One Thousand and One Conjectures

The astragalus — the ankle bone of a sheep or goat — was the everyday die of Greek and Roman gaming, and unlike a cube it is honest about its dishonesty: its four usable faces land with very different frequencies, broad faces…

Roman dice are notoriously irregular — surviving cubes are often visibly asymmetric, with face dimensions and pip placement far from the modern standard — and the crookedness has usually been waved off as indifference to fairness. Asymmetry, though, is measurable: 3D scanning…

The Shang kings at Anyang divined by heating cattle scapulae and turtle plastrons until they cracked, reading the cracks as answers — and the inscriptions often record the sequence in which the questions were put. Crack formation, driven by micro-structural accidents of…

The two classical ways of casting the I Ching generate hexagrams with different statistics: the older yarrow-stalk procedure is asymmetric — its line-types fall with unequal probabilities, making some changing lines markedly likelier than others — while the later coin method is…

A forecaster facing a high-stakes, high-uncertainty question hedges — wide intervals, conditional phrasing — because a confident miss is fatal to credibility; options markets price the same logic as implied volatility. The oracle at Delphi faced the identical institutional problem for a…

Spanish chroniclers report that Inca accounting was run by paired officials who kept independent records of the same stocks so that each could check the other — and the surviving corpus does contain matching khipus, cord records whose numerical content substantially overlaps.…

Bachet's classic weight problem asks for the smallest set of standard weights that can weigh out every required quantity on a balance — a combinatorial optimisation whose solutions depend on whether weights may sit in one pan or both. The Indus Valley…