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Egg-Dropping Problem

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Latest News   Have you purchased a copy of my brand new book Dynamic Programming: Foundations and Principles? If not, now is the right time! Have you read the preview of my forthcoming book Foolded by Robustness: A Perspective from Down Under -- the Land of the Black Swan? If not, now is the right time! December 11, 2011 Added a review report on Adaptation in the UK: a decision-making process. November 19, 2011 Added a review report on Ecological-economic optimization of biodiversity conservation under climate change. November 17, 2011 Slides of my ASOR Recent Advances in Operations Research 2011 presentation On the Power of the (peer-reviewed) Word are now available. November 8, 2011 Slides of my recent Alex Rubinov Memorial Lecture The Art, Science, and Joy of (mathematical) Decision-Making are now available. July 27, 2011 Added a discussion on a CSIRO report on Uncertainty and Uncertainty Analysis Methods July 27, 2011 Added a Second Call for the Reassessment of Info-Gap Decision Theory June 27, 2011 Added a review report on Investment decisions under uncertainty --- A methodological review on forest science studies June 27, 2011 Added a review report on Integrating info-gap decision theory with robust population management: a case study using the Mountain Plover June 27, 2011 Added a review report on Linear bounds on an uncertain non-linear oscillator: an info-gap approach June 22, 2011 Added a review report on Robust climate policies under uncertainty: A comparison of Info-­-Gap and RDM methods May 9, 2011 Added a review report on Precaution with the Precautionary Principle: How does it help in making decisions April 9, 2011 Added a review report on The Precautionary Approach and the Role of Scientists in Environmental Decision-Making April 3, 2011 Added a review report on Frequently asked questions about info-gap decision theory October 23, 2010 Added a review report on Info-gap robustness for the correlation of tests and simulations of a non-linear transient October 23, 2010 Added a review report on Applying information-gap reasoning to the predictive accuracy assessment of transient dynamics simulations October 23, 2010 Added a review report on Information-gap robustness for the test-analysis and optimization of a nonlinear transient simulation October 16, 2010 Added a review report on Computing Robustness of FlexRay Schedules to Uncertainties in Design Parameters October 10, 2010 Added a review report on What Makes a Good Decision? Robust Satisficing as a Normative Standard of Rational Decision Making September 22, 2010 Added an Official Mobile Debunker of info-gap decision theory September 14, 2010 Added a review report on Robust satisficing voting: why are uncertain voters biased towards sincerity? September 12, 2010 Added a review report on Robustness of Risk Maps and Survey Networks to Knowledge Gaps About a New Invasive Pest September 11, 2010 Added a review report on Securing the Border from Invasives: Robust Inspections under Severe Uncertainty September 7, 2010 Added a review report on Allocating monitoring effort in the face of unknown unknowns September 6, 2010 Added a Mobile Maximin Theorem for the convenience of the traveling info-gap scholar. September 6, 2010 Added a Mobile Radius of Stability Theorem for the convenience of the traveling info-gap scholar. September 1, 2010 Added a review report on Interpreting Null Results from Measurements with Uncertain Correlations: An Info-Gap Approach August 17, 2010 Added a review report on Information gap analysis of flood model uncertainties and regional frequency analysis August 16, 2010 Added a review report on Info-Gap Economics: An Operational Introduction April 18, 2010 Added a discussion entitled When the Info-Gap Spin Goes Marching in! February 19, 2010 Added a review report on Robust decisions for declaring eradication of invasive species by Tracy M. Rout, Colin J. Thompson, and Michael A. McCarthy. November 26, 2009 Created a new directory called Info-Gap Economics July 12, 2009 Added a review report on Heterogeneous uncertainties in cholesterol management by Yakov Ben-Haim, Clifford C. Dacso, Jonathon Carrasco, and Nithin Rajan. June 30, 2009 Added a discussion on Myth and Facts about Info-Gap Decision Theory June 27, 2009 Added a PDF file of a presentation entitled How do you recognize a Voodoo decision theory? (RMIT June 26, 2009). June 22, 2009 Added a discussion on Reinventing a Square Wheel. June 15, 2009 Added a preliminary review report on the recently published article An info-gap approach to managing portfolios of assets with uncertain returns by Beresford and Thompson. June 11, 2009 Added a discussion on the Too Good to be True! syndrome. June 6, 2009 Added a tutorial A Guided Tour of Info-Gap's Robustness Model and Some of its Fundamental Flaws June 1, 2009 Added a review report on Do we know how to set decision thresholds for diabetes? by Y. Ben-Haim, M. Zacksenhouse, C. Keren, and C.C. Dacso. May 23, 2009 Added a short discussion entitled A Short Second Opinion on Info-Gap Decision Theory. May 10, 2009 Added a short discussion on Laplace's Principle of Insufficient Reason. May 9, 2009 The PDF file of the seminar entitled Black Swans, Modern Nostradamuses, Voodoo Decision Theories, Info-Gaps, and the Science of Decision-Making in the Face of Severe Uncertainty is now available. May 5, 2009 Added a short discussion on Bayes Rule vs Maximin Rule, indicating that the latter is not an instance of the former. April 28, 2009 Added a review report on a new refereed Info-Gap paper by Lior Davidovitch, Richard Stoklosa, Jonathan Majer, Alex Nietrzeba, Peter Whittle, Kerrie Mengersen, Yakov Ben-Haim. April 24, 2009 Added a page providing some criticism of Bueno de Mesquita's theories. It is linked to the page New Nostradamuses. April 14, 2009 The bug in the "Comments" facility was fixed. You can now post your comments anonymously. April 9, 2009 New Decision-Making Under Severe Uncertainty site launched. April 5, 2009 A page dedicated to reviews of publications on Info-Gap decision theory is now available. March 31, 2009 An experimental Your comments, please! facility is now onboard. Give it a try. March 26, 2009 A very interesting Freudian Info-Gap slip! March 24, 2009 First major sign of progress! See comments on Hall and Harvey's (2009) new intriguing assumption regarding the likelihood of deviations from the estimate. March 24, 2009 A new section on Modern alchemy, Freudian slips, quick-fixes and suchlike was added.

Supplementary Material   Mobile Maximin Theorem Mobile Radius of Stability Theorem Reinventing a Square Wheel Too Good to be True! The Spin Stops Here! Black Swan New Nostradamuses Criticism of Bueno de Mesquita's Theories Modern alchemy, Freudian slips, quick fixes and suchlike Satisficing vs optimizing 2-envelope puzzle Counterfeit coin puzzle Egg-dropping puzzle

This page is under construction.

For the time being ....

This puzzle is extremely popular with math teachers. Indeed, it appears on a number of very popular lists!

Suppose that we wish to know which windows in a 36-story building are safe to drop eggs from, and which will cause the eggs to break on landing. We make a few assumptions:

• An egg that survives a fall can be used again.
• A broken egg must be discarded.
• The effect of a fall is the same for all eggs.
• If an egg breaks when dropped, then it would break if dropped from a higher window.
• If an egg survives a fall then it would survive a shorter fall.
• It is not ruled out that the first-floor windows break eggs, nor is it ruled out that the 36th-floor windows do not cause an egg to break.

If only one egg is available and we wish to be sure of obtaining the right result, the experiment can be carried out in only one way. Drop the egg from the first-floor window; if it survives, drop it from the second floor window. Continue upward until it breaks. In the worst case, this method may require 36 droppings. Suppose 2 eggs are available. What is the least number of egg-droppings that is guaranteed to work in all cases?

Which Way Did the Bicycle Go? (page 53)
Konhauser J.D.E., D. Velleman, and S. Wagon (1996)
Dolciani Mathematical Expositions - No. 18
The Mathematical Association of America.

I decided to take a formal look at it whereupon it became amply clear that the treatment of this puzzle in the official literature only scratches the surface. This puzzle is a real treasure trove!

So .... during my 2002 sabbatical I decided to investigate this gem more thoroughly. I published the results in an electronic paper entitled "The Joy of Egg-Dropping in Braunschweig and Hong Kong". This paper contains a number of on-line computational modules.

I strongly recommend this paper to lecturers who teach subjects that are related to robust decision-making under severe uncertainty.

This puzzle is interesting because it has a solution that is optimal with respect to both Laplace's Principle of Insufficient Reason and Wald's Minimax paradigm.

Disclaimer: This page, its contents and style, are the responsibility of the author (Moshe Sniedovich) and do not represent the views, policies or opinions of the organizations he is associated/affiliated with.

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