“Don’t you ever make mistakes?”

Munoz slowly shifted his eyes from the painting to Julia. The suggestion of a smile hovering on his lips was utterly without humour.

“Not in chess.”

“How do you know?”

“When you play, you’re confronted by an infinite number of possible situations. Sometimes they can be resolved by using simple rules and sometimes you need other rules in order to decide which of the simple rules you should apply. Or completely unfamiliar situations arise and you’ve got to imagine new rules that either include or discount the previous ones. The only time you make mistakes is in choosing one rule over another, when you’re deciding which option to take. And I only make a move after I’ve discounted all the rules that don’t apply.”

“I find such confidence astonishing.”

“I don’t know why. That’s precisely why you chose me.”

The doorbell rang, announcing the arrival of Cesar with a dripping umbrella and sodden shoes, cursing the season and the rain.

“I hate the autumn, my dear, I really do. Season of mists and all that rubbish.” He sighed and shook Munoz’s hand. “After a certain age, some seasons seem horribly like a parody of oneself. Can I pour myself a drink? Silly me, of course I can.”

He served himself a large measure of gin, ice and lemon, and a few minutes later joined them, just as Munoz was setting out his chess set.

“Although I haven’t got as far as the move involving the white knight,” he explained, “I think you’ll be interested to know what progress I’ve made so far.” With the small wooden pieces, he reconstructed the positions depicted in the painting. Julia noticed that he did so from memory, without looking at the Van Huys or at the sketch he’d made the night before, which he now took out of his pocket and placed to one side on the table. “If you like, I can explain the reverse reasoning process I’ve followed so far.”

“Retrograde analysis,” said Cesar, sipping his drink.

“That’s right,” said Munoz. “And we’ll use the same system of notation that I explained yesterday.” He leaned towards Julia with the

sketch in his hand, indicating to her the situation on the board.

“According to the way the pieces are distributed,” Munoz went on, “and bearing in mind that Black has just moved, the first thing to find out is which of the black pieces made that last move.” He pointed at the painting with a pencil, then at the sketch and finally at the situation reproduced on the real board. “The easiest way to do that is to discount the black pieces that could not have been moved because they’re blocked or because of the particular position they’re in. It’s clear that none of the three black pawns, on a7, b7 or d7 could have moved, because they’re all in the position they occupied at the start of the game. The fourth and last pawn, on a5, couldn’t have moved either, because it’s between a white pawn and its own black king. We can also discount the black bishop on c8, still in its initial position, because the bishop moves diagonally and both of his two possible diagonal paths are blocked by the black pawns that have not as yet been moved. As for the black knight on b8, that wasn’t moved either, because it could only have got there from a6, c6 or d7 and those three squares are already occupied by other pieces. Do you understand?”

“Perfectly,” said Julia, who was leaning over the board following his explanation. “That means that six out of the ten pieces could not have moved.”

“More than six. The black rook on c1 couldn’t move, since it only moves in a straight line and its three surrounding squares are all blocked. So none of those seven black pieces could have made the last move. And we can also discount the black knight on d1.”

“Why?” asked Cesar. “It could have come from squares b2 or e3.”

“No, it couldn’t. On either of those squares, that knight would have had the white king on c4 in check; in retrograde chess that’s what we might call an imaginary check. And no knight, or any other chess piece for that matter, with a king in check is going to abandon that position voluntarily; that’s simply impossible. Instead of withdrawing, it would capture the enemy king, thus ending the game. Since such a situation is impossible, we can deduce that the knight on d1 could not have moved either.”

“That,” said Julia, who had kept her eyes glued to the board, “reduces the possibilities to two pieces then, doesn’t it?” She put a finger on each of them. “The king and the queen.”

“Right. That last move could have been made only by the king or the queen.” Munoz studied the board and gestured in the direction of the black king, without actually touching it. “First, let’s analyse the position of the king, which can move one square in any direction. That means he could have arrived at his present position on a4 from b4, b3 or a3… in theory.”

“Even I can sec what you mean about b4 and b3,” remarked Cesar. “No king can be on a square next to another king. Isn’t that right?”

“Right. On b4 the black king would have been in check to the white took, king and pawn. And on b3, he’d have been in check to rook and king. Both of which are impossible positions.”

“Couldn’t he have come from below, from a3?”

“No, never. It would then be in check to the white knight on b1, which, given its position, is clearly not a recent arrival, but must have got there several moves ago.” Munoz looked at them both. “So it’s another case of imaginary check showing us that it wasn’t the king that moved.”

“Therefore the last move,” said Julia, “was made by the black queen.”

The chess player looked noncommittal.

“That, in principle, is what we must assume,” he said. “In terms of pure logic, once we’ve eliminated the impossible, what remains, however improbable or difficult it may seem, must be right. Moreover, in this case we can prove it.”

Julia looked at him with new respect.

“This is incredible. Like something out of a detective novel.”

Cesar pursed his lips.

“I’m afraid, my dear, that’s exactly what it is.” He looked at Munoz. “Go on, Holmes,” he added with a friendly smile. “We’re on tenterhooks.”

One corner of Munoz’s mouth twitched humourlessly, a mere polite reflex action. It was clear that all his attention was taken up by the chessboard. His eyes seemed even more deeply sunk in their sockets and there was a feverish gleam in them: the expression of someone absorbed in contemplating imaginary, abstract spaces that only he could see.

“Now,” he suggested, “let’s look at the possible moves the queen could have made, positioned as she is on square c2. I don’t know if you’re aware, Julia, that the queen is the most powerful piece in the game. She can move across any number of squares, in any direction, imitating the movement of all the other pieces except for the knight. As we can see, the black queen could have come from four possible squares: a2, b2, b3 and d3. By now, you can see for yourself why she couldn’t have come from b3, right?”

“I think so.” Julia frowned in concentration. “I presume she would never have left a position where she had the white king in check.”

“Exactly. Another case of imaginary check, which discounts b3 as her possible origin. And what about d3? Do you think the queen could have come from there, for example, to avoid the threat from the white bishop on fl?”

Julia considered that possibility for a while. At last her face lit up.

“No, she couldn’t, for the same reason as before,” she exclaimed, surprised to have reached that conclusion on her own. “On d3, the black queen would have been holding the white king in another one of those imaginary checks, right? That’s why she couldn’t have come from there.” She turned to Cesar. “Isn’t this fantastic? I’ve never played chess in my life.”