3 deleted 40 characters in body
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Note: This solution is DISPROVED. But the reasoning might be correct.

Here's what I think:

enter image description here
I have shown three configurations A, B, C. The main thing I kept in mind while making these grids is that the board should be closely packed with pawns. Now you may ask how is B different than C and A that it have more pawns.

I think the reason is

enter image description here
their slopes. The magnitude of slope for the line joining the pawns in slant position is more for B than A and it is more in such a way that the line contains most number of pawns by the given rules. In situation C, it makes both kind of angles, therefore having mixture of both kind of slopes.

Therefore

Professor Halfbrain's corrected first theorem:

It is possible to place 16 pawns according to the rules.

Professor Halfbrain's corrected second theorem:

It is not possible to place 17 pawns according to the rules.

Note: This solution is DISPROVED. But the reasoning might be correct.

Here's what I think:

enter image description here
I have shown three configurations A, B, C. The main thing I kept in mind while making these grids is that the board should be closely packed with pawns. Now you may ask how is B different than C and A that it have more pawns.

I think the reason is

enter image description here
their slopes. The magnitude of slope for the line joining the pawns in slant position is more for B than A and it is more in such a way that the line contains most number of pawns by the given rules. In situation C, it makes both kind of angles, therefore having mixture of both kind of slopes.

Therefore

Professor Halfbrain's corrected first theorem:

It is possible to place 16 pawns according to the rules.

Professor Halfbrain's corrected second theorem:

It is not possible to place 17 pawns according to the rules.

Note: This solution is DISPROVED.

Here's what I think:

enter image description here
I have shown three configurations A, B, C. The main thing I kept in mind while making these grids is that the board should be closely packed with pawns. Now you may ask how is B different than C and A that it have more pawns.

I think the reason is

enter image description here
their slopes. The magnitude of slope for the line joining the pawns in slant position is more for B than A and it is more in such a way that the line contains most number of pawns by the given rules. In situation C, it makes both kind of angles, therefore having mixture of both kind of slopes.

Therefore

Professor Halfbrain's corrected first theorem:

It is possible to place 16 pawns according to the rules.

Professor Halfbrain's corrected second theorem:

It is not possible to place 17 pawns according to the rules.

2 added 81 characters in body
source | link

Note: This solution is DISPROVED. But the reasoning might be correct.

Here's what I think:

enter image description here
I have shown three configurations A, B, C. The main thing I kept in mind while making these grids is that the board should be closely packed with pawns. Now you may ask how is B different than C and A that it have more pawns.

I think the reason is

enter image description here
their slopes. The magnitude of slope for the line joining the pawns in slant position is more for B than A and it is more in such a way that the line contains most number of pawns by the given rules. In situation C, it makes both kind of angles, therefore having mixture of both kind of slopes.

Therefore

Professor Halfbrain's corrected first theorem:

It is possible to place 16 pawns according to the rules.

Professor Halfbrain's corrected second theorem:

It is not possible to place 17 pawns according to the rules.

Here's what I think:

enter image description here
I have shown three configurations A, B, C. The main thing I kept in mind while making these grids is that the board should be closely packed with pawns. Now you may ask how is B different than C and A that it have more pawns.

I think the reason is

enter image description here
their slopes. The magnitude of slope for the line joining the pawns in slant position is more for B than A and it is more in such a way that the line contains most number of pawns by the given rules. In situation C, it makes both kind of angles, therefore having mixture of both kind of slopes.

Therefore

Professor Halfbrain's corrected first theorem:

It is possible to place 16 pawns according to the rules.

Professor Halfbrain's corrected second theorem:

It is not possible to place 17 pawns according to the rules.

Note: This solution is DISPROVED. But the reasoning might be correct.

Here's what I think:

enter image description here
I have shown three configurations A, B, C. The main thing I kept in mind while making these grids is that the board should be closely packed with pawns. Now you may ask how is B different than C and A that it have more pawns.

I think the reason is

enter image description here
their slopes. The magnitude of slope for the line joining the pawns in slant position is more for B than A and it is more in such a way that the line contains most number of pawns by the given rules. In situation C, it makes both kind of angles, therefore having mixture of both kind of slopes.

Therefore

Professor Halfbrain's corrected first theorem:

It is possible to place 16 pawns according to the rules.

Professor Halfbrain's corrected second theorem:

It is not possible to place 17 pawns according to the rules.

1
source | link

Here's what I think:

enter image description here
I have shown three configurations A, B, C. The main thing I kept in mind while making these grids is that the board should be closely packed with pawns. Now you may ask how is B different than C and A that it have more pawns.

I think the reason is

enter image description here
their slopes. The magnitude of slope for the line joining the pawns in slant position is more for B than A and it is more in such a way that the line contains most number of pawns by the given rules. In situation C, it makes both kind of angles, therefore having mixture of both kind of slopes.

Therefore

Professor Halfbrain's corrected first theorem:

It is possible to place 16 pawns according to the rules.

Professor Halfbrain's corrected second theorem:

It is not possible to place 17 pawns according to the rules.