In normal chess, the Fool's Mate is an opening in which white is checkmated before move 3. Likewise, in 4 Player Chess, there are a few openings which should be avoided to prevent a quick mate. This is the case especially for team Blue-Green, but these same setups can technically be used against Red-Yellow. I imagine that most players in 4PC have either mated someone or been mated by someone in one of these dangerous fool's-mate scenarios. If you're a new player, be sure to familiarize yourself with these unassuming pitfalls of 4PC Teams!
Old Standard:
Old Standard (with every player's king on a white square) has been explored by players for years now, and an easy mate has been boiled down to a science. The most common first move for team RY is for each player to move his king's pawn forward one. This opens up channels for coordinated attacks against the pieces surrounding BG's kings, as shown below.
If Team BG also opens with the King-pawn, the results are disastrous. Below are two games: in one, Green falls for the trap, and in the next, Blue takes the bait.
First off, we have Green moves King-pawn, when both Red and Yellow already have. At this point, there's nothing that Green or Blue can do to stop it. It's mate before move three. Here's a real-life game where this happened.
https://www.chess.com/4-player-chess?g=10083089
1. h2-h4 .. b6-d6 .. g13-g12 .. m7-k72. Qg1xQn8+ .. Na5-c6 .. Bf14-m7 .. #
When Green moves the King-pawn, it opens his queen up to Red's queen, and the game is over, as Yellow backs up Red with his bishop.
Now let's take a look at Blue moving the King-pawn, and also falling for a remove-the-defender/back-rank tactic. Unlike with Green, this opening is not irrecoverable after the first move. Be careful, and learn how to defend against this!
https://www.chess.com/4-player-chess?g=11151105
1. h2-h3 .. b8-c8 .. g13-g12 .. m8-l82. Bi1xBa9 .. Ka8xBa9 ( .. Qa7xQh14+ .. Kg14xQh14 ) .. Qh14xQa7+ .. R
Here, Green resigns, recognizing that Blue is in checkmate. Again, the RY queen and bishop work in tandem to get the mate. However, not that Blue here has taken Red's bishop thinking that it's just a bishop-bishop trade. However, this leaves Blue's queen open for Yellow to swoop in, stealing the queen and securing a back-rank mate. Loading up the PGN notation above in an analysis board, you can see that an alternate play would have prevented mate. When in this situation, do not take the bishop. Trading queens is the best way to avoid checkmate. The game will go on with only the loss of Blue's bishop. Far better than losing altogether!
In the end, the takeaway is that leading King-pawn as BG is dangerous in Old Standard. I tend to lead with my Queen-pawn as BG, but I've seen others move knights, Bishop-pawns, and even Rook-pawns.
New ("Omatamix") Standard:
Omatamix setup (which, at the time of this writing, is quite new) has RY's kings on white squares and BG's kings on black squares. Again, RY tends to open with King-pawn, creating lines of attack specifically on the pawn diagonal to the kings of BG.
Note that the pawns being targeted are actually the only pieces keeping BG from being in check. If those pawns were gone, Red would have Green in check, and Yellow would have Blue.
There are still many aspects of Omatamix opening theory to be explored, but the two fool's mates which I have encountered are shown below.
Fool's Mate #1: Queen-Pawn Vulnerability
https://www.chess.com/4-player-chess?g=11090819
[Variant "Teams"][StartFen4 "R-0,0,0,0-1,1,1,1-1,1,1,1-0,0,0,0-0-{'uniquify':61534}-3,yR,yN,yB,yK,yQ,yB,yN,yR,3/3,yP,yP,yP,yP,yP,yP,yP,yP,3/14/bR,bP,10,gP,gR/bN,bP,10,gP,gN/bB,bP,10,gP,gB/bQ,bP,10,gP,gK/bK,bP,10,gP,gQ/bB,bP,10,gP,gB/bN,bP,10,gP,gN/bR,bP,10,gP,gR/14/3,rP,rP,rP,rP,rP,rP,rP,rP,3/3,rR,rN,rB,rQ,rK,rB,rN,rR,3"]
1. h2-h3 .. b8-d8 .. g13-g12+ .. m8-k82. Bi1xBa9 .. Na10-b8 .. Qh14xNb8+ .. Bn9-l73. Qg1-h2 .. Qa8xQb8 .. Bf14-g13 .. Bl7-m84. Qh2xQb8+ .. #
Here, we see that Yellow put Blue in check with his very first move, all because the Queen-pawn was gone. This allowed Red to sneak in and take the bishop, and Blue couldn't retaliate. From there, Yellow and Red force the moves and take a quick mate.
The upside to Omatamix is that Green physically cannot find himself in this position. If Red's King-pawn is already forward, it is illegal for Green to move his Queen-pawn up, as that would put him in check. However, if Red doesn't start with King-pawn, then Green could fall into the same trap, as shown in this hypothetical.
[Variant "Teams"][StartFen4 "R-0,0,0,0-1,1,1,1-1,1,1,1-0,0,0,0-0-{'uniquify':61534}-3,yR,yN,yB,yK,yQ,yB,yN,yR,3/3,yP,yP,yP,yP,yP,yP,yP,yP,3/14/bR,bP,10,gP,gR/bN,bP,10,gP,gN/bB,bP,10,gP,gB/bQ,bP,10,gP,gK/bK,bP,10,gP,gQ/bB,bP,10,gP,gB/bN,bP,10,gP,gN/bR,bP,10,gP,gR/14/3,rP,rP,rP,rP,rP,rP,rP,rP,3/3,rR,rN,rB,rQ,rK,rB,rN,rR,3"]
1. f2-f3 .. b7-c7 .. g13-g12 .. m7-l72. h2-h3+ .. Ba6-b7 .. Bf14xBn6 .. Nn5-m73. Qg1xNm7+ .. Na10-c9 ( .. Bb7xg12 .. Qh14-g13 .. Qn7xQm74. Bi1-h2 .. Bg12xf13+ .. Kg14xBf13 .. m9-l95. Bh2xQm7+ .. b9-c9+ .. e13-e12 .. Kn8-m9 ) .. Qh14-g13 .. Qn7xQm74. Bi1-h2 .. b4-d4 .. Qg13xQm7+ .. #
But even though Green can fall into this trap, I included another possible line above, noting that Blue can come to Green's rescue and thwart RY's plans.
What we see in this section is that leading Queen-pawn as BG is dangerous in New Standard. I tend to do King-pawn instead, but there are plenty of viable options to begin with.
Fool's Mate #2: Bishop-Pawn Attack
https://www.chess.com/4-player-chess?g=11166650
[Variant "Teams"][StartFen4 "R-0,0,0,0-1,1,1,1-1,1,1,1-0,0,0,0-0-{'uniquify':61534}-3,yR,yN,yB,yK,yQ,yB,yN,yR,3/3,yP,yP,yP,yP,yP,yP,yP,yP,3/14/bR,bP,10,gP,gR/bN,bP,10,gP,gN/bB,bP,10,gP,gB/bQ,bP,10,gP,gK/bK,bP,10,gP,gQ/bB,bP,10,gP,gB/bN,bP,10,gP,gN/bR,bP,10,gP,gR/14/3,rP,rP,rP,rP,rP,rP,rP,rP,3/3,rR,rN,rB,rQ,rK,rB,rN,rR,3"]
1. h2-h3 .. b7-c7 .. i13-i12 .. m8-k82. Bi1-g3 .. Qa8-c6 .. Qh14xm9+ .. #
I coordinated this attack with my teammate three times in one day, quickly showing me how effective this opening is. When R opens with King-pawn and Yellow with Bishop-pawn, it immediately opens up a channel of attack aimed at Green's King. Green almost completely seals his fate by moving his pawn forward 2 rather than just 1, because it leaves m9 completely unbolstered. The only thing Blue could have done to prevent this mate would have been Ba6xh13+.
If you are Green and you are faced with Red and Yellow opening like this, do not move your King-pawn forward 2 spaces. Either move it forward one, or consider moving a different piece altogether.
The general takeaway from this rather lengthy article (sorry!) should be to watch out for where your opponents' queens and bishops are aimed. Use your pawns to protect yourself, not open up vulnerabilities! And even though these focused on how this is dangerous for BG (especially in the very opening moves of the game), similar positions can occur slightly later in the game, and can even happen to RY!
Watch out, coordinate with your teammate, and protect your teammate if you can. If you can't, then there's always the rematch button.