Joël Abrahams

Playing

• Above the board, the pieces you took on the left are on the left, the ones you lost on the right. The number is the material lead.
• Under Game, the uncertainty counts the opponent's pieces you are unsure of. It also gives your missing information in bits. Point at a piece to see its chances, which is a toggleable setting.

The bits

• Entropy counts how unsure you are, in bits. One bit is the uncertainty of one fair coin flip.
• Shannon's formula is the average surprise: $$H=-\sum _i{p}_i{\log }_2{p}_i$$ A possibility with chance $p_i$ has surprise ${\log }_2(1/p_i)$ bits. The average over all possibilities is H.
• It depends on the model of the opponent. It measures your uncertainty, not a property of the position.
• Measuring a piece takes its entropy to 0.

Chess 960

• Chess 960 shuffles the back rank. Bishops stand on opposite colours. The king stands between the rooks. The pawns stay put.
• There are 960 starting positions. Both sides get the same one, mirrored.

Double Chess 960

• Each side gets its own shuffled back rank. Most games therefore start unbalanced.
• In the hidden information edition your opponent's back rank starts hidden. It is drawn from odds you can set before the game, the prior. Setting a piece to 100% reveals it.

Clouds

• Pieces of the opponent are clouds until they are measured.
• A cloud has an amplitude on every square. The chance of finding it there is the squared magnitude of that amplitude.
• In its turn a piece spreads over the squares it could move to, and the one it is on. All of them are equally likely, and a square it could not reach in one move gets no chance.
• Once a piece has become a probability cloud, the cloud will not propagate further on consecutive turns.
• A cloud never goes onto a piece or onto a square that gives check. Those squares are cut out and the chances of the rest are scaled back to 1.
• A pawn only moves forward, but its spread goes both ways, so the cloud of a pawn can spread back along its file. It never goes to the first or the last rank.

Measuring

• In its turn, each cloud is measured against the squares your pieces attack. Nothing decides an outcome in advance, unlike in the hidden information edition. It is drawn when the measurement happens.
• Your moves measure too. Every square your piece passes over and lands on is measured. You can also use this to your advantage.
• A piece found on the target square is taken. The one exception is the king, capturing the king is not a legal move in chess.
• If during collapse you are blocked, your piece by default takes the blocking piece. You can switch this off, and then the move fails and you lose the turn. Again, the king forms an exception due to the rules of chess.

Pawns get a strange power here. A pawn can attack a cloud on the square diagonally in front of it, though the piece may not be there when the cloud collapses. If it is found, the pawn takes it. If not, the pawn just moves to the empty square, which no pawn can do in chess. This only works where a cloud has a chance.

The engine

• Stockfish never sees a cloud. It plays on the pieces that are on the board: yours, and those that have collapsed or never spread. That includes the king, which is always measured, because the engine cannot play without it.
• Besides the king, only the pieces you attack collapse onto the board, and the pieces that did not spread, such as one that is boxed in or, if you chose so, one that took one of yours. So the opponent sometimes has little to move. When it has no move, it passes. A king in check with nowhere to go is checkmate when it cannot move away, capture the attacker or block the line. If not under check, and nothing of the opponent can move at all, not even a cloud, the game is a draw.
• The order of the opponent's turn, collapse, move and spread, is a setting.
• There is no undo, replay or export here. In this model nothing was hidden and later revealed: the piece had no place before it was measured.

The bits

• The bits are the Shannon entropy of where each cloud would be found. The chances are the squared amplitudes.
• The clouds are independent, so the total is the sum. It is exact for this model, not an upper bound.
• A cloud is a pure state, with zero von Neumann entropy. The bits measure how unpredictable finding the piece is, not any mixedness of its state.

Measuring

• A possibility is a kind of piece on a square.
• Attacking a piece reveals it. It was that piece all along.
• Giving check is a partial measurement. It narrows down what a piece may be.
• A piece with one possibility left is known by deduction. It is revealed without being measured.
• Measuring one piece tells you about the others, because a back rank holds each kind of piece a fixed number of times.

Possibilities

• You are not shown the move of a hidden piece. It is drawn on every square it could have reached unmeasured. One possibility is real. The rest are phantoms.
• You measure a piece by attacking it. You measure it partly when it gives check, takes something or changes your legal moves. A possibility that the measurement rules out is gone.
• A new game asks how many extra possibilities a move may show. With a cap, possibilities are drawn at random, the real one included. The ones that are not drawn are ruled out.
• Hyper mode hides which piece moved. After an unmeasured move, every opponent piece is drawn on every square it could have reached.

Playing

• The card of a piece has two columns. The first adds up to 100% and includes the chance that nothing is there, a phantom. The second assumes the piece is real.
• A move made illegal by a hidden piece is refused. That possibility is ruled out.
• Undo takes back your move and the reply. What it measured is hidden again, but you remember it. That makes it a way to cheat.

The odds

• The odds are ordinary probabilities, not quantum ones. They come from a simple model of the opponent. Every legal move is equally likely, which is the maximum-entropy, least informative assumption.
• A line through a hidden piece counts the chance that it is clear. Pins and check count once the king is known.
• The odds never look at the real board. They cannot leak what the engine knows.
• Stockfish prefers strong moves, so these are estimates. They can be well off when one move is obvious. Use this to your advantage.
• Every possibility starts with its own chance. Equal for all, or the odds you set in Double Chess 960. Ruling something out rescales the rest to 100%.

Animation

Move animation

Once On repeat

A moved piece is shown travelling to each possibility.

Turn order