Using the diagram below, a seven digit integer can be created as follows: trace a path that uses each line segment exactly once and use the labels on the line segments as digits. For example, the path that goes from $C$ to $A$ to $B$ to $C$ to $E$ to $B$ to $D$ and finally to $E$ gives the positive integer 3264715. What is the largest possible positive integer that can be created in this way?

![](https://cdn.mathpix.com/cropped/2024_04_17_d13f9eaf1d614231018ag-1.jpg?height=608&width=612&top_left_y=1905&top_left_x=819)
Since every line segment must be used exactly once, the integer will have seven digits with each digit from 1 to 7 used exactly once.

Observe that the path that goes from $E$ to $B$ to $C$ to $E$ to $D$ to $B$ to $A$ to $C$ creates the integer 7645123 .

We will now argue that this is the largest integer that can be created.

Since every digit from 1 through 7 is used exactly once, if we start with any digits other than 7 and 6 , in that order, the integer formed must be smaller than 7645123 .

This means that the path forming the largest possible integer must start with $E$, then go to $B$, and then $C$.

From $C$, the options are to go to $A$ or $E$. If the path continues to $A$, then the integer would start with 763 , which is guaranteed to be smaller than 7645123 , regardless of how the path is completed.

The path must start with $E, B, C, E$, so the first three digits of the largest inter are 764 .

The line segments connecting $E$ to $B$ and $E$ to $C$ have already been used, which means that the path must continue to $D$ since otherwise we would have to reuse a line segment. We now have shown that the first four digits of the largest integer are 7645 .

Since the path arrived at $D$ from $E$, there is nowhere to go from $D$ but to $B$, and then by similar reasoning, it must continue to $A$ and finally to $C$, so the largest integer must be 7645123 .

ANSWER: 7645123

#