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Think you still got what it takes?

Techers are known for their collaboration and extraordinary problem solving skills. Here are a few problem sets from alumni who are currently professors at Caltech. Use the comment section for each problem set as a discussion board to collaborate with your fellow alumni.

When you're ready to check your answers, solutions to the problem sets can be found at the bottom of this page.

Chances Are


Kim C. Border (BS '74), Professor of Economics
MA3/103 Introduction to Probability and Statistics

The answer to the following question is more interesting than the question itself.

Consider the experiment of tossing a fair coin until you get two consecutive heads. What is the probability distribution of the number of tosses?

(Just to be clear, the sequence HH requires two tosses and HTTHH requires five tosses.)



Ready for the answer? Click here.



Paul Asimow (PhD '97), Eleanor and John R. McMillan Professor of Geology and Geochemistry

Consider a He-filled balloon with a volume of 1 cubic meter. At 25°C and 1 atmosphere pressure, can you hold it without floating away?

What if you tied it to your cat?

What if you tied it to your hamster?

In general, how much mass can 1 cubic meter of He lift in air (the answer is a satisfyingly round number)?


BONUS: See if you can do this in your head, from memory, without looking up any constants or equations.



Ready for the solution? Find it here.

Self-Heating Container


OnTech, a San Diego startup company, has developed self-heating containers for meals and beverages. These containers will heat coffee to an optimal drinking temperature of 145 degrees F (63 degrees C) within minutes after pressing a button, and the containers are designed to keep beverages hot for 20 minutes (and warm for about an hour).

When you’re about ready for coffee, turn the can over and remove the “tamper proof foil”. Push the “activation button” to break the “foil seal." Gravity causes water (“heat generating liquid”) to flow from the “water container” into the “heat generating material”, which is calcium oxide (=3.34 g/cm3; =42 J/K/mol). Water reacts with calcium oxide, producing heat and calcium hydroxide (=2.21 g/cm3; =87.5 J/mol/K):

Screen Shot 2017-11-27 at 3.54.21 PM.png

(The “internal heat generation cone” contains no calcium oxide; it is a finned heat transfer surface that is designed to expedite heat transfer to the coffee by increasing the heat transfer surface area. You may assume that this finned cone requires negligible volume since it’s comprised of very thin, extended surfaces.)

After five seconds, turn the can over. Wait about six to eight minutes, pop the top, and enjoy your hot coffee without ever leaving your desk.

You must design a standard 16-oz (470 mL) beverage container to hold coffee and all components of the heat source (water, calcium oxide). Check the feasibility of this concept by calculating the largest volume of coffee that the container can hold while still meeting the requirement that the coffee be heated from room temperature to 145 degrees F (63 degrees C). Does this volume of coffee seem reasonable for a single serving? (You do not need to worry about the rate of heat transfer at this time; so you don’t need to consider the requirement that the coffee reaches 145 degrees F (63 degrees C) in six minutes.)

Ready for the answer?