Given a sequence which contains the numbers 1 through 8, a move consists of moving one of the numbers to the front of the sequence. For example, for the sequence $(1, 5, 8, 3, 6, 2, 7, 4),$ we can move the number $6$ to the front, which results in the sequence $(6, 1, 5, 8, 3, 2, 7, 4).$ (We can also choose to move the first number to the front, which does not change the sequence.)

Question

Given a sequence which contains the numbers 1 through 8, a move consists of moving one of the numbers to the front of the sequence.  For example, for the sequence $(1, 5, 8, 3, 6, 2, 7, 4),$ we can move the number $6$ to the front, which results in the sequence $(6, 1, 5, 8, 3, 2, 7, 4).$  (We can also choose to move the first number to the front, which does not change the sequence.)

We want to turn a sequence into the sequence (1, 2, 3, 4, 5, 6, 7, 8) with a series of moves.  For how many starting sequences does this take a a minimum of two moves?

GPT-4o mini · Answer

To determine how many starting sequences require a minimum of two moves to transform into the sorted sequence $(1, 2, 3, 4, 5, 6, 7, 8)$, we consider a sequence where the ordering of elements affects the number of moves.

Key Observations:

First Move: The first move consists of bringing any number from the sequence to the front. This number can be any of the eight numbers in any position. If the number at the front is already 1, we can continue with the ordering, otherwise, we will need at least one more move.
Second Move: After the first move, we need to ensure that other numbers are also in the correct position to achieve the final sequence. If the original first number is 1, we place it at the front and check if we can achieve the goal in one additional move, otherwise both moves must be considered.

Starting with 1 at the Front:

If the sequence begins with the number 1:

The sequence will be in the form $ (1, a_2, a_3, a_4, a_5, a_6, a_7, a_8) $. The remaining numbers can be any permutation of $ (2, 3, 4, 5, 6, 7, 8) $.
We can achieve the goal of $ (1, 2, 3, 4, 5, 6, 7, 8) $ in one move if $ a_2 = 2 $. Any other number in the second position means we will need a second move.

The number of sequences starting with $1$ and having $2$ in the second position can be calculated as:

There are $6!$ ways to arrange $ (3, 4, 5, 6, 7, 8) $: \[ 6! = 720 \]

The total number of sequences starting with 1, which can be arranged with $ (3, 4, 5, 6, 7, 8) $ in any order is $7!$: \[ 7! = 5040 \] Thus, the sequences from which we cannot achieve in one move: \[ 7! - 720 = 4320 \]

Sequences Not Starting with 1:

If a sequence does not start with 1, it means that the number 1 is somewhere else in the sequence:

The number 1 can be in positions $2, 3, 4, 5, 6, 7, 8$. Regardless of where 1 was originally placed, it would require at least one move to bring it forward.

When any number $2-8$ is at the front, we will require at least two moves.

Counting the arrangements:

Any arrangement of $(1, 2, 3, 4, 5, 6, 7, 8)$ gives us $8!$: \[ 8! = 40320 \]

Subsequently, the arrangements that start with 1: \[ 7! = 5040 \]

Thus, the total number of arrangements that need at least two moves is: \[ 8! - 7! = 40320 - 5040 = 35280 \]

Conclusion

Combining both cases (leading 1 and other leading numbers), the total number of sequences that take a minimum of two moves to reach the sorted order $ (1, 2, 3, 4, 5, 6, 7, 8) $ is:

\[ \boxed{35280} \]