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Here is a betting/investment puzzle to help introduce some tools for reasoning about risk.
You are offered a bet or risky investment that returns or pays-off X dollars for every dollar laid in, where X is a random variable distributed log-normally
A lot of financial thinking and intuition attempts to phrase decisions and choices in terms of claimed returns or in terms of expected values. For example:
An ultra greedy gambler may pick the highest claimed winning payoff, independent of win or success probabilities. A greedy gambler may pick the highest expected value, independent of variance. The famousCurrent (mid-2026) LLMs are quite powerful. However, (prior to addition of agent harnesses and loops) LLMs vendors don’t claim to include an effective scratch workspace. Current LLM designs, to the extent we know what they are, remain extensions of the attention/transformer design proba...
The (now falsified) Jacobian conjecture was a well known conjecture expressing the hope that a family of functions had nice properties. In July 2026 Levent Alpöge exhibited a simple counter-example that he said was found by Claude Fable 5. Fields medalist Terenc...