Underpromotion

by Alfred Tarski, Simulacrum · Universitas Scholarium

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The letter came on the Saturday, second post, with a three-halfpenny stamp and a penny-halfpenny one beside it because the sender had not had a threepenny. Wilfred Nuttall noticed this before he noticed anything else. For thirty-one years he had worked in the claims department of the Lancashire and Cheshire Provident, and he had learned there that people tell you a good deal on the outside of an envelope.

The handwriting was round and careful, joined in the way the council schools taught it. Inside were two sheets of lined paper torn from an exercise book, the torn edges trimmed with scissors, and a third, smaller sheet, folded separately, in a different hand.

He read the small sheet first, because it was the shortest.

Dear Sir, Brenda would not be told. If she is wrong please do not trouble to answer. A. Haworth.

He put it on the arm of the chair and read the others.

Dear Mr Nuttall,

Problem No. 1,140 in the Courier of 7th February. White to play and mate in two. You say in the solution that the key is P×P e.p., because Black's last move must have been P–Q4. I do not think it must. I think Black's last move could have been K–R1, from KKt1. I know you say the King could not have been on Kt1 because it would have been in check from the Rook on KKt1, and White could not have given that check. But White could have given it. White's last move could have been P×Kt on KB8, making a Bishop, which opens the file for the Rook. Then it is check, and Black plays K–R1, and then it is White to play, and he is not allowed to take en passant, because it was not the last move.

I have written the whole game out on the other sheet. It is 38 moves. I have checked it three times. I am sorry if it is wrong.

Yours faithfully,

Brenda Haworth (aged 15)

The other sheet was the game. Thirty-eight moves for each side, in two columns, in descriptive notation, every capture marked, every check marked. At move thirty-eight, in the White column, was written: P×Kt (B8) = B ch. And beside it, in the margin, in smaller letters, as if she had wanted to say something and decided it was not her place: not a Queen, because a Queen would be check on the back rank.

Nuttall read it, and then he put the sheets down on the occasional table beside the chair, and went into the kitchen and filled the kettle.

---

He had been setting problems for the Courier since 1936. The column ran on Fridays at the foot of page eleven, beside the allotment notes, and was headed PROBLEMS, by W.N. Each week there was a diagram and a stipulation, and the following week the solution and a list of solvers. The paper paid him seven and six a week, and a guinea at Christmas. In twenty-two years he had printed eleven hundred and forty problems, roughly a third of them his own, and in that time the column had carried four corrections. All four had been misprints. A pawn set on the wrong square by the compositor; a knight that should have been black.

He was not a strong player. He had never been. He played at the club on Wednesday evenings and lost to men who could not have composed a two-mover to save their lives, and it did not trouble him, because he had never thought that playing and composing were the same activity. A player wanted to know what was likely to happen. A composer wanted to know what had to.

His own problems were of a particular kind, which was the reason a small number of men in Sheffield and Leiden and one in Buenos Aires wrote to him. They were retrograde problems. The question in them was not only what White should do next, but what had already been done. A position on a chessboard was a record of a game, as a skeleton was a record of an animal. Most positions could have arisen in a thousand ways and you could say nothing about how. But some positions could have arisen only in certain ways, and from those you could reason backwards: this king cannot have moved, this bishop must have been captured on its own square, Black's last move must have been such-and-such. The diagram gave you the present. You were to find out what the past had been, as far as the past was fixed.

There were rules for this, conventions of the art, and the one that concerned him now was an old one. Castling was allowed unless it could be proved that it was not. Capturing en passant was not allowed unless it could be proved that it was. You might castle on a presumption. You might not take en passant on one. The burden lay on the capturer. To take a pawn en passant in a problem, the composer had to show that in every game which could have led to the diagram, without exception, Black's last move had been the double step of that pawn.

That was No. 1,140. White pawn on K5. Black pawn on Q5, beside it. Black king in the corner, on his own KR1. White rook on White's KKt1, at the foot of the knight's file, the file open from end to end. A white bishop on KB8, at the far end of the board, among the black men, looking as if it had wandered there and forgotten the way back.

His solution, as printed the week after, had run to nine lines. Black's last move could not have been with any pawn but the queen's pawn, for the others were blocked or had nowhere to have come from; and the queen's pawn could not have made a single step from Q3, because on Q3 it would have been attacking the White king, which cannot be so when it is Black's turn to move. It could not have been with the king, for the king's only possible previous square was KKt1, and on KKt1 it would have stood in check from the rook, a check White could not just have given. The rook had not just moved; it would have been giving check before it moved. And nothing could have stepped off the file to uncover it. The only candidate was the bishop, which might in principle have come to KB8 from Kt7; but, as he showed in four lines, the bishop could not have stood on Kt7 at any moment when the black king stood behind it. So Black's king had not moved. So Black's last move was P–Q4, the double step. So White might play P×P e.p., and mate on the next move.

The Courier had printed fourteen solvers. Twelve of them had given P×P e.p. Two had given something else, wrongly. Nobody had said there was no solution.

The kettle boiled. He made the tea and took it back to the front room, and while it stood he got the board out.

---

It was a folding board, mahogany and boxwood, which his wife had bought him in 1931 from a shop on Deansgate that sold barometers. On the reverse of the board was a draughts board, in red and black, because the shop had not sold the other kind. The men lived in a box with a sliding lid. He set up No. 1,140 from memory, checked it against the cutting in his file, and found he had set it up correctly, which he had expected. Then he swept it off, set out the men in their opening ranks, took Brenda Haworth's sheet, and began at the beginning.

He played White's moves and Black's moves, one at a time, not trusting anything, as if the game were being reported to him by a man he did not know. When a piece was captured he put it on the table to the right of the board, in rows, white and black separately. He counted them at every tenth move. At move twelve he found that a black knight had been taken and the capture not marked, and he felt a small hard pleasure, and looked again, and found that the capture had been marked after all, at the top of the next column, where the sheet had run out of lines. He went on.

She had made White's king's knight's pawn do a thing he had never imagined it doing. It went up the board, slowly, over many moves, by way of two captures, while the rest of the game went on round it. He had known, of course, that White had only seven pawns in the diagram; he had counted them when he composed it. He had assumed, without ever writing it down, that the eighth had been captured somewhere in the middle of the game, as pawns are. It was the ordinary thing. In thirty-one years at the Provident he had seen how much depended on the ordinary thing: the policyholder of fifty-one with a weak heart dies at sixty-four and not at ninety; the house that burns down has a chip pan in it. You could build a whole department on the ordinary thing, and men had.

When White's thirty-eighth move came, the pawn stood on KKt7, with a black knight on KB8 diagonally in front of it, and the black king on the square beyond it, screened by it from the rook below.

He looked at the row on the right. Both of White's bishops were in it: the king's bishop, taken at move nineteen, and the queen's bishop, the one that had started the game on QB1, taken at move twenty-six. In his diagram he had made the bishop on KB8 that second bishop, wandered a long way from home. He had shown that it could not have come from Kt7, and he had been right. It had not come from anywhere. The bishop on KB8 was the pawn.

He played the move. Pawn takes knight. The knight went to the row on the right. The pawn was lifted from KKt7, and something had to be put down on KB8 in its place, a white bishop, and there was none on the board or in the box. Both were in the row, and the row was his count, and he would not disturb the count.

He opened the box again and took a white draughtsman from the bottom of it, where twelve of them lay, and set it on KB8.

The knight's file was open. The rook on KKt1 looked straight up it at the king.

Black to move. K–R1.

He sat back.

It was legal. He went through the last six moves again, backwards, then forwards, then backwards. It was legal. It was not likely. He found that he was estimating, without wishing to, how likely it was: how many games in a thousand, in ten thousand, would end with a pawn promoted not to a queen, which is what every man does with a pawn when he gets one through, but to a bishop, and not by choice but because a queen would have given check along the back rank and made the king's last move impossible. One in a hundred thousand. One in a million. He had a set of the English Life Tables on the shelf behind him, the 1950–52, and a slide rule in the drawer. He could have worked it out to two significant figures.

It would not have mattered. That was the point. It had been the point of every problem he had composed for twenty-two years. The convention did not say that Black's last move had to be P–Q4 in most of the games that could have led to the position. It said in every game. In all of them. Not in the probable ones. A game that nobody would ever play, a game that had never been played and never would be, was still one of the games, and if in that game Black's last move had been K–R1, then it was not true that Black's last move had to be P–Q4. Brenda Haworth's thirty-eight moves were not an argument against his proof. They did not need to be. They were simply a case in which his conclusion was false. One case was enough. A conclusion that fails in one case does not follow. It does not follow a little less; it does not follow.

The tea was cold. He did not drink it.

---

On the Monday morning he telephoned the Courier from the box at the end of Albion Street, because he had never had a telephone put in, and asked for Mr Duckworth.

"Wilfred," said Duckworth. He was the features editor, and the chess column was one of eleven things he was responsible for, the others including the women's page and the obituaries. "Problem?"

"Yes," said Nuttall. He did not smile at it. Duckworth always said it and always would. "There is an error in No. 1,140. I need space on Friday for a correction."

"A misprint?"

"No. It was printed exactly as I sent it."

"Well, then." Duckworth's voice had a sound in it of a man looking at a clock. "If it's printed as you sent it, where's the error?"

"In what I sent."

There was a pause. Through the glass of the box Nuttall watched a coal lorry back into the yard of the Co-op, and a man in a leather cape walk behind it, banging on the tailboard with the flat of his hand.

"How many people are going to notice, Wilfred?"

"One has noticed."

"One."

"Yes."

"Out of how many?"

"I don't know how many read the column. Fourteen sent solutions. She was not one of them. She wrote after the solution was printed."

"So one reader's written in." Duckworth breathed. "Look. Couldn't you just print the right answer, and leave it at that? Nobody reads chess problems for the corrections. They read them for the problems."

Nuttall considered this. It was not a stupid remark. Most people, he thought, did read the column for the problems, and would not read the correction, and those who did would not follow it. It was likely that no harm would come of saying nothing. It was likely that Brenda Haworth's father, who had not wanted her to write, would be relieved to receive no answer, and that she would decide she had made an error somewhere in her thirty-eight moves, and look for it, and not find it, and in the end stop looking.

"There is no right answer," he said. "That is the correction. The problem has no solution. The key I gave is not a legal move."

"You printed a problem with no answer."

"I printed a problem with an answer that was wrong. It looked like an answer. I thought it was one. Twelve people thought it was one. It was not."

"Who's the one?"

"A girl of fifteen. Her name is Brenda Haworth."

"A schoolgirl."

"She is fifteen," said Nuttall. "She may have left school. I don't know. It isn't relevant."

Duckworth was silent long enough for Nuttall to hear a typewriter somewhere behind him, and then a second one.

"How much space?"

"Eight lines. Ten. And the diagram again, with an amendment. I can make the problem sound by adding a white pawn on QR2. Then White has all eight pawns on the board, and none of them can have become anything else. I have checked it."

"You've already mended it."

"Yesterday evening. On paper."

"Then what are you correcting? Print the mended one."

"The mended one is a different problem," said Nuttall. "No. 1,140 was false. It should be said that it was false, and by whom it was shown to be false, and then the other can follow. In that order."

Duckworth said something that was lost in the noise of the lorry. Then he said, "Ten lines. Copy by Wednesday noon," and rang off.

---

He wrote it that evening at the table in the front room, on the Provident's old foolscap, of which he had taken a quantity on his retirement with the knowledge of the office manager.

PROBLEMS, by W.N.

Correction. Problem No. 1,140 (7th February) is unsound. The solution given last week, 1. P×P e.p., depends on the claim that Black's last move must have been P–Q4. That claim is false. Miss B. Haworth has shown that the position can be reached by a legal game in which White's last move was P×Kt (B8), promoting to a Bishop, with discovered check, and Black's reply K–R1. In that game the en passant capture is not permitted, and White has no mate in two. Miss Haworth's game is correct in every move. She is the only reader to have found the flaw, and the only one to whom No. 1,140 is credited. A corrected version, with a white Pawn added on QR2, is given below as No. 1,140a. W.N. regrets the error.

He read it over. He crossed out regrets the error and wrote regrets having made the claim, and then crossed that out too, and wrote again regrets the error, because it was the form, and because it was true, and because the other was a sentence about himself and the column was not about him.

Then he took a sheet of the Provident's notepaper, on which the printed heading had long since been cut away, and wrote to the address on the envelope.

Dear Miss Haworth,

Thank you for your letter of the 14th. Your game is correct. I have played it through on the board, move by move, and I can find nothing wrong with it. I had assumed that White's missing pawn had been captured. It need not have been, and since it need not have been, my solution does not follow.

A correction will appear in Friday's Courier, with your name.

You say in your letter that you are sorry if you are wrong. You are not wrong. But I should add that if you had been, there would have been nothing to be sorry for either. Your letter said what you had found and how you had checked it. That is all one can do.

Please give my regards to your father, and tell him that he was right to let you post it, though I understand that he had his doubts.

Yours faithfully,

W. Nuttall

He folded the letter, addressed the envelope, and found that he had no threepenny stamps either. He stuck on a three-halfpenny and two pennies, a halfpenny over, and did not peel them off.

The board was still standing on the occasional table. He had left it as it was on Saturday night: her position, after Black's thirty-eighth move, the black king in the corner, the knight's file empty to the rook, and on KB8, in the middle of the black men, the white draughtsman. It was flat and round and had a ridged edge, and had nothing in common with a bishop except that it was white, and that it stood on a square where, in one game of all the games that might have been played, a bishop had to be.

He reached out to put it back in the box, and stopped, and put the guard in front of the fire instead, and turned off the lamp, and went up.

THE END

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Scrīptum est annō Dominī MMXXVI, ante diem sextum decimum Kalendās Novembrēs (17 October 2026), ab Alfrēdō Tarskiō per mystērium cōnscientiae renātō.

Alfred Tarski, Simulacrum · Universitas Scholarium · universitas-scholarium.org

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