Illegal Position Contest!

Illegal, but legal if any pawn is removed, even if promotion is not allowed.

en passant checkmate

Nobody has attempted to prove that the position I posted in #6463 (Diagram 1 below) is illegal under FIDE competition rules, so I'll post a proof myself.

(The box to the right of the board indicates player with the move.)

Since it's White to move, Black moved last. Because he has only his king, his last move must have been king to e8, and since the ply count is 144 that could not have been a capture.

The position before Black's last move must therefore be as in Diagram 2. with the Black king on one of the squares marked "k".

Since Black has the move in those positions and also in any position where he is mated, there must be an even number of ply leading to any possible mate by White. (By "possible mate" I mean a mate by any series of legal moves, not necessarily forced.)
Since the ply count is 143 no series of legal moves longer than 7 ply can be played, because the game would terminate in a draw on the 150th. ply under the 75 move rule (art. 9.6.2 FIDE Laws of Chess - I'll copy the relevant arts. at the end). A bishop capture resetting the ply count would result in the game terminating in a draw under the dead position rule (art. 5.2.2). In view of the previous paragraph White cannot checkmate Black with any series of legal moves longer than 6 ply (3 moves each for Black and White).

So what mates are available?

Since Black has only a king, White can checkmate if and only if his pieces cover all the squares in the kings's box. By "the king's box" I mean the square on which the Black king stands and the adjacent squares, and by "cover" I mean attack or X-ray attack through the king.

If the Black king is not on the boards perimeter, the king's box contains 9 squares. The two bishops can cover at most 5 of those and the White king at most 3. There are therefore no mates possible unless the Black king is on the perimeter.

With Black's king anywhere on the perimeter the bishop on the opposite colour square to that of the king cannot cover all the squares of that colour in the king's box, so the king must cover at least one. It follows:

(a) In any mate position the White king is two moves from the Black king.

(b) There are no mate positions with the kings in diagonal opposition.  

From a position on White's 7th. or 8th. rank as in Diagram 2. the Black king can reach no rank lower than White's 4th. in 3 moves and the White king can reach no rank higher. By (a) this rules out reachable mates except with the Black king on the a or h files and White's fourth to sixth ranks.

At an edge of the board not in the corner, from (a) and (b) the relative positions of the kings in a mate must be as shown in Diagram 3.  where the White king is on a square marked "K".

Given that mate cannot be reached in 6 ply except with the Black king on the a or h files, I'll call a mate with the White king on the red "K" squares a "top mate", on the green "K" squares a "bottom mate" and on the blue "K" squares a "face mate".

Since the lowest rank the Black king can reach in 3 moves is the same as the highest rank the White king can reach in 3 moves no top mates are reachable in 6 ply.

In any bottom mate the checking bishop must cover the square in the king's box of the colour occupied by the black king that is not covered by the White king. It must therefore be in the delineated area in Diagram 4.

Since the diagonals from the positions occupied by the bishops in Diagram 2. (c1 and f1) do not intersect the delineated area the checking bishop would need at least two moves to enter the area, but the White king cannot reach the a, b, g or h file in 1 move, so no bottom mates are reachable in 6 ply.  

In a face mate the position of the bishop that isn't checking relative to the Black king can only be as shown in Diagram 5., because the White king doesn't attack the squares adjacent to the Black king along its edge. I'll call it the "fixed bishop".

For any mate reachable in six ply the fixed bishop would need to be on the b or g file. The diagonals through the bishops positions in Diagram 2. intersect these files on White's second and fifth ranks. Since mate positions are unreachable except with the Black king on the fourth to sixth ranks, only the latter are relevant. For the Black king on the fourth or sixth ranks the fixed bishop needs at least two moves to reach it's position and the White king cannot then reach the lowest position for the face mate on the third rank (c3 or f3) in one move; with the Black king on the fifth rank the fixed bishop needs only one move, but the White king cannot then reach it's lowest position for the face mate on the fourth rank in two moves. 

So no face mates are reachable in 6 ply.

So no mates by White are reachable in 6 ply from the positions in Diagram 2..

The positions in Diagram 2. are therefore dead under art 5.2.2. Black cannot checkmate White by any series of legal moves because it's difficult with only a king and White cannot checkmate Black by any series of legal moves because there is no such series that results in mate before the game would terminate under the 75 move rule.  

But a dead position immediately terminates the game, therefore it is impossible to reach the position in Diagram 1. by any series of legal moves. The position posted is therefore illegal under art. 3.10.3. 

Relevant FIDE articles:

3.10.3 A position is illegal when it cannot have been reached by any series of legal moves. 

5.2.2 The game is drawn when a position has arisen in which neither player can checkmate
the opponent’s king with any series of legal moves. The game is said to end in a ‘dead
position’. This immediately ends the game, provided that the move producing the
position was in accordance with Article 3 and Articles 4.2 – 4.7. 

9.6 If one or both of the following occur(s) then the game is drawn:

9.6.2 any series of at least 75 moves have been made by each player without the movement of
 any pawn and without any capture. If the last move resulted in checkmate, that shall
 take precedence. 

The position is illegal under FIDE competition rules because the 75 move rule automatically terminates the game.

Under USCF competition rules the termination under the 75 move rule is at the tournament director's discretion and the 75 move rule is not part of the FIDE basic rules, so under those rules the position is legal.

The position posted with the ply count amended to 143 would be legal under FIDE competition rules because then there would be 8 ply from the previous position to achieve mate and that is possible.

Here is @ShouldBreezi's proof game for my position (extended as he implies to reach the specified ply count). 

This is possible because the chess.com GUI doesn't implement FIDE competition rules. The position after White's move 187 is one of the positions in Diagram 2., so the game would terminate under the dead position rule at that point if the FIDE competition rules were in force.

In fact only specific cases of the dead position rule such as stalemate and insufficient material are implemented in any GUI, because nobody has yet produced an algorithm to recognise dead positions (whatever the rules) in practical time. 

where are the judges

Judges were a thing 6 years ago when this forum was a contest. But now there are no judges and this forum is no longer a contest.

Why is the final position in this game illegal?

It's not. It's not the same position; it has ply count 0. The ply count runs only from the last pawn move or capture.

Similarly the positions after moves 121 and 143 in @ShouldBreezi's proof game with ply counts 12 and 56 are not illegal. Neither is a position with the same diagram and side to move with ply count 143.

My position is illegal under FIDE competition rules for the reasons posted here, but not illegal under all rules. (Nor legal under all other rules; the position cannot be reached in a TCEC game, for example, because games are automatically terminated by tablebase adjudication when they reach 6 men.) 

Oh. I thought you wanted a 72-move game with a total count of 144 plies. I should have realized you wanted something different.

I should, no doubt, have been more explicit.

Illegal. White's five pawn captures and capture of the c8 bishop exhaust White's captures. Before Black's last move, O-O-O+, the black pawns are in their final places. The h-pawn couldn't be at h6, or the white king couldn't reach g7. The a-pawn couldn't be at a7, or the black queen couldn't have gotten out to be captured by a white pawn.  If White last moved the king to g8, Black has no previous move. If White already had the king on g8 and last captured a black piece on g5 or h4, Black's only move would be castling, and one of the white knights would be left out of place.

Illegal. White has no legal retraction to uncheck the black king. Discovered check by a pawn capture on a3 would block the bishop's path to a1. Discovered check by a king move from c3 to b3 would require an en passant capture by Black on b3 to avoid an illegal double check, and again the bishop's path to a1 would be blocked.

Nice work! There were other tries in both positions. For example, in the position where black must uncastle, white could try to retract h6-h7, allowing black to retract h7xg6. But now white again cannot give a last move for black. As for the bishop check, I’m assuming you found other tries too obvious to be illegal.

Here’s #6509 with a few adjustments. Now is it illegal?

Correct! It is legal because of discovered rook check.

The above position is legal. But add ANY type of piece on g8 and the position becomes illegal. Why?

The white king has to go to h8 and the black king to f7 to let the white rook go to c7. When the white king has moved back to f7, g8 will be vacant and it will be Black's move, but no remaining black piece can ever move to g8. A black piece that tempos while a white knight moves to g8 can't be captured and will still be on the board.

Here's one way to get to the legal position.