Conway Sequence Mary Beth Helms Jennifer Turner

John Horton Conway Born in Liverpool, England on Dec. 26, 1937 Born in Liverpool, England on Dec. 26, 1937 BA in 1959 from Cambridge BA in 1959 from Cambridge Ph.D. in 1964 from Cambridge Ph.D. in 1964 from Cambridge Professor of Mathematics at Princeton University Professor of Mathematics at Princeton University

John H. Conway

Conway Sequence A.k.a- the Look and Say Sequence A.k.a- the Look and Say Sequence The Method 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: 1 Start with 1: 1 –There is one 1: 1,1 –Now, there are two 1’s: 2,1 –There are one 2 and one 1: 1,2,1,1 –There are one 1, one 2, two 1’s: 1,1,1,2,2,1

Ordered Conway Sequence The Method 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 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. The Ordered Sequence does not have to begin with a single digit. We can begin with two digits. We can begin with two digits. For example, For example, –we can begin with: 1,3 –We have one 1 and one 3: 1,1,1,3 –Three 1’s and one 3: 3,1,1,3 –Two 1’s and two 3’s: 2,1,2,3

Limiting Pattern for n<5 The limiting pattern for any single n<5 is the same: The limiting pattern for any single n<5 is the same: The limiting pattern for nay single n>4 is: n The limiting pattern for nay single n>4 is: n

Limiting Pattern for (n<5, n<5) The limiting pattern for any two digits where both digits are less than 5 vary 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 (the same as n<5) –(2,4) and (4,4) have the same limiting pattern of –(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 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 The limiting pattern always reaches a three cycle pattern m m m m m m

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) 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 limit always reaches a cyclical pattern The number of ones in the limiting pattern is either n-2 and n-1 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 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 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 limit always reaches a cyclical pattern The number of ones is either n and n-1 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 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 All Ordered Conway Sequences have a limiting sequence when digits are less than 50 In the future, more work could be done: 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