Behind every calculation is a person.

01 / Idai challenge problems

Unanswered problems inspired the next book.

The 1641 edition of Jinkōki included twelve problems without solutions. Readers who solved them could publish answers and pose harder questions of their own. A book for learning arithmetic could therefore become a starting point for further research.

This resembles an open mathematical challenge in one respect, but it was not the same institution as a modern contest: schools, secret transmission, and manuscript teaching also existed alongside print culture.

02 / Sangaku

Mathematics was displayed at temples and shrines.

Sangaku are votive tablets carrying mathematical problems, often with colorful circles and triangles. They preserve traces of solved problems, challenges, and networks of learning within the wider culture of dedication at temples and shrines.

Some originals were lost to fire, damage, or disappearance. An original tablet, a later reconstruction, and a transcription in a book are different kinds of evidence and should not be treated as interchangeable.

03 / Itinerant mathematicians

A travel diary preserved 324 problems.

Some mathematicians travelled from place to place teaching and exchanging problems. Yamaguchi Kazu, also known by the sobriquet Kanzan, left a Dōchū Nikki recording his travels; the National Diet Library notes that he copied 324 sangaku problems he encountered along the way. People moving between communities were themselves a medium for mathematical transmission.

04 / π

To calculate faster, study the error.

Inscribed polygons approach a circle as their number of sides increases. Seki and Takebe also studied how successive approximations changed, looking for ways to improve the calculation rather than merely extending a list of values.

This site uses the cautious phrase “around forty decimal places” for Takebe’s work because written digits, correct digits, and working precision are not the same thing. Seki’s acceleration from three approximations and Takebe’s later numerical procedure are also kept distinct.

05 / Six-sphere chain

After six spheres, the chain returns.

Inside a large sphere, place two mutually tangent fixed spheres that also touch the outer sphere. Then continue adding spheres that touch the outer sphere, both fixed spheres, and the preceding sphere without immediately reversing direction. In a nondegenerate configuration, the chain closes after six spheres; it is not a claim that any arbitrary choice of six spheres will close.

Irisawa Shintarō Hiroatsu’s 1822 sangaku problem is preserved in the 1832 Kokon Sankan. Frederick Soddy’s first Nature publication on the hexlet appeared on December 5, 1936: 114 years after the 1822 dedication. The later 1937 discussion should not be confused with that first report.

Repeating radii and returning to the same spatial position are separate claims. Modern inversion geometry can explain the closure, but that modern proof must not be attributed to Irisawa unless a historical source supports it.

06 / Sequences

The same coefficients appeared in Edo and Basel.

Power-sum formulas in the Japanese and European traditions contain coefficients corresponding to what are now called Bernoulli numbers. Seki’s Katsuyō Sanpō was published posthumously in 1712 and Jacob Bernoulli’s Ars Conjectandi in 1713. A one-year difference in publication dates does not by itself establish a one-year difference in discovery.

07 / Hyakugo gensan

Can remainders identify a number?

If division by 3 leaves 2, division by 5 leaves 3, and division by 7 leaves 2, the smallest non-negative solution is 23. But 23 is not the only solution: adding 105 preserves all three remainders because 3 × 5 × 7 = 105.

The problem has roots in the Chinese Sunzi Suanjing and was also cultivated in Japan as hyakugo gensan. Learning about Japanese mathematics therefore includes tracing the routes by which knowledge arrived and changed.

08 / Continuity

Wasan did not vanish the day school policy changed.

Meiji education shifted institutional teaching toward Western mathematics, but people trained in wasan continued teaching, local schools persisted, and later researchers collected and studied the surviving books and tablets. Institutional replacement and cultural disappearance are not the same event.

09 / Genjikō

Five incense samples give 52 patterns.

Genjikō diagrams classify which of five incense samples are judged to have the same scent. There are 52 possible equality patterns. In modern combinatorics, 52 is also the Bell number B₅, the number of partitions of five distinct objects into nonempty unlabeled groups.

Edo-period practitioners did not use the modern name “Bell number.” The connection is a later mathematical comparison, not evidence that the historical notation or terminology was identical.