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THE DYING RABBIT PROBLEM
V. E. HOGGATT, JR., and D. A. LIIMD
San Jose State College, San Jose, Calif., and University of Cambridge, England
1. INTRODUCTION
Fibonacci numbers originally arose in the answer to the following problem posed by Leonardo de Pisa in 1202. Suppose there is one pair of rabbits
in an enclosure at the 0
month, and that this pair breeds another pair in
each of the succeeding months. Also suppose that pairs of rabbits breed in the
second month following birth, and thereafter produce one pair monthly.
What
is the number of pairs of rabbits at the end of the n
month? It is not diffi
th
cult to establish byJ induction that the answer is F n+2
l 0 , where F
n is the n
Fibonacci number. In [l] Brother Alfred asked for a solution to this problem
if, like Socrates, our rabbits are motral, say each pair dies one year after
birth. His answer [2], however, contained an e r r o r .
The mistake was noted
by Cohn [3], who also supplied the correct solution. In this paper we generalize the dying rabbit problem to arbitrary breeding patterns and death times.
2. SOLUTION TO THE GENERALIZED DYING RABBIT PROBLEM
Suppose that there is one pair of rabbits at the 0
time point, that this
pair produces Bf pairs at the first time point, B2 pairs at the second time
point, and so forth, and that each offspring pair breeds in the same manner.
We shall let B0 = 0, and put
B(x) = £
n=0
B x11
n
so that B(x) is the birth polynomial associated with the birth sequence
(Bj
n=0
The degree of B(x), deg B(x), may be finite or infinite. Now suppose a pair
of rabbits dies at the m
time point after birth (after possible breeding), and
let D(x) = x
be the associated death polynomial. If our rabbits are immortal,
482
Dec. 1969
put D(x) = 0.
THE DYING RABBIT PROBLEM
483
Clearly deg D(x) > 0 implies deg D(x) > deg B(x), unless
the rabbits have strange mating habits.
pairs of rabbits at the n
Let T
be the total number of live
time point, and put
T(x) = £ Tx11
n=0
where T 0 = 1. Our problem is then to determine T(x), where B(x) and D(x)
a r e known.
Let R
be the number of pairs of rabbits born at the n
time point
assuming no deaths. With the convention that the original pair was born at the
0
time point, and recalling that B0 = 0, we have
R0 - 1,
Rj
:=
BQR^ + BJRQ ,
R 2 = B0R2
+
B-^i +
B 2 RQ
•
and in general that
(1)
R
n
= £
po
B.R
.
J n"J
(n > 1) .
Note that for n = 0 this expression yields the incorrect R0 = 0. Then if
RW = E Rnxn ,
n=0
equation (1) is equivalent to
R(x) = R(x)B(x) + 1 ,
so that
484
THE DYING RABBIT PROBLEM
tota] number T* of pairs at the n
The total
[Dec
time point assuming no deaths is
given by
n
T*n = Y, R. ,
• n
3
and we find
*».l±±){i^)
(1 - x) [1 - B(x7)
1 -
(2)
00
•
A
k=Q
/ n
\
°°
n=0\j=0
7
n=0
E H. xn = D Tnxn = T*(x) •
= E
Hoggatt [4] used slightly different methods to show both (1) and (2).
We must now allow for deaths.
afterbirth, the number of deaths D
ber of births R
at the (n-m)
Since each pair dies m time points
at the n
time point.
time point equals the numTherefore
W = »«JV'
= ^Bn=0
n=0
Letting the total number of dead pairs of rabbits at the n
n
c n = y\
-*
D.
3
time point be
,
we have
^=(lo*%foV°Ho(NX"
D(x)
(1 - x) [1
= E Cxn 11 = C(x)
n=0n
1969]
THE DYING RABBIT PROBLEM
Now the total number of live pairs of rabbits T
T
n " Cn'
so
at the n
time point is
that
T(x) = T*(x) - C(x) =
(3)
485
T r
^ ^ L
M
_
3. SOME PARTICULAR CASES
To solve Brother Alfred f s problem, we put B(x) = x2 + x 3 + • • • + x12
and D(x) = x12 in (3) to give
1 - x12
1 - x12
T(x) - (1 - x)(l - x - x 3 - . . . - x 12 ) _ 1 - xl - xx2 + x13
X
2
x
It follows that the sequence { T } obeys
T 1 - 0 = T_L10 + T J _ 1 - - T
n+13
n+12
n+11
n
together with the initial conditions T
T12
= F
= F
(n > 0)
—
,
- for n = 0, 1, ••• , 11, and
i3 - 1J which agrees with the answer given by Cohn [3].
As another example of (3), suppose each pair produce a pair at each of
the two time points following birth, and then die at the m
birth (m> 2). In this case, B(x) = x + x
2
and D(x) = x .
see
-i
T(x) -
JH
l
" x
(1 - x)(l - x - x 2 )
Making use of the generating function
v
1 - x - X-
we get
n
t Fn+1
n+1 x ,
n
n=0
time point after
From (3), we
486
THE DYING RABBIT PROBLEM
. ,
T(x) =
—
—
1 - x - x2
m - 1 / °°
(4) = E
n
= 2^ —
n
3=0 1 - x - x2
m-1 / n
\
,. \
E F n+1 x ^
j=0 \ n =0
XJ
V^
1 + X + ••• + X
[Dec.
X
= E
J
E F k+1 )x
K+i
n=0 \ k=0
m-1
°°
n=0
n=m
= E ft (F n+3 - Dx11 - E
n
°° / m - 1
E
+
/
\
E Fn k + 1 x 1
n=m \ k=0
n
K+1
/
(F n+3 - F n _ m + 3 )x n
For m = 4r it is known [5] that
F
where L
n
is the n
n+3
F
F
n-4r+3
2rLn-2r+3 '
Lucas number, while for m = 4r + 2,
F
n+3 " F n - 4 r + l
=
L
2r+lFn-2r+2
'
which may be used to further simplify (4). In particular, for m = 2,
T(x) = 1
+
2x
+
E Fn+2xn = E Fn+2xn
n=0
n=0
while for m = 4 we have
T(x) = 1 + 2x + 4x2 + 7x3 + T L .x 11
~ , n+1
n=4
^
n=0
n+l
Thus for proper choices of B(x) and D(x) we a r e able to get both Fibonacci
and Lucas numbers as the total population numbers.
The second-named author was supported in part by the Undergraduate
Research Participation Program at the University of Santa Clara through NSF
Grant GY-273.
1969
]
THE DYING RABBIT PROBLEM
487
REFERENCES
1.
B r o t h e r U. Alfred, " E x p l o r i n g Fibonacci N u m b e r s , " Fibonacci Q u a r t e r l y ,
1 (1963), No. 1, 5 7 - 6 3 .
2.
B r o t h e r U. Alfred, "Dying Rabbit P r o b l e m Revived," Fibonacci Q u a r t e r l y ,
1 (1963), No. 4 , 53-56.
3.
John H. E. Cohn, " L e t t e r to the E d i t o r , " Fibonacci Q u a r t e r l y , 2 (1964),
108.
4.
V. E . Hoggatt, J r . , " G e n e r a l i z e d Rabbits for G e n e r a l i z e d Fibonacci N u m b e r s , " F i M n a c £ i ^ Q u £ r ^ r ] ^ , 6 (1968), No. 3 , 105-112.
5.
I. Dale R u g g l e s ,
"Some F i b o n a c c i
Results U s i n g
Fibonacci - Type
S e q u e n c e s , " Fibonacci Q u a r t e r l y , 1 (1963), No. 2 , 75-80.
• • • •
•
[Continued from page 4 8 1 . ]
3.
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M o r d e l P s E q u a t i o n , " Doctoral D i s s e r t a t i o n ,
Arizona State
University,
1968.
7.
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y2 - f2 = x 3 , "
Math. Scand. , 4 (1956), 95-107.
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H. London and R. F i n k e l s t e i n ,
"On M o r d e l P s Equation
y 2 - k = x 3 , " to
appear.
10.
H. London and R. F i n k e l s t e i n , "On M o r d e l P s Equation y2 - k = x 3 : II. The
11.
C a s e k < 0 , " to be submitted.
C. Siegel, " Z u m Beweise d e s S t a r k s c h e n S a t z e s , " Inventiones Math. ,
5 (1968), 180-191.
• * • •
•