The answer is

Here x represents position where a movement is made to the outer block

Pre Steps

You need to see the possible number of inward and outward movements you can make from the box at center. This is 22(Because you need to get in inside box 11 times and get out of it 11 times). Now observe that the perimeter of inside box is 34 units, of which 22 will be used for making in and out movements. So we have 12(34-22) units in perimeter left where we should not see any movements. But wait, there a catch. If you look at the squares circled, you can only use atmost 2 of the 3 sides(give this a thought). Since there are 10 such squares where you can only use 2 of the 3 sides, this reduces the total places where we can make in and out movements by 10 more places. So we can afford to not make in-out movement in atmost 2 (12-10) places.(Or make 1 movement in 2 of the regions circled in purple) ,

Step 1

So you can see that we already got 1 the moment we step out of S. Now we need to join it to 3

Step 2

Similarly go about other numbers 5, 7, and 9

Step 3

Now we connect 12 to G. Now we just need to make movements in-out in the remaining squares such that the total movements when it reaches 12 is 12.

The answer is

Here x represents position where a movement is made to the outer block

Pre Steps

You need to see the possible number of inward and outward movements you can make from the box at center. This is 22(Because you need to get in inside box 11 times and get out of it 11 times). Now observe that the perimeter of inside box is 34 units, of which 22 will be used for making in and out movements. So we have 12(34-22) units in perimeter left where we should not see any movements. But wait, there a catch. If you look at the squares circled, you can only use atmost 2 of the 3 sides(give this a thought). Since there are 10 such squares where you can only use 2 of the 3 sides, this reduces the total places where we can make in and out movements by 10 more places. So we can afford to not make in-out movement in atmost 2 (12-10) places.(Or make 1 movement in 2 of the regions circled in purple) ,

Step 1

So you can see that we already got 1 the moment we step out of S. Now we need to join it to 3 .

Not Step 1

The below can't be a correct line diagram for step 1. The reason is simple. If you look at blue circles in the diagram, we have already missed those. And we can only afford to skip 2 of these places. So when we reach the red circle and only use 1 of its sides(instead of 2), we know that this map cannot be completed.

Step 2

Similarly go about other numbers 5, 7, and 9

Step 3

Now we connect 12 to G. Now we just need to make movements in-out in the remaining squares such that the total movements when it reaches 12 is 12.

The answer is

Here x represents position where a movement is made to the outer block

Pre Steps

You need to see the possible number of inward and outward movements you can make from the box at center. You will see it is close to 12. So you need to make in-out movement a lot

Step 1

So you can see that we already got 1 the moment we step out of S. Now we need to join it to 3

Step 2

Similarly go about other numbers 5, 7, and 9

Step 3

Now we connect 12 to G. Now we just need to make movements in-out in the remaining squares such that the total movements when it reaches 12 is 12.

The answer is

Here x represents position where a movement is made to the outer block

Pre Steps

You need to see the possible number of inward and outward movements you can make from the box at center. This is 22(Because you need to get in inside box 11 times and get out of it 11 times). Now observe that the perimeter of inside box is 34 units, of which 22 will be used for making in and out movements. So we have 12(34-22) units in perimeter left where we should not see any movements. But wait, there a catch. If you look at the squares circled, you can only use atmost 2 of the 3 sides(give this a thought). Since there are 10 such squares where you can only use 2 of the 3 sides, this reduces the total places where we can make in and out movements by 10 more places. So we can afford to not make in-out movement in atmost 2 (12-10) places.(Or make 1 movement in 2 of the regions circled in purple) ,

Step 1

So you can see that we already got 1 the moment we step out of S. Now we need to join it to 3

Step 2

Similarly go about other numbers 5, 7, and 9

Step 3

Now we connect 12 to G. Now we just need to make movements in-out in the remaining squares such that the total movements when it reaches 12 is 12.