How Did Water Solve the 1800-Year-Old Talmudic Bankruptcy Problem? @Mathologer

Uploaded February 2025 | Updated September 2026, 2 weeks ago

Today we are solving an ancient bankruptcy puzzle from the Talmud using the principle of communicating vessels. A very nice visual way of making sense of an otherwise tricky problem.

"Game theoretic analysis of a bankruptcy problem from the Talmud" by Robert J Aumann and Michael Maschler
sciencedirect.com/science/article/pii/0022053185901024

" ‘Hydraulic’ rationing" by Marek M. Kaminski
sciencedirect.com/science/article/pii/S0165489699000451
Also check out his book about his time as a political prisoner
Games Prisoners Play: The Tragicomic Worlds of Polish Prison

"Notes on the Bankruptcy Problem: an Application of
Hydraulic Rationing" by Tamás Fleiner and Balázs Sziklai
https://www.econstor.eu/bitstream/10419/108227/1/MTDP1123.pdf

Robert Aumann with an anecdote of fair division from his childhood
youtube.com/watch?v=rHDUMQfDkmQ

Andrew Kepert contributed this nice geogebra app: geogebra.org/classic/bfkqsmbk

I did not mention this but the three creditors are actually the three wives of the man who dies.

The Talmud online:
Hebrew and English
sefaria.org/texts/Talmud
Scanned pages in Hebrew
e-daf.com

Talmud references:
The bankruptcy case: Ketubot 93a
sefaria.org/Ketubot.93a.1?lang=bi
(on e-daf Ketubot is spelled Kesuvos)
Contested garment: Bava Metzia 2a
sefaria.org/Bava_Metzia.2a.1?lang=bi
The brother who dies childless case: Yevamot 38a
sefaria.org/Yevamot.37b.1?lang=bi
(on e-daf Yevamot is spelled Yevamos)

A few of the images in this video are from the wiki page dedicated to the Talmud.

There is something called the Water filling algorithm in information theory en.wikipedia.org/wiki/Water_filling_algorithm

Music: "Here to fight" by Roman P.
T-shirt: One of my own (I think) from a long time ago.

Enjoy!

Burkard

How Did Water Solve the 1800-Year-Old Talmudic Bankruptcy Problem?

$The golden ratio spiral: visual infinite descent NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) So you all know the golden (ratio) spiral. But did you know that not only the golden ratio but really every number has such a spiral associated with it? And that this spiral provides key insights into the nature of a number. Featuring more proofs by contradiction by infinite descent (my current obsession), infinite continued fractions, etc. Here are some articles that debunk a lot of the golden spiral in nature nonsense: Clement Falbo http://web.sonoma.edu/Math/faculty/falbo/cmj123-134 George Hart http://www.georgehart.com/rp/replicator/replicator.html https://www.youtube.com/channel/UCTl0dASnxto6j2wlVs5Bs2Q Keith Devlin http://devlinsangle.blogspot.com.au/2017/04/fibonacci-and-golden-ratio-madness.html Here is a very good website devoted to everything to do with the golden ratio and Fibonacci http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html Thank you very much to Michael (editing), Marty (nitpicking) and Danil (translating). Enjoy!$

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$Visualising irrationality with triangular squares NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) Get ready for some brand new and very pretty visual proofs of the fact that root 2, root 3, root 5 and root 6 are irrational numbers. Root 2 being irrational also translates into the fact that the equation x^2+x^2=y^2 has no solutions in positive integers, root 3 being irrational translates into the fact that the equation x^2+x^2+x^2=y^2 has no solutions in positive integers, etc. What I find very attractive about these proofs is that the destructive core of these proofs by contradiction lead a second secret constructive life, giving birth to infinitely many nearest miss solutions of our impossible equations like for example 15^2+15^2+15^2=26^2-1. Here is the paper by Steven J. Miller and David Montague which features the basic root 3 and pentagonal root 5 choreographies. https://arxiv.org/abs/0909.4913 Footnotes: -our nearest miss solutions like, for example, 15^2+15^2+15^2=26^2-1 correspond to the solutions of the equation y^2 - n x^2 = 1 with n=2, 3, 5 and 6. This is the famous Pells equation, which happens to have solutions for all integers n that are not squares. -there is also a second type of nearest miss solutions like 4^2+4^2+4^2=7^2+1 (a plus instead of a minus at the end). Starting with one of these our choreographies also generate all other such nearest misses. -the original Tennenbaum square choreography and the first puzzle root three choreography generate both types of nearest misses from any nearest miss solution. -The close approximations to the various roots corresponding to our nearest miss solutions are partial fractions of the continued fraction expansion of the roots. -lots more things to be said here but we are getting close to the word limit for descriptions and so I better stop :) Thank you very much to Marty for all his nitpicking of the script for this video and Danil for his ongoing Russian support. Todays t-shirt is the amazing square root t-shirt (google square root tshirt). Note that the tree looks like a square root sign AND that the roots of the tree are really square. Enjoy!$

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