One hundred birds for one hundred coins: the puzzle that toured the medieval world

# One hundred birds for one hundred coins: the puzzle that toured the medieval world

A servant is sent to market with a hundred coins and strict instructions, in Mahāvīra's *Gaṇita-sāra-saṅgraha* (850 CE):

> "Pigeons are sold at the rate of 5 for 3 paṇas, sārasa birds at > the rate of 7 for 5, swans at the rate of 9 for 7, and peacocks > at the rate of 3 for 9. A certain man was told to bring at these > rates 100 birds for 100 paṇas for the amusement of the king's > son… What amount does he give for each?" > > — *Gaṇita-sāra-saṅgraha*, as translated in Datta & Singh (1938)

Exactly one hundred birds, exactly one hundred coins, four prices. Two equations, four unknowns — an *indeterminate* system, solvable only because bird-counts must be whole numbers (and, at these package rates, whole multiples: pigeons come in fives, sārasas in sevens). One solution the reader can check: **15 pigeons (9 paṇas), 28 sārasas (20), 45 swans (35), 12 peacocks (36)** — 100 birds, 100 paṇas on the nose. Mahāvīra's method generates the whole family of valid purchases.

The same puzzle, everywhere

Now the remarkable part, and the reason this claim exists. Almost this exact problem — one hundred creatures, one hundred coins, mixed prices, integer solutions — appears across the medieval world:

- **China, c. 475 CE** (and this is the earliest known instance — priority is Chinese, stated plainly): Zhang Qiujian's *Suanjing* poses cocks at 5 coins, hens at 3, chicks at 3 for 1 — buy 100 fowls for 100 coins. The "hundred fowls problem," a fixture of Chinese mathematics thereafter. - **India, 850 CE**: Mahāvīra's birds, above — with the same 100-for-100 dress and the same integer logic. - **Carolingian Europe, c. 800**: Alcuin of York's *Propositiones ad acuendos juvenes* — the puzzle collection tradition associates with Charlemagne's court school — runs the same structure with various casts of buyers and beasts. - **The Islamic world, c. 900**: Abū Kāmil of Egypt wrote an entire treatise, *Kitāb al-ṭair* ("Book of the Birds"), on this problem class, complaining that people passed such puzzles around and answered them without understanding.

One mathematical species, four plumages. Whether the puzzle travelled the trade routes (commercial arithmetic moved with merchants, and [the numerals themselves were making the same journey](/c/52c04b7b-4540-59e7-a538-16ebfe83fd7f) in these very centuries) or was reinvented where markets and arithmetic met, is genuinely debated — the claim frames the clustering as a *marker of circulation*, which is the standard scholarly reading of the shared 100-for-100 costume, while flagging the debate. Either answer is a good story: mathematics travelling like a folk-tale, or market arithmetic being so universal that four civilizations independently posed the same brain-teaser. On this platform's evidence standard, what is *documented* is each text, each date, each cast of birds.

Why the problem matters mathematically

Underneath the poultry is serious structure: linear Diophantine systems — equations whose unknowns must be non-negative integers. The bird constraint does real work: without integrality the system has infinitely many solutions and the problem is trivial; with it, solutions are scarce and *finding the family* requires method. Mahāvīra, whose [combinatorial rules](/c/50f870a0-ac17-52a9-ad76-d91de7ac23f6) appear elsewhere in this corpus, treats the problem inside a tradition already possessing the strongest integer-equation machinery of the age — [the kuṭṭaka algorithm](/c/a0ac1d5a-b8a4-5228-bb4e-e69636d8d613) descending from Āryabhaṭa. In China the hundred-fowls problem likewise pushed method forward; in Europe, Alcuin's puzzles carried integer reasoning through centuries with little other mathematical infrastructure. A joke problem, load-bearing.

Legacy

The hundred-birds family never went extinct. It survives in Fibonacci's *Liber Abaci* (1202), in Renaissance problem books, and in every modern recreational-mathematics collection — and its serious descendant, integer linear programming, now schedules airlines and allocates cloud servers: two equations, many unknowns, solutions required to be whole. The servant buying birds for the king's son is solving, in miniature, the same class of problem your delivery app solves at breakfast.

For this corpus, the claim does one more job. Most transmission-history here runs outward from India — numerals, siddhāntas, series. This claim runs the other way: the earliest hundred-fowls is Chinese, and Indian mathematics appears as one brilliant participant in a Eurasian conversation, receiving as well as giving. That is what the platform's multi-civilization honesty looks like in practice — and the conversation itself, four traditions batting one puzzle around for centuries, may be the most cheering fact in the whole file.

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Sources

- [Datta & Singh, *History of Hindu Mathematics: A Source Book*, 1938](https://archive.org/details/wg143) — the Gaṇita-sāra-saṅgraha hundred-birds problem in translation (the same problem appears in the enrolled Rangacarya 1912 GSS). - Katz, *A History of Mathematics*, 3rd ed. — the hundred-fowls problem across China, India, Islam, and Europe (secondary synthesis for the comparative dates). - Singmaster, "Sources in Recreational Mathematics" — the problem-family's documented instances (referenced for context only).

Related claims

- [Mahāvīra's combinations formula (850 CE)](/c/50f870a0-ac17-52a9-ad76-d91de7ac23f6) - [Aryabhata's kuṭṭaka algorithm (499 CE)](/c/a0ac1d5a-b8a4-5228-bb4e-e69636d8d613) - [The thousand-year road trip of the digits 0–9](/c/52c04b7b-4540-59e7-a538-16ebfe83fd7f)