2)How many arrangements of the integers 1,2,3,...,n are there such that each integer differs by one (except for the first integer) from some integer to the left of it?
Let's construct the arrangement.
First consider 1, it can only have two placed next to it so it has to be at the end or beginning. Do you see why?
Ok, so let's suppose the arrangement starts with 1, then the only number that can come after it is 2, then 3 then 4...
Now suppose 1 is at the end. Then the only number that can come before is 2, then 3 then 4...
The only two arrangements are
1 2 3 4 5 ... n-1 n and
n (n-1) (n-2) ... 3 2 1
By the way, we are not constructing a digit, but rather an arrangement for the first n integers.
In order to get the answer to the question, we first need to understand the pattern and constraints of the arrangement. We are given the integers 1,2,3,...,n and we need to arrange them in a way that each integer differs by one from some integer to the left of it (except for the first integer).
To start, we consider the first integer, which is 1. Since each integer should differ by one from some integer to the left of it, the only possible option for 1 is to be either at the beginning or at the end of the arrangement. This is because there are no integers that would satisfy the condition of differing by one before the first integer.
Let's consider the case where 1 is at the beginning of the arrangement. In this case, the only number that can come after 1 is 2. Then, the only number that can come after 2 is 3, and so on, until we reach n-1, which is the integer right before n. Finally, n is placed at the end of the arrangement.
So, if the arrangement starts with 1, the sequence would look like this: 1 2 3 4 ... n-1 n.
Now, let's consider the case where 1 is at the end of the arrangement. In this case, the only number that can come before 1 is 2. Then, the only number that can come before 2 is 3, and so on, until we reach n-1. Finally, n is placed at the beginning of the arrangement.
Therefore, if the arrangement ends with 1, the sequence would look like this: n (n-1) (n-2) ... 3 2 1.
These are the only two possible arrangements that satisfy the condition of each integer differing by one from some integer to the left of it (except for the first integer).
So, to answer the question, there are two arrangements of the integers 1,2,3,...,n such that each integer differs by one (except for the first integer) from some integer to the left of it.