The Purse and the Promise
In the market town of Harrow-on-Tally, where every stall kept a ledger and even the children could reckon change, a stranger set up a small table at the edge of the fair. He laid on it a single copper coin and a slate, and on the slate he wrote a promise.
“Toss my coin,” he said, “until it shows the crown. If the crown comes on the first toss, I pay you two shillings. On the second, four. On the third, eight — and so, doubling, with no ceiling and no end. There is no largest prize. Name me a fair entrance fee, and if it is fair, we play.”
A hush went round, for the people of Harrow-on-Tally loved a fair price above all things, and here was a game whose fairness they could compute. The schoolmaster came first, chalk already moving. “One time in two I win two — that is one shilling of worth. One time in four I win four — another shilling. One time in eight, eight — a shilling again.” He looked up, pale. “Every rung of the ladder is worth exactly one shilling, and the ladder has no top. The fair price is every shilling there is. The fair price is everything.”
The stranger turned his palm to the sky, as if to say: you see.
And the town, being honest, began to loosen its purses — for the number did not lie. The average winning was beyond all counting, and to refuse a game worth infinity for the sake of a few coins seemed the act of a fool, or a coward.
But at the back stood old Maren, who had kept the fair’s strongbox forty years and had paid out every prize the fair had ever owed. She did not touch her chalk. She walked to the table, lifted the stranger’s copper coin, and weighed it in her hand.
“Your ladder has no top,” she said. “But your chest does. Show me the gold behind the promise.”
The stranger’s smile thinned. He opened a strongbox. It held a thousand shillings, and a score and four besides — a great sum, but a countable one.
“Then the promise ends where the gold ends,” said Maren. “A prize you cannot pay is not a prize; it is a story. Count only the rungs your chest can honor, and the whole forever folds up into eleven shillings — twelve, if you are feeling generous.” She reckoned on her fingers and showed the town the sum. “That is what an honest chest can promise. Not infinity. Eleven.”
“But the average—” the schoolmaster began.
“The average is a promise made by a thousand players at once,” said Maren, “and you are one. You will toss this coin as many times as one life allows, not as many as a thousand lives allow. Play it every night you have, and your winnings will never settle on any sum — they will only climb, slow and jagged, hauled up each time by a crown so rare you may wait your whole life to meet it. That is not a fortune. That is a fever.” She set the coin down.
“There is a truer average,” she said, “the one your own purse meets — the sum you’d have to live on, night after night, counting only what you actually carried home. It is small. It is finite. It is four shillings, near enough. Pay the man four, and you have paid him fair.”
The people of Harrow-on-Tally put their purses away. And the stranger — who had counted on the infinity, and not on the widow — folded his slate and was gone before the lamps were lit.
The moral, or near enough to one:
An average can be a promise the world is not able to keep. So ask always who is meant to pay it, and out of how deep a chest — for you will play the game only as many times as one life allows, and the fortune that lives in the arithmetic will never reach your hand. Do not price the promise. Price the purse.
