SOME RESULTS CONCERNING POLYOMfNOES
DAVID A. KLARNER
University of Alberta, Edmonton, Alberta
INTRODUCTORY REMARKS
An n - o m i n o is a plane figure composed of
s q u a r e s joined edge on edge.
n
connected unit
In the e a r l y nineteen h u n d r e d s , H e n r y
Dudeney, the famous B r i t i s h puzzle expert, and the F a i r y Chess Review
p o p u l a r i z e d p r o b l e m s involving n - o m i n o e s which they r e p r e s e n t e d a s
figures cut from c h e c k e r b o a r d s ,
Solomon Golomb s e e m s to have been
the f i r s t m a t h e m a t i c i a n to t r e a t the subject s e r i o u s l y when as a g r a d uate student at H a r v a r d in 1954,
he published " C h e c k e r b o a r d s and
P o l y o m i n o e s " in the A m e r i c a n M a t h e m a t i c a l Monthly.
Since 1954,
s e v e r a l a r t i c l e s have a p p e a r e d (see R e f e r e n c e s ) ; in p a r t i c u l a r , R„ C„
Read [9] and M u r r a y Eden [2] have d i s c u s s e d the p r o b l e m of finding or
e s t i m a t i n g the n u m b e r
p(n) of n - o m i n o e s for a given n.
r e s u l t s it is now known that for l a r g e
n
cx
where
c,
and c ?
, x
< p(n)
From their
n
n
< c2
a r e c e r t a i n positive constants g r e a t e r than 1.
In
the f i r s t p a r t of this p a p e r we e n u m e r a t e a subset of n - o m i n o e s and
provide an i m p r o v e d lower
bound for
p(n); l a t e r we d i s c u s s other
p r o b l e m s of this s o r t and conclude with a brief exposition of p r o b l e m s
dealing with configurations of n - o m i n o e s .
MOSER'S BOARD P I L E P R O B L E M
In the following it will be convenient to have c e r t a i n conventions,,
We say the region between y = n-1 and y = n is the n
a r e c t a n g l e of width one a s t r i p .
row and call
The f i r s t s q u a r e on the left in a s t r i p
located in a row is called the initial s q u a r e of the s t r i p ; an n - o m i n o is
located in the plane when s o m e s q u a r e in the n - o m i n o exactly c o v e r s
a s q u a r e in the plane l a t t i c e . The set of all incongruent n - o m i n o e s will
be denoted by P(n) and for convenience we think of the e l e m e n t s of
P(n) located in a r b i t r a r y r e g i o n s of the plane.
position due to t r a n s l a t i o n s ,
Ignoring changes in
each e l e m e n t of P(n) h a s eight or l e s s
9
10
SOME RESULTS CONCERNING POLYOMINOES
p o s i t i o n s w i t h r e s p e c t t o 90
February
rotations about the origin and reflections
a l o n g t h e x o r y a x e s ; t a k i n g t w o n - o m i n o e s t o be d i s t i n c t if o n e c a n n o t
be t r a n s l a t e d t o c o v e r t h e o t h e r ,
w e find a n e w s e t
by i n c l u d i n g r o t a t i o n s a n d r e f l e c t i o n s
S(n)
of n - o m i n o e s i n
from
P(n)
P(n)
in
S(n).
T h e p r o b l e m w h i c h i s n o w t o be d i s c u s s e d w a s p r o b a b l y f i r s t
p o s e d by L e o M o s e r i n p r i v a t e c o r r e s p o n d e n c e w i t h t h e p r e s e n t a u t h o r ;
l a t e r h e p o s e d it in a d i f f e r e n t
f o r m a t t h e 1963 N u m b e r T h e o r y C o n -
f e r e n c e h e l d a t t h e U n i v e r s i t y of C o l o r a d o .
the p r o b l e m ,
but h i s r e s u l t s
The p r o b l e m
is to e n u m e r a t e a s u b s e t
E d e n [2] a l s o d i s c u s s e s
a r e not as c o m p l e t e as those given h e r e .
B(n)
of
S(n)
which
contains
n - o m i n o e s h a v i n g t h e p r o p e r t y t h a t t h e y c a n be t r a n s l a t e d i n s u c h a w a y
that t h e y a r e e n t i r e l y in the f i r s t and s e c o n d q u a d r a n t s with e x a c t l y
one s t r i p in the f i r s t r o w with its i n i t i a l s q u a r e at the o r i g i n and e a c h
r o w a f t e r t h e f i r s t h a s no m o r e t h a n o n e s t r i p i n i t .
Such n - o m i n o e s
m a y b e v i s u a l i z e d a s s i d e e l e v a t i o n s of b o a r d p i l e s c o n s i s t i n g of b o a r d s
of v a r i o u s
lengths which g e n e r a l l y h a v e not been s t a c k e d
carefully,
s e e F i g u r e 1.
M o s e r n o t e d t h a t if
b(n)
d e n o t e s t h e n u m b e r of e l e m e n t s i n
B(n),
then
(1)
b(n)
= X ( a 1 + a 2 - l ) ( a 2 + a 3 - 1) . . . ( a . ^ + a. - 1)
w h e r e t h e s u m m a t i o n e x t e n d s o v e r a l l c o m p o s i t i o n s a, + a ? . 0 . + a. = n
of n .
T h e r e l a t i o n i n (1) c a n be e s t a b l i s h e d b y t h e f o l l o w i n g c o m b i natorial argument.
F o r e a c h c o m p o s i t i o n a , + a~ + , . . + a. of n
°
1
Z
I
t h e r e i s a s u b s e t of B(n) c o n s i s t i n g of n - o m i n o e s w h i c h h a v e a s t r i p
of
a
s q u a r e s in the
t
o m i n o e s i n e a c h of t h e s e
row
(t = 1, 2, . . . , i); t h e n u m b e r
s u b s e t s i s 1 if
of n -
i = 1 w h i c h c o r r e s p o n d s to
t h e v a l u e of t h e e m p t y p r o d u c t i n t h e s u m (in t h i s t h e r e i s a s t r i p
u n i t s long in the f i r s t
row) and
(a, + a~ - 1) ( a ? + a~ - 1) . . .
n
(a. , +
+ a . - 1) if i > 2„ T h i s f o l l o w s s i n c e t h e r e a r e e x a c t l yJ (a^ , + a , - 1)
l
.,
t-1
t
w a y s to j o i n t h e s t r i p of a
s q u a r e s in the t
r o w to t h e s t r i p of
a
,
s q u a r e s i n t h e r o w b e l o w a n d t h e t o t a l n u m b e r of w a y s t o c o n n e c t
u p t h e s t r i p s t o f o r m a n n - o m i n o w o u l d be t h e p r o d u c t of a l l of t h e s e
alternatives,,
T h e s u b s e t s c o r r e s p o n d i n g to t h e c o m p o s i t i o n s of
n
are
1965
SOME RESULTS CONCERNING POLYOMINOES
11
exhaustive and disjoint in B(n), so that b(n) is the sum of the n u m b e r
of e l e m e n t s in each subset s which is (1).
The r e l a t i o n for
b(n) given by (1) does not furnish a v e r y handy
device for computing
mating
b(n).
b(n), but as Eden h a s shown it is helpful in e s t i -
Rather
than a t t e m p t to sum
(1) by p u r e l y a l g e b r a i c
m a n i p u l a t i o n s , we r e t a i n the g e o m e t r i c i n t e r p r e t a t i o n of the p r o b l e m
so that c o m b i n a t o r i a l a r g u m e n t s can be m o r e e a s i l y applied toward
finding a r e c u r s i o n r e l a t i o n for
b(n)Q
To find a r e c u r s i o n r e l a t i o n for
(r = 1, 2, . . . , n)
exactly r
of
B(n)
b(n) we define s u b s e t s
B (n)
which contain n - o m i n o e s with a s t r i p of
s q u a r e s in the f i r s t row and let b (n) denote the n u m b e r
of e l e m e n t s in B (n).
It is obvious that the s u b s e t s
B (n) (r = 1, 2,
. . . , n) a r e exhaustive and disjoint in B(n) s o t h a t w e have i m m e d i a t e l y
n
(2)
b(n) -
X b r (n)
r=l
.
Bv definition of
B (n), b (n) = 1.
Consider the e l e m e n t s of
n
n
B (n) with r < n; each e l e m e n t of B (n) c o n s i s t s of a s t r i p of r
r
r
s q u a r e s in the f i r s t row with some e l e m e n t of B(n-r) located in the
rows above the f i r s t . The situation can be a p p r a i s e d m o r e c o n c i s e l y
J
when one c o n s i d e r s the n u m b e r of ways an e l e m e n t of the subset
of B(n-r)
can be attached to the s t r i p of r
B.(n-r)
s q u a r e s in the f i r s t row
so that the n - o m i n o e s formed will be an e l e m e n t of B (n).
Clearly
this can be done in r + i - 1 ways, so that exactly (r + i - 1) b. ( n - r )
of the e l e m e n t s of B (n) have an element of B.(n-r)
r
s t r i p of
r
i
s q u a r e s in the f i r s t row.
(i = 1, 2, . . . , n - r )
of
B(n-r)
connected to the
Since the s u b s e t s
a r e exhaustive,
B-(n-r)
disjoint s u b s e t s ,
it
follows that
n- r
(3)
b (n) =
1
(r + i - 1) b . ( n - r )
for
r < n
.
i=l
It will be s e e n p r e s e n t l y that the r e l a t i o n s in (2) and (3) a r e
enough to find the d e s i r e d r e c u r s i o n r e l a t i o n for
b(n).
Before this
12
SOME RESULTS CONCERNING POLYOMINOES
February
r e s u l t can be given, we have to p r o v e a few l e m m a s .
L e m m a 1:
Proof:
If n > 1, b (n) - b T (n-1) = b ( n - r ) 0
r-1
r*
Using (3) it is s e e n that
n-r
b r (n) - b r _ x ( n - l )
=
2
n-r
(r+i-l)b.(n-r)
-
i=l
2
(r+i-2)b.(n-r)
i=l
n-r
=
2
b.(n-r)
,
i=l
but a c c o r d i n g to (2), the l a s t e x p r e s s i o n is p r e c i s e l y
b ( n - r ) , so the
proof is finishedo
L e m m a 2:
Proof:
that
If n > 1, b(n) = 2 b ( n - l ) + b (n) - b ^ n - 1 ) .
Using r e l a t i o n s for b(n) and b ( n - l )
n
(5)
b(n) - b ( n - l ) =
2
given by (2), it is s e e n
n-1
b (n) -
2
i=l
b.(n-l)
i=l.
n-1
= b x (n) +
2 jb.(n) - b . ^ n - l ) !
i=2
;
but a c c o r d i n g to L e m m a 1, b(n-i) can be substituted for b.(n) - b.
(n-1)
i n the l a s t m e m b e r of (5) so that making this substitution and t r a n s posing
- b ( n - l ) from the f i r s t to the l a s t m e m b e r gives
n-1
(6)
b(n) = b x (n) +
2
b(n-i)
.
i=l
Now using r e l a t i o n s given by (6) for b(n) and b ( n - l ) we have
n-1
(7) b(n) - b ( n - l )
= b^n) +
2
n-2
b(n-i) - b ^ n - 1 )
i=l
= b^n) - b,(n-l) + b(n-l);
-
2
i=l
b(n-l-i)
1965
SOME RESULTS CONCERNING POLYOMINOES
13
the d e s i r e d r e s u l t is obtained by adding b ( n - l ) to the f i r s t and last
m e m b e r s of (7).
L e m m a 3:
Proof:
t^ (n) = 4 b ^ n - 1 ) - 4 b ^ n - 2 ) + b ^ n - 3 ) + 2 b(n-3).
Taking
r = 1 in (3) gives an e x p r e s s i o n for
b, (n); namely,
n-1
(8)
b x (n) =
X
ib.(n-l)
.
i=l
Using r e l a t i o n s for
stituting b(n-2-i) for
b, (n)
and
b, (n-1)
given by (8) and s u b -
b . + , ( n - 1 ) - b.(n-2) and b ( n - l )
for
n-1
2
b.(n-l)
i=l
when they o c c u r , it is s e e n that
n-1
(9) b x (n) - b ^ n - 1 ) =
2
n-2
i b.(n-l) -
i=l
2
i b.(n-2)
i=l
n-1
=
2
n-2
b (n-1) + 1
i=l
n-2
ib.+1(n-l)-
i=l
2
i b.(n-2)
i=l
n-2
= b(n-l) + 2
i ) b . + 1 ( n - l ) - b.(n-2)j
i=l
n-2
= b(n-l) + 2
i b(n-2-i)
.
i=l
Adding b, (n-1) to each m e m b e r of the equality and dropping the
l a s t t e r m in the sum in the right m e m b e r of (9) (since b(0) - 0) a new
r e l a t i o n for
b, (n) is obtained:
n-3
(10)
b^n)
= b ^ n - 1 ) + b(n-l)
+
2
i=l.
(n-2-i) b(i)
.
14
SOME RESULTS CONCERNING POLYOMINOES
This t i m e using e x p r e s s i o n s for
and again writing a r e l a t i o n for
b, (n) and b , ( n - l )
February
givenby(lO)
b, (n) - b (n-1), one obtains after a
few a l g e b r a i c manipulations
n-1
(11)
b^n)
= 2^(11-1) - b (n-2) - 2b(n-2)
+
1
b(i)
.
i=l
Repeating the s a m e p r o c e d u r e as before only this t i m e using e x p r e s s i o n s for
(12)
b, (n) and b ^ n - l )
given by (11) yields
b x (n) = S b ^ n - 1 ) - 3b x (n-2) + ^ (n-3) + b(n-1) - 2b(n-2) + 2b(n-3);
but by L e m m a 2, b ( n - l ) - 2b(n-2) = b , ( n - l ) - b, (n-2)
so that s u b s t i -
tuting the l a t t e r quantity for the f o r m e r in (1 2) gives the d e s i r e d r e s u l t .
T h e o r e m 1:
Proof:
b(l) = 1, b(2) = 2, b(3) = 6, b(4) = 19, and
b(n) = 5 b ( n - l ) - 7b(n-2) + 4b(n-3) for n > 4.
The values of b(i) (i = 1,.2,
3, 4) can be computed d i r e c t l y
from (1) or by taking b(l) = b, (1) = 1 the r e l a t i o n s in (2) and (3) can
be used together for the s a m e p u r p o s e .
l i n e a r difference equations involving
L e m m a s 2 and 3 provide the
b, (n)
and
b(n)
which can be
u s e d to find
(13)
(14)
b(n) = 5 b ( n - l ) - 7b(n-2) + 4b(n-3) ,
b x (n) = 6b (n-1) - 12b (n-2) + l l b ( n - 3 ) - 4b(n-4),
which c o m p l e t e s the proof.
The a u x i l i a r y equation for (13) has one r e a l root g r e a t e r than 3„ 2
so that for
n sufficiently l a r g e
b(n) > ( 3 . 2 ) n .
(15)
We conclude from e a r l i e r r e m a r k s that
b(n)/8 incongruent n - o m i n o e s ,
B(n) contains at least
so that we can a l s o r e p l a c e
b(n) in
(15) with p(n).
Having disposed of the m o r e difficult p r o b l e m f i r s t , we now t u r n
attention to solving an e a s i e r and r e l a t e d p r o b l e m which was posed and
solved by M o s e r .
1965
SOME RESULTS CONCERNING POLYOMINOES
15
Let C(n) be the subset of B(n) which contains all n - o m i n o e s
having the p r o p e r t y that the initial s q u a r e of the s t r i p in the k
row
is no further to the left than the initial s q u a r e of the s t r i p in the (k-1) st
row.
R e c a l l from the definition of B(n) that the initial s q u a r e of the
s t r i p in the f i r s t row is always located at the origin.
Using a c o m b i -
n a t o r i a l a r g u m e n t s i m i l a r to the one provided for the proof of (1), it is
e a s y to prove
(16)
c(n) =
.
2
a. a 2 . . . a. ^
a, +a~+. ... +a.=n
1 Z
i
where
c(n) denotes the n u m b e r of e l e m e n t s in C(n).
Applying the
m e t h o d s he gave in [8] , M o s e r was able to show from (16):
T h e o r e m 2:
c(n) is equal to the
(2n-l)
st
Fibonacci n u m b e r .
We will give an a l t e r n a t e proof using the s a m e idea used in the
proof of T h e o r e m 1. Let C.(n)
be the subset of C(n) which contains
all n - o m i n o e s having s t r i p s of exactly
C l e a r l y the s u b s e t s
i
s q u a r e s in the f i r s t row.
C.(n) (i = 1, 2, . „ . , n) a r e exhaustive and d i s -
joint in C(n) so that letting c.(n) denote the n u m b e r of e l e m e n t s in
C.(n) we have
i
n
(17)
c(n) =
2
i=l
c.(n)
.
Next, it is e a s y to see that c (n) = 1, and for i < n, c.(n) = i c(n-i)
since e a c h element.of
C(n-i) can be joined exactly i ways to the s t r i p
of i s q u a r e s in the f i r s t row so as to form an e l e m e n t of C.(n); the
n - o m i n o e s thus formed obviously c o m p r i s e all the e l e m e n t s of
Substituting the e x p r e s s i o n s just found for
C.(n).
c.(n) into (17) we obtain
n-1
(18)
c(n) = 1 + 2
i c(n-i)
.
i=l
Using e x p r e s s i o n s for c(n) and c ( n - l )
bine the s u m s in c(n) - c ( n - l )
to find
given by (1 8) we can c o m -
16
SOME RESULTS CONCERNING POLYOMINOES
February
n-1
c(n) - c ( n - l ) =
£
c(i) ,
i=l
or
n-1
c(n) = c ( n - l ) + £'
c(i)
.
i=l
Now using e x p r e s s i o n s for c(n) and c ( n - l )
combine the s u m s in c(n) - c ( n - l ) and deduce
(20)
given by (19) we can
c(n) = 3 c ( n - l ) - c(n-2).
It is e a s y to p r o v e that the Fibonacci n u m b e r s with odd indices
satisfy the r e c u r r e n c e r e l a t i o n i n (20). Also, using (16) we find c(l) = f,
and c(2) = f~
(f. denotes the i
the s e q u e n c e s
Fibonacci n u m b e r as usual) so that
{c.} and {f ? . ,} m u s t be identical. E d i t o r i a l Note:
See H-50 Dec. 1964 and note notational d i f f e r e n c e s .
N-OMINOES ENCLOSED IN RECTANGLES
R0 C. Read [9] h a s t r e a t e d the p r o b l e m of e n u m e r a t i n g the nominoes which "fit" into a p x q r e c t a n g l e . An n - o m i n o is said to fit
in a p x q r e c t a n g l e if it is the s m a l l e s t r e c t a n g l e in which the n - o m i n o
can be d r a w n with the s i d e s of its s q u a r e s p a r a l l e l to the sides of the
r e c t a n g l e . R e a d ' s m e t h o d s give exact counts of the n - o m i n o e s in the
s e t s c o n s i d e r e d ; however, it is p o s s i b l e to obtain lower bounds for t h e s e
n u m b e r s with l e s s effort using s i m i l a r i d e a s . To i l l u s t r a t e we will
c o n s i d e r the p r o b l e m of e s t i m a t i n g from below the n u m b e r s ? (n) of
n - o m i n o e s which fit in a 2 x k r e c t a n g l e ; we call this set of n - o m i n o e s
S ? (n). Two e l e m e n t s a r e distinct if they a r e incongruent, so S~(n) is
a subset of P(n).
F i r s t , we o b s e r v e that each e l e m e n t of S ? (n) can be located ent i r e l y in the f i r s t q u a d r a n t in rows 1 and 2 with a s q u a r e Located at the
origin. If each e l e m e n t of S ? (n) is situated in the way j u s t d e s c r i b e d
in e v e r y w a y p o s s i b l e , a new set U(n) is obtained w h e r e two e l e m e n t s
a r e distinct if one does not exactly cover the other 0 C l e a r l y , u(n), the
1965
SOME RESULTS CONCERNING POLYOMINOES
17
n u m b e r of e l e m e n t s in U(n), is l e s s than or equal to 4 s ? ( n ) .
U(n) can be divided into two s e t s
U"(n) and
Now
U'(n) consisting r e s p e c -
tively of n - o m i n o e s having and not having a s q u a r e in the second row
attached to the s q u a r e at the origin.
Let the n u m b e r of e l e m e n t s in
U'(n) and U"(n) be u'(n) and u"(n)
respectively.
Now it is e a s y to
see that
u f {n) = u ' ( n - l ) + u " ( n - l )
(21)
since e v e r y e l e m e n t of U"(n-1)
and U ' ( n - l )
can be t r a n s l a t e d a unit
to the right of the origin and a s q u a r e located at the origin to give an
e l e m e n t of U'(n) and e v e r y e l e m e n t is obviously obtained in this fashion.
It is a l s o e a s y to p r o v e
(22)
u"(n) = 2u'(n-2) + u"(n-2)
since e v e r y e l e m e n t of U"(n-2) and e v e r y e l e m e n t of U'(n~2) and its
h o r i z o n t a l reflection can be t r a n s l a t e d a unit to the right of the origin
and two s q u a r e s added (one at the origin,
the other attached above it)
to form e v e r y e l e m e n t of U"(n).
Using (21) and (22) we can find
(23)
u'(n)
= u'(n-l) +u'(n-2)
+u'(n-3)
and
uM(n) = u " ( n - l ) + u"(n-2) + u M (n-3),
(24)
so that it b e c o m e s evident from
(25)
u(n) = u ( n - l ) + u(n-2) + u(n-3)
Since
s?(n).
u(n) = u'(n) + u"(n)
u(n)/4 < s ? (n) s
(25) p r o v i d e s
that
.
a r e l a t i o n for
estimating
The s a m e p r o c e d u r e can be used for e s t i m a t i n g the n u m b e r of
e l e m e n t s in S,(n) consisting of n - o m i n o e s which fit in k x q r e c t a n g l e s .
N-OMINO CONFIGURATIONS
P r o b l e m s involving n - o m i n o configurations have enjoyed a g r e a t
p o p u l a r i t y among m a t h e m a t i c a l r e c r e a t i o n i s t s [4] , [6] .
We plan to
devote a s m a l l amount of space to giving an exposition of p r o b l e m s
which m a y be of i n t e r e s t to the m a t h e m a t i c i a n . G e n e r a l l y t h e s e p r o b l e m s
18
SOME RESULTS CONCERNING POLYOMINOES
have the following form:
February
given a region of a r e a A and a set of n-
ominoes having a combined a r e a a l s o A; can one cover the region with
the s e t ?
We say a set exactly c o v e r s a region when t h e r e is no o v e r l a p
and no p a r t of the region is left u n c o v e r e d .
It would be i n t e r e s t i n g to
know n e c e s s a r y conditions that an n - o m i n o be such that an unlimited
n u m b e r of copies could be used to exactly cover the plane 0
A related
p r o b l e m is to d e t e r m i n e n e c e s s a r y conditions that some n u m b e r of
copies of a given n - o m i n o could be used to exactly cover a r e c t a n g l e .
Thus, some e a s i l y proved n e c e s s a r y conditions a r e given by:
(i)
if an n - o m i n o has two lines of s y m m e t r y and a set of t h e s e nominoes exactly c o v e r s a r e c t a n g l e , then the n - o m i n o is itself a
rectangle.
(ii)
if an n - o m i n o fits in a p x q r e c t a n g l e and c o v e r s diagonally opposite c o r n e r s of the r e c t a n g l e , and a set of t h e s e n - o m i n o e s can
be used to exactly cover a r e c t a n g l e ,
then the n - o m i n o is itself
a rectangle.
A r e c t a n g l e exactly c o v e r e d with a set of congruent n - o m i n o e s is
m i n i m a l when no r e c t a n g l e of s m a l l e r a r e a can be exactly c o v e r e d with
a set of the s a m e n - o m i n o e s containing fewer e l e m e n t s .
It is e a s y to
p r o v e that t h e r e is an unlimited n u m b e r of m i n i m a l r e c t a n g l e s involving
e i t h e r two or four n - o m i n o e s .
F i g u r e s 2, 3, 4 and 5 show i n s t a n c e s of
m i n i m a l r e c t a n g l e s involving m o r e than four n - o m i n o e s .
Are there
infinitely m a n y c a s e s of m i n i m a l r e c t a n g l e s which involve m o r e than
four n - o m i n o e s (no two c a s e s involving s i m i l a r n - o m i n o e s ) ? A r e t h e r e
m i n i m a l r e c t a n g l e s involving an odd n u m b e r of n - o m i n o e s which a r e not
themselves rectangles?
Note that the configurations depicted in F i g u r e s 1, 2, 3 and 4 a r e
s y m m e t r i c with r e s p e c t to the c e n t e r s of the r e c t a n g l e s .
Can this a l -
ways be done in m i n i m a l r e c t a n g l e s ?
GENERALIZATIONS OF N-OMINOES
In [5] , Golomb s u g g e s t s that one could t r y to d e t e r m i n e or e s t i m a t e the n u m b e r of distinct ways n e q u i l a t e r a l t r i a n g l e s or n r e g u l a r
1965
SOME RESULTS CONCERNING POLYOMINOES
hexagons could be simply connected edge on edge.
19
Using 1, 2, 3, 4,
5 or 6 hexagons 1, 1, 3, 7, 22 or 83 combinations respectively result;
so far no upper or lower bounds for the terms of this sequence have
been given.
There is no reason why regular k-gons could not be used for
cells in such combinatorial problems; overlapping of cells could be
permitted so long as no cell exactly covered another.
Thus, where
at most four squares or three hexagons might have a vertex in common, at most ten pentagons might have a vertex in common.
The num-
ber of distinct ways to join two regular k-gons is one; the number of
ways to join three regular k-gons is the greatest integer in k/2.
Per-
haps it would not be difficult to determine in how many ways four or
five regular k-gons could be joined together edge on edge so that distinct simply connected figures are formed.
Still another generalization of n-ominoes which seems not to
have been considered is joining squares together edge on edge in three
or more dimensions.
The number of ways of joining k cubes face on
face in three dimensions (including mirror images of some pieces) is
1, 1, 2, 8, 29j and 166 for
k = 1, 2, 3, 4, 5, and 6 respectively; no
bounds have been given for the terms of this sequence nor has much
been done in a serious vein connected with the packing of space with
these three dimensional analogues of polyominoesD
REFERENCES
1.
T. R. Dawson and W. E. Lester,
"A Notation for Dissection
Problems, " The Fairy Chess Review, Vol. 3, No. 5 (April 1957),
pp„ 46-47.
2.
M. Eden, "A Two-Dimensional Growth Process, "Proceedings of
the Fourth Berkeley Symposium on Mathematical Statistics and
Probability, Berkeley and Los Angeles, University of California
Press, (1961), pp0 223-239.
3.
M. Gardner, The 2nd Scientific American Book of Mathematical
Puzzles and Diversions, New York, Simon and Schuster, (1961),
pp. 65-880
SOME RESULTS CONCERNING POLYOMINOES
February
Mo G a r d n e r , " M a t h e m a t i c a l G a m e s , " Scientific A m e r i c a n , Vol.
196,
( D e c , 1957), pp e 126-134.
S. Wo Golomb,
" C h e c k e r b o a r d s and P o l y o m i n o e s , " A m e r i c a n
M a t h e m a t i c a l Monthly, Vol„ 61, (1954), pp. 675-82.
, "The G e n e r a l T h e o r y of P o l y o m i n o e s , " R e c r e a t i o n a l
M a t h e m a t i c s Magazine, No. 's 4, 5, 6 and 8, (1962).
, "Covering a Rectangle with L - t e t r o m i n o e s , " A m e r ican M a t h e m a t i c a l Monthly, Vol. 70, (1961), pp 0 3 9 - 4 3 .
L. M o s e r a n d E o L. Whitney, "Weighted Compositions, " Canadian
M a t h e m a t i c a l Bulletin, Vol 0 4, No. 1, (1961), pp. 3 9 - 4 3 ,
R. C. Read,
"Contributions to the Cell Growth P r o b l e m , " Ca-
nadian J o u r n a l of M a t h e m a t i c s , Vol. 14, (1962), pp. 1-20.
W. Stead, "Dissection, " The F a i r y Chess Review, Vol. 9, No. 1,
(Dec.
1954), pp. 4 6 - 4 7 .
S„ Wo Golomb, " P o l y o m i n o e s , " C h a r l e s S c r i b n e r , N. Y., 1965 0
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