NRM 340 Problem Set #1

Heuristics for Problem-Solving

Background: Heuristics and Fermi Problems

(a.k.a sanity tests and back-of-the-envelope calculations)

A heuristic method is an experience-based technique that assists problem-solving, particularly when a quick estimation of an optimal solution is needed. Heuristics are "rules of thumb", educated guesses, intuitive judgments or simply common sense. Perhaps the most fundamental heuristic is "trial and error", which can be used in everything from matching bolts to bicycles to finding the values of variables in algebra problems.

Here are a few other commonly used heuristics:

·  If you are having difficulty understanding a problem, try drawing a picture.

·  If you can't find a solution, try assuming that you have a solution and seeing what you can derive from that ("working backward").

·  If the problem is abstract, try examining a concrete example.

·  Try solving a more general problem first (the "inventor's paradox": the more ambitious plan may have more chances of success).

Heuristics are useful in many branches of research because they allow you to check the sanity of your results either before or after embarking on a full-scale calculation. Having a ballpark estimate of what the right answer should be can help you avoid wasting time, and can help you get back on track if you make a mistake. Can a forest really have 30,000 board-feet per acre? Can Can the temperature of a lake really be -150°C? It is almost always worth the small amount of time and effort needed to check the sanity of an answer before submitting it.

A Fermi problem is an estimation problem based on heuristic methods. Named for 20th century physicist Enrico Fermi, such problems typically involve making justified guesses about quantities that seem impossible to compute given limited available information.

Fermi was known for his ability to make good approximate calculations with little or no data. One well-documented example is his estimate of the strength of the atomic bomb detonated at the Trinity test, based on the distance traveled by pieces of paper dropped from his hand during the blast. Fermi's estimate of 10 kilotons of TNT was remarkably close to the now-accepted value of around 20 kilotons.

The classic Fermi problem is "How many piano tuners are there in Chicago?" A typical solution might include the following assumptions:

1. There are approximately 5,000,000 people living in Chicago.

2. On average, there are two persons in each household in Chicago.

3. Roughly one household in twenty has a piano that is tuned regularly.

4. Pianos that are tuned regularly are tuned on average about once per year.

5. It takes a piano tuner about two hours to tune a piano, including travel time.

6. Each piano tuner works eight hours a day, five days a week, and 50 weeks a year.

From these assumptions:

(5,000,000 persons in Chicago) / (2 persons/household) × (1 piano/20 households) × (1 piano tuning per piano per year) = 125,000 piano tunings per year in Chicago.

(50 weeks/year)×(5 days/week)×(8 hours/day)×(1 piano tuning per 2 hours per piano tuner) = 1000 piano tunings per year per piano tuner.

(125,000 piano tuning per year in Chicago) / (1000 piano tunings per year per piano tuner) = 125 piano tuners in Chicago.

Is this the right number? Almost certainly not. But the answer probably is correct within one degree of magnitude (meaning that there are more than 12 and fewer than 1250 piano tuners). For some purposes, this level of accuracy can be enormously helpful.

Problems

Try your hand at the following problems. Don’t agonize over your estimates. Remember, Fermi problems are supposed to be very quick. You are not expected to get the “right” answer, and you won’t be graded on accuracy, but you do need to clearly show your reasoning at each step. Use round numbers for your estimates to make the math simpler.

  1. How many square inches of pizza are consumed by all the students at the UAF during one semester?
  2. What is the total amount of time 20-year-olds worldwide spent during this past semester studying for exams in college?
  3. Assuming the U.S. national debt were divided equally among every man, woman, and child in the country, what is your share? (after doing this problem by estimation, use the Web to find the “real” answer.)
  4. If the total land area for the state of Alaska were divided up among Alaskans, how much would each person get? (after doing this problem by estimation, use the Web to find the “real” answer.)
  5. How much milk is produced in the US each year? (after doing this problem by estimation, use the Web to find the “real” answer.)
  6. How many pounds of cabbage are grown by home gardeners in Alaska annually?
  7. How many pounds of moose does the average Alaskan eat annually? (after doing this problem by estimation, use the Web to find the “real” answer.)
  8. How many rolls of toilet paper could be made from all the trees in the Fairbanks North Star Borough?

Please type or use legible hand-writing. Make sure that your answers show all the steps in your reasoning, including all estimations and calculations performed. This assignment is due in one week, on Wednesday September 16 by 10:30 a.m.