399.
3.
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designs," San.k.hy'E, Vol. 6 (1942)" pp. 105-110.
4.
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R. C. Bose and W. S. Connor, "Combinatorial properties of group diVisible incomplete block designs, II Ann. of Math. Stat., Vol. 23
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9.
R. C. Bose and S. S. Shrikhande, liOn the falsity of Euler's conjecture
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- 30 10.
K. A. Bush, "A generalization of a theorem due to MacNeish,"Ann. of
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F. W. Levi,
Finite geometrical systems, University of Calcutta,
(1942) •
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H. B. Mann, "The construction of orthogonal Latin squares, Ann. of
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T. Parker, "Construction of some sets of pairwise orthogonal Latin
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F. Yates, "Incomplete randomized blocks," Ann. of Eugen. London,
Vol. 1 (193 6 ), pp. 121-140.
- 31 -
Appendix
22 x 22 Orthogonal Latin Squares
1
1
2
3
4
5
6
7
Aa Dp Gv Pb Ft Te Vc
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
oJ
1
Sn Um Lr Rk
Ju Is Qh Bd Ho K1 Ni Ef Mjf Cg
I
~
Pd Bb Eq Ap Qe Gu Uf Jt Or Th Vn Ms S1 Kv Ri Da Ce Io Lm Hj Fg NIt
.5
Vg Qe Cc Fr Bq Rd Av Lp Ku Os Ui Ph Nt '1DJ Sj H1
4 Bp Pa Rf Dd Gs Cr Be UD Mq Lv
5 Tf Cq
Qb Sg
ot
Eb
Df Jo
Vj Qi Hu T~ Ab Im Fe Eg Ko
Fu
Va Es Sd Ub Gg
8 Qo Tj P1,Nu Vi Ks
9 Ns Ro
Uk Qm
Ik Ga
Nh
Jl
Ee At Ds Iv Vh Nr Mp Ou Pk Rj U1 KIn Be In Gd Fa 10 Hi
6 Et Ug Dr Re Ta Ff Bu Sk Jp Pi Hs Nq Ov Ql VIIl Ij Ln Cd
7 Cv
Mn
Fm'
Tl Kg Qj It Hr
Mr Hh Ev
Bt Sf Cp
Rg
Op PD
Kh
Ae Gb Mo
No Jk Mh De Li Bf Ac
Ud Aq Lc Oa Gm Fk Jb
Dn
Ie
Hv Pj Lt Ve I i Fp Cu Tg Dq Sa Br Jf Md Ob An Gl Ke Eh
10
Mu Ht So VI Rn Ip Qk Tb Pf Jj Gq
11
R1
Nv Iu To Pm Sh Jq Fs Uc ~ Kk Ar
12
Kr
an
13
Uj Ls Tn Iq Kp Vo Rh Gr Qd Au Pe Sb MIn Ct Fv Ek Di
14
Si Vk Mt tJh Jr Lq Po Du
15
Hq Ir Js Kt Lu Mv Np Qa Rb Se Td Ue Vf Pg 00 Gn Ail Bi Cj Dk El FIn
16
Db Ad Lh Ba Mk Ji On Cl Fj If Em He Ke Go Ng Pp Rv
t
17
Oh Ee
l
18
Lk Oi Fd Cf Nj De Hm Mg Bo ED A1 Ka Gh Je Ib Ut Ps Rr Tq Vp Qv Su
19
In
20
Fe Jh Nm Ok At Ea II Bj
21
Jm Gf Ki lin 01 Bg Fb Nd Ck Me Ie Eo Aj Dh
22
Gc Kn
p
Hp Jv Uo
Be M1 Cb
Qn
Ti Pe Gt
As
va
Dv
Ua Er Cs Fi
Ep Vb
Qf
Ne Oc Bh Am
Ld
Dt Me Gj La Hf Od Ci En
Ra Ll Bs Fq Eu
Re Bv
Kg
Ch
Nf
Ak Mb Ig
Hg
Oe Dj
Bb Ne Ja Of
Tc Nn Gp Og F1 Ej Ia em Hd Kb
Tu
Vt Qs Sr Uq
N1 Kj Ao Dm GlI Jg Fn Id Lf Ha Vr Qq Sp Uv
Pu
Rt Ts
f
~.
M1 OJ Ge Dg Hk Ed Kf Na Co Fh Em Lb Ai Jc Tv Vu Qt Ss Ur Pq
Lg lib
Do G1 Cn Me Kd Sq Up Pv Ru
Le
Rs Tr Vq
Qp
Tt
Rp
Vs Qr
Sv Uu Pt
Ag Lj Ih om Ca Ei He D1 Nb Jd Fo Bk Mf Qu St Us Pr Rq Tp VV
I
!