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The grid with numbers given as below:

enter image description here

You can start wherever square you want and move left, right, up or down. You cannot pass the same square again. After ending your route, write those numbers next to each other in the route order.

What is the biggest number you can have?

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It's

594736281

because

complete tours are the only ones to give 9 digit numbers, so we need a complete tour. If you color the grid like a checkerboard, starting with a black square in the corner, you'll see that with every step the color will change, there are 5 black and 4 white squares, so we'll have to start at a black square. That means we cannot start at the high numbers in the center of the edges (6, 7, 8, 9). The next largest number is 5; we'll visit the 9 next and then the 4 because those are the highest orthogonally adjacent numbers. From there, there's only one way to complete the tour.

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