The death of Mr Auslander turns a request about pear blossoms into a test of what Duckling can do with numbers—and what he can discover without being taught. Asked to calculate how many days his father had lived so that the same number of flowers could be buried with him, the twelve-year-old works through a long calculation with only the mathematics he has picked up in daily life. The result is mistaken, but the path toward it reveals an independent mathematical intelligence that will soon outgrow the problem that first called it forth.

The flowers and the final request

Mr Auslander’s attachment to pear blossoms had grown from an unexpected recovery: after a period of constipation, he escaped the stomach inflammation and diarrhoea that had usually followed. He came to see the blossoms’ medicinal properties as more than an accident, brewed them like tea, and felt renewed joy when the trees flowered each year. During his years of pain, he had dreamed of the flowers opening in sunlight and floating through wind and rain; he also hoped to die as they bloomed.

As death approached, his wishes for his burial became progressively more exact. One morning he asked Duckling to write down that pear flowers should be placed in his coffin. That evening he specified eighty-nine flowers for his eighty-nine years. The next morning, he asked the boy to work out how many days were contained in those years and to use that number instead. In giving these increasingly elaborate instructions, Mr Auslander seemed to forget that he had never formally taught Duckling mathematics.

The request was not entirely without preparation. Each year, Mr Auslander collected the falling blossoms and had Duckling count them; the totals were written on the wall and sometimes counted again. Through this repeated household task, Duckling had practiced addition and subtraction and encountered numbers and decimal places. That experience supplied a practical, limited foundation, but not the formal instruction the new calculation appeared to demand.

A day of calculation, and an answer left uncertain

Duckling spent a whole day working from the birth information Mr Auslander had prepared for his tombstone. By dusk he brought his result to the bedside. His father no longer had the strength to nod: he touched the boy’s hand gently and closed his eyes for the last time. Duckling never learned from him whether the answer was right.

Young Lillie later examined the three sheets Duckling had used. The first showed that he had added year by year, arriving at 32,485 days and subtracting 253 to reach 32,232. He had not used multiplication, and the painstaking method had taken a twelve-year-old a full day. When Duckling asked whether he was right, Young Lillie knew that he was not: the calculation failed to account for leap years. Still, rather than distress him, Young Lillie praised the way he had counted from the birthday, which avoided having to handle incomplete years at both ends of a count beginning on January 1. The reassurance recognized the boy’s effort and his sound choice of approach, while leaving the factual error uncorrected.

The answer’s uncertainty became part of the event’s emotional weight. Duckling had worked for his dying father, yet the person whose approval might have confirmed the result was already beyond response. When he noticed Young Lillie studying the calculations, he felt nervous and uncomfortable, only beginning to sense that their relationship might matter deeply to him.

Inventing multiplication

After the funeral, Duckling showed Young Lillie another set of pages. Their different size and texture, and the altered handwriting, marked them as work made at another time. One column continued the addition; in the other, Duckling had devised a dot to stand for multiplication, not knowing the conventional sign or even the method’s name. He had worked out the first twenty years using both procedures, then changed their order from the twenty-first year onward: multiplication came first, with addition serving as a check.

The corrections on the pages trace the learning itself. An early multiplication result for 7,665 had been revised from something like 6,565, and other products also showed corrections. For the first twenty years, addition remained his principal method while the dot calculation seemed almost decorative. Later, multiplication became primary and addition its proof; as Duckling mastered the new operation, the corrections gradually disappeared. He continued year by year to forty, then leapt to eighty-nine, again reaching 32,485 before subtracting 253 and circling 32,232 so that it stood apart from the surrounding figures.

A final, confused-looking page made the discovery visible in another form. Young Lillie recognized that Duckling was trying to work out the principles of multiplication, and the rules were set down at the bottom. The old man recited them aloud; the passage establishes that what Duckling had reached was multiplication, not merely a private symbol or a rearrangement of addition. The pages preserve an improvised route from counting blossoms to a general operation, with the corrections showing knowledge being built rather than simply displayed.

From a lifespan to a formula

The calculation did not end with the burial request. Later, Jinzhen revisited Mr Auslander’s lifespan through the problem of leap years and the exact times of birth and death. According to the account, Mr Auslander had been born at noon and died at nine in the evening. Counting whole days therefore included non-existent time at the beginning and end of his life, as well as 14 hours, 41 minutes, and 32 seconds accumulated during it. Jinzhen argued that a day had been added to the lifespan, turning the leap-year error into a further question about what a day and a year actually counted.

Master Rong recalled that she and her father were amazed by this precision. A couple of days later, her father was in Zhendi’s room when the family learned that the boy seemed to have arrived at a mathematical theorem; the news left him stunned. Jinzhen laid out the leap-year discrepancy through successive calculations. The annual error was 6,421 seconds, while the relevant 44 minutes and 56 seconds amounted to 2,696 seconds. He expressed the successive adjustments as –6,421 + 2,696, then –6,421 + 2 × 2,696, continuing through –6,421 + 22 × 2,696. He worked the missing time into twenty-three lines of calculation, starting from the earlier figure of 88 years and 112 days, or 32,232 days.

From this work, without anyone’s instruction, Jinzhen derived the formula X = [(first value + last value) × number] / 21. The account presents the achievement not as the application of a lesson but as the boy’s independent reframing of a problem into a formula. The first calculation had been a child’s laborious response to his father’s final wish; the later work turns its unresolved arithmetic into a route toward abstraction. Mr Auslander’s request thus links a deathbed act of care, an answer the boy could never verify with his father, and the emergence of a mathematical method that no one had taught him.