there are 120 seven digit numbers which can be formed by starting from any number in the diagram. 1,2,3,4,5,6,7. And 7 is in the middle of the diagram. And proceeding to any neighboring number, using each number only once and only once. Of these, how many are divisible by 11 and what are the number values?

Given the problem, it describes forming seven-digit numbers using specific rules about movement and number uniqueness. However, to solve this, we need more detailed information about the "diagram" where these numbers (1, 2, 3, 4, 5, 6, 7) are placed, and which neighboring numbers each can proceed to. Since there is a lack of clarity about the exact structure of the diagram, we will approach the problem using some assumptions and a mathematical method to check divisibility by 11.

### Divisibility by 11
The rule for divisibility by 11 is that the alternating sum of the digits (starting from the first, subtracting the second, adding the third, and so on) must be divisible by 11. For example, if the number is \( N = a_1a_2a_3a_4a_5a_6a_7 \), it is divisible by 11 if:

\[
a_1 - a_2 + a_3 - a_4 + a_5 - a_6 + a_7 \equiv 0 \pmod{11}
\]

Thus, for each of the 120 seven-digit numbers, we need to compute this alternating sum and check if it is divisible by 11.

### Steps to Solve:
1. Clarify the "neighboring" number rules in the diagram.
2. Generate all 120 valid seven-digit numbers.
3. For each number, calculate the alternating sum \( a_1 - a_2 + a_3 - a_4 + a_5 - a_6 + a_7 \).
4. Count how many of these sums are divisible by 11.
5. List out the numbers that satisfy this condition.

Can you clarify the diagram or provide further details, particularly the layout of numbers and the allowed movements between neighbors? This will allow us to generate the correct permutations for analysis.

There are 120 seven-digit numbers which can be formed by starting from any number in the diagram. 1, 2, 3, 4, 5, 6, 7. And 7 is in the middle of the diagram. And proceeding to any neighboring number, using each number only once and only once. Of these, how many are divisible by 11 and what are the number values?

Learn how to find how many seven-digit numbers, formed using digits 1 to 7 with movement rules, are divisible by 11. Explore the alternating sum method to check for divisibility by 11 in number theory. Suitable for high school students, Grades 9-12.

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