Explorations of the Collatz Conjecture - Moravian College

Chapter 4
Blocks and Strings
In the previous chapter, we discussed a special type of parity sequence called
stems. Recall that when a set of these corresponding stems is applied to a pair of
real numbers x and x + 1, the result of both applications is the same real number
y. We will now look at two more special types of parity sequences that we will
call blocks and strings. These blocks and strings will take real numbers x and
x + 1 to some real numbers y and y + 1 (not necessarily respectively). Blocks and
strings are very helpful because we can append them to a parity sequence before
a stem, and doing so will give us another set of consecutive integers of the same
total stopping time.
4.1 Corresponding Blocks and Strings
Definition 2. A block is a pair of parity sequences of length k, b and b ′ , such that
for all positive integers x, T b (x) + 1 = T b ′(x + 1) and for any initial subsequences
v and v ′ of b and b ′ of equal length, |T b (x) − T b ′(x + 1)| 1.
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