7 x 7 filled with odd integers

 

We want to fill a 7 x 7 grid with the numbers 1 to 29, one digit per square.
We start somewhere on the grid with 1 and successively place the integers 2, 3, 4, ...until 29.
At each step a number must "touch" the existing structure at least by an edge.
The "touching" number we place will thus form with the existing digits at least one new integer.
All such new integers must be odd.
No "isolated" even integer can be visible on the grid.

+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   | 1 |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+

We start randomly with 1
 
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   | 2 | 1 |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+

We form 21 with 2 – and 21 is odd (don't mind the colors)
[we don't read vertically the digit 2 as an even "new integer"]

+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   | 2 | 1 |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   | 3 |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+

A new genuine odd integer is formed with 3: 13
 
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   | 4 | 2 | 1 |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   | 3 |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+

Two odd integers with the placement of 4: 421 and 13
 
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   | 4 | 2 | 1 |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   | 3 |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   | 5 |
+---+---+---+---+---+---+---+

Still two odd integers with 5: 421 and 135
 
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   | 6 | 4 | 2 | 1 |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   | 3 |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   | 5 |
+---+---+---+---+---+---+---+

Still two odd integers: 6421 and 135
 
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   | 6 | 4 | 2 | 1 |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   | 3 |
+---+---+---+---+---+---+---+
|   |   |   |   |   | 7 | 5 |
+---+---+---+---+---+---+---+

With 7 we get the third genuine odd integer 75 
 
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   | 6 | 4 | 2 | 1 |
+---+---+---+---+---+---+---+
|   |   |   |   |   | 8 | 3 |
+---+---+---+---+---+---+---+
|   |   |   |   |   | 7 | 5 |
+---+---+---+---+---+---+---+

With 8 we have five odd integers:
horizontally 6421, 83 and 75;
vertically 287 and 135 

+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   | 6 | 4 | 2 | 1 |
+---+---+---+---+---+---+---+
|   |   |   |   |   | 8 | 3 |
+---+---+---+---+---+---+---+
|   |   |   |   | 9 | 7 | 5 |
+---+---+---+---+---+---+---+

With 9 we turn 75 into 975: still five odd integers
 
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   | 6 | 4 | 2 | 1 |
+---+---+---+---+---+---+---+
|   |   |   | 1   0 | 8 | 3 |
+---+---+---+---+---+---+---+
|   |   |   |   | 9 | 7 | 5 |
+---+---+---+---+---+---+---+

10 (above) produces seven odd integers
 
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   | 6 | 4 | 2 | 1 |
+---+---+---+---+---+---+---+
|   |   |   | 1   0 | 8 | 3 |
+---+---+---+---+---+---+---+
|   |   | 1   1 | 9 | 7 | 5 |
+---+---+-------+---+---+---+

11 turn 61 into 611 and 975 into 11975; the overall count doesn't change: seven odd integers
 
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   | 1 | 6 | 4 | 2 | 1 |
+---+---+   +---+---+---+---+
|   |   | 2 | 1   0 | 8 | 3 |
+---+---+---+---+---+---+---+
|   |   | 1   1 | 9 | 7 | 5 |
+---+---+-------+---+---+---+

Above, a sound placement for 12 (forming 16421, 21083 and 121);
hereunder a forbidden one – as we can see an "isolated" even integer: the 12 itself

+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+
|   |   |   | 1   2 |   |   |
+---+---+---+---+---+---+---+
|   |   |   | 6 | 4 | 2 | 1 |
+---+---+---+---+---+---+---+
|   |   |   | 1   0 | 8 | 3 |
+---+---+---+---+---+---+---+
|   |   | 1   1 | 9 | 7 | 5 |
+---+---+-------+---+---+---+
 

The grid below is completed – and here are the questions: 
can you fill such a 7 x 7 grid that starts with 1 in the exact middle ?

+---+---+---+---+---+---+---+
| 2 | 2 | 2 | 2 | 2 | 2 | 1 |
|   |   |   |   |   |   |   |
| 9 | 8 | 7 | 6 | 5 | 4 | 7 |
+---+---+---+---+---+---+---+
| 2   1 | 2   3 | 1 | 2 | 1 |
+-------+-------+   |   |   |
| 1   9 | 2   2 | 6 | 0 | 5 |
+-------+---+---+---+---+---+
| 1   8 | 1 | 6 | 4 | 2 | 1 |
+-------+   +---+---+---+---+
| 1   4 | 2 | 1   0 | 8 | 3 |
+-------+---+---+---+---+---+
| 1   3 | 1   1 | 9 | 7 | 5 |
+-------+-------+---+---+---+

Can you fill a 9 x 9 grid starting also with a 1 in the middle 
(the last placed integer will be 45)?







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