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Conway Sequence
Mary Beth Helms
Jennifer Turner
John Horton Conway
 Born
in Liverpool, England on
Dec. 26, 1937
 BA in 1959 from Cambridge
 Ph.D. in 1964 from Cambridge
 Professor of Mathematics at
Princeton University
John H. Conway
Conway Sequence
A.k.a- the Look and Say Sequence
 The Method

–
–
–
–
Look at the first digit.
Count it as 1.
If the next digit is the same as the first, count as 2.
Continue counting until the digit is not the same and
record number of first digit.
– Repeat with the new digit and continue until end of
sequence.
Example

Start with 1:
– There is one 1:
– Now, there are two 1’s:
– There are one 2 and one 1:
– There are one 1, one 2, two 1’s:
1
1,1
2,1
1,2,1,1
1,1,1,2,2,1
Ordered Conway Sequence
 The
Method
–Look at lowest digit in sequence
–Count the number of times it
appears
–Record the number of times
–Repeat with the next lowest digit in
sequence
Example

Start with 1:
1
– There is one 1:
1,1
– There are two 1’s:
2,1
– There are one 1 and one 2:
1,1,1,2
– There are three 1’s and one 2:
3,1,1,2
– There are two 1’s, one 2, and one 3: 2,1,1,2,1,3
Different Starting Terms
The Ordered Sequence does not have to begin
with a single digit.
 We can begin with two digits.
 For example,

–
–
–
–
we can begin with:
We have one 1 and one 3:
Three 1’s and one 3:
Two 1’s and two 3’s:
1,3
1,1,1,3
3,1,1,3
2,1,2,3
Limiting Pattern for n<5
 The
limiting pattern for any single
n<5 is the same: 21322314
 The limiting pattern for nay single
n>4 is: 411213141n
Limiting Pattern for
(n<5, n<5)

The limiting pattern for any two digits
where both digits are less than 5 vary
– (1,1), (1,2), (1,3), (1,4), (2,3), (3,4) all have
the same limiting pattern of 21322314 (the
same as n<5)
– (2,4) and (4,4) have the same limiting pattern
of 31123314
– (2,2) has a limiting pattern of (2,2)
Limiting Patterns for
(3<n<50, 3<m<50)

There are four possible outcomes for
limiting sequence
– When a pair consists of (5, m)
– When a pair consists of (n, n)
– When a pair consists of consecutive numbers
(n, n+1)
– When a pair consists of non-consecutive
numbers (n,m) where m>n+1
Limiting Pattern for (5, m)

The limiting pattern always reaches a
three cycle pattern
5122133415161m
5122231425161m
4142131425161m
Limiting Pattern for (n, n)

If the number is the same, then the
limiting pattern would be the same as
beginning the sequence with (2, n)
Limiting Pattern for (n>5,n+1)
The limit always reaches a cyclical pattern
 The number of ones in the limiting pattern
is either n-2 and n-1
 Example 1: Given a pair (9,10) the
number of ones alternates between 7 and
8
 Example 2: Given a pair (6, 7) the limiting
pattern is a cycle of three where the
number of ones is either 4 and 5

Limiting Patterns for
(n>5, m>n+1)
The limit always reaches a cyclical pattern
 The number of ones is either n and n-1
 For example, given a pair (23, 35) the
number of ones in the alternating pattern
is 23 and 22

Conclusion
All Ordered Conway Sequences have a limiting
sequence when digits are less than 50
 In the future, more work could be done:

– To understand why a pair with 5 gives a three cycle
pattern
– To see if a limiting pattern can be a cycle of 4 or more
– To understand Ordered Conway Sequences where
n>50