Science
Olympiad
2 Trans Am Plaza Drive, Suite 415, Oakbrook Terrace,
Illinois, 60181
Tel: 630-792-1251
FAX: 630-792-1287
The information below should not be interpreted as an
extension of the rules. The official rules in the current Coaches Manual and Rule Book take precedence.
Fermi Questions is named after Enrico Fermi, a Nobel Laureate in Physics, who
was famed for being able to do order-of-magnitude calculations in his head. For
example, after watching the first atomic bomb explosion, he immediately
calculated that the strength of the explosion was equivalent to the explosion of
20 kilotons of TNT. It took another three weeks for a panel of the Manhattan
Project's best scientific brains to do an 'exact' calculation; the answer that
they came up with, yes, you guessed it - 20 kilotons. Such calculations are
sometimes called 'ballpark estimates' or 'back-of-the-envelope' calculations.
While these calculations were important, years ago, because one had to keep
track of decimal places when using a slide rule, these calculations are still
very important because an approximate answer will often dictate the amount of
resources required to attack a problem. For example, when you consult a wedding
consultant to plan the affair, they often ask the question, "How many people
will attend the dinner?". Your approximate answer will allow them to estimate
the amount of food required, the number of tables and their layout, the size of
the hall to be rented, etc., etc. Fundamental to the solution of these problems
is a skill called Critical Thinking - essentially a method of attacking such
problems in an orderly, logical way. This skill can be learned and it is the
underlying basis for the event.
Why this event?
Numbers (when you think about it) are a measure of our surroundings and life.
There
are several types of questions that can be answered by this procedure:
* math (straight) – where the answer can be calculated using a calculator or
computer but, since such aids are not allowed in the competition, it forces the
student to consider other routes to provide a reasonable answer
* how answers from one problem relate to other problems – as with many facets of
life, an answer to one problem leads to many other choices and problems.
* having solutions to problems relate to 'real life', for example, a problem
might ask for an estimate of the amount of gasoline used by passenger cars in
the U.S., how an increase in gas mileage would relate to a decrease in
green-house gas production, and how the amount of water produced by same relates
to other items such as rainfall or filling of swimming pools.
In short, if
something has numbers associated with it, that subject is fair game for a Fermi
Question.
Underlying Considerations
* Behind each problem set that I create is the tacit assumption that the
contestants have a reasonable knowledge of mathematics, specifically, the use
and operations of exponential notation. The lack of math skills is not too
apparent when the answers to the problems are in the range 0.001 to 1000 (Fermi
Question notation –3 to +3). But when I ask the students to calculate the number
of iron atoms on the head of a pin, the inability to handle exponents readily
shows (there are approximately 3*1013 iron atoms – Fermi Question
answer +13; see Example xiv. below). I can't count the number of times that I've
seen students cover the scrap paper (that I distribute for them to use in their
deliberations) with zeroes. For that reason, it is imperative to stress the use
of exponential notation (which also serves as the basis for the metric system).
Not only does the use of exponential notation make calculations faster, but it
also helps avoid problems with writing, transcribing, and counting the correct
number of zeroes. So much so, that in some branches of science there are
specially named units that have very large (or very small) numbers associated
with them, such as, one Angstrom = 10-8 cm, one Light Year = 5.9*1012
miles, Avogadro's Number = 6.023*1023.
* As I noted previously, an important component of the event is logical,
critical thinking. Reading and understanding the problem is one important
component; the other important component is to develop a plan to provide the
answer in the requested units.
* And finally, time is a critical parameter. The ability to think and calculate
rapidly can be learned – the keywords are, in the immortal words of a
Hall-of-Fame football coach, "practice, practice, and more practice". I have
watched students (when I was a coach) significantly lower the times required to
solve these problems. In fact, some of them have returned from college and told
me that the same skills (required to solve Fermi Questions) permitted them to
handle tests and problem sets much faster than their contemporaries.
Typically, the first time that a team tries to solve a problem, they try to be
too exact. For example, if the Fermi Question is "how many toothpicks are
equivalent to the perimeter of Colorado?", they discuss the length of a
toothpick ("is it 2.0, 2.25, 2.45 inches?); then they try to estimate the
perimeter of the state; and finally, they calculate a value. Any time there is a
discussion, time is lost. Since the answer to any question is the correct order
of magnitude, an error of a factor of two or three will probably yield the
correct exponent (the Fermi Question answer). Hence, they should pick a value
and work up their answer. The time that they save will be needed to solve other
problems.
Examples
- how many air
molecules are in this room (where I was presenting this lecture)?
- how many
pounds of CO2 and H2O does the U.S. population expel in a
year?
- how many tons
of food are consumed in Chicago during the course of a day?
- how many
people are involve in delivering and preparing that food?
- how many
gallons of paint do you need to paint the walls of your school?
- how many
baseballs are used during the course of a Major League season?
- how many pizzas were eaten last year in the U.S.?
Scoring
The scoring for the event is like horseshoes:
5 points for the correct exponent
3 points for the correct exponent ± 1
1 point for the correct exponent ± 2
The answer to a
Fermi Question is the correct exponent of 10 (if an answer is 5*10n,
round the answer up to the next power of 10; I try to manage the problems so
that answers are not 5*10n). Generally, if a team averages 3 points
per problem and there are 30 problems, the 90 points that they will have
achieved will garner them first or second place. Calculators, computers, or any
other device, including crib sheets, lists of constants, formulae, etc., are not
permitted. All the contestants need are pencils (with erasers) and a good
night's sleep – The Olympiad supplies scratch paper (to simulate the
'back-of-envelopes'). Positive exponential values are the default; negative
exponents MUST have the - (minus) sign as part of the answer.
Some
considerations involved when learning to solve these problems:
1. Exponents are short-hand notation
(knowledge of which makes it easier and faster to solve the problems). The
notation used below is: Ex. = example; Ans. = Answer; FA = Fermi Answer.
Ex. What is the population of New York City? Ans. 7,000,000 = 7*106 ~
107
FA 7
Ex. What is the distance, in miles, from the Earth to the Sun? Ans. 100,000,000
= 108
FA 8
2. Properties of exponents.
500 = 5*102; 5 is the coefficient, 10 is the base, 2 is the exponent
3. Round off values BEFORE doing a
calculation. This makes it much easier and faster to do the problems. Why?
because the FA is the correct order of magnitude which means that there is a
large range that yields the correct answer. For example, the FA for the
distance, in miles, from the Earth to the Sun is 8 (shown previously in 1.) but
the range of values giving the same answer is 5*107 to 4.99*108!!
In this context, I suggest using the values below which are somewhat different
from the exact values:
Item
Exact Value
Fermi Value (for ease of calculation)
1 day
24 hours
25 hours
1 mile
5280 feet
5000 feet
1 yard
0.9144 meter
1 meter
1 foot
30.48 cm
30 cm
1 pound 453.6
g
500 g
1 hour
3600 seconds
4000 seconds
4. Always
keep the units as part of working a problem. In some instances, keeping
track of the units will lead to the correct answer. The U.S. uses both metric
and English systems of units. Sometimes the units get left off solutions to real
problems with tragic, unforeseen results. As an example, most US cooks know what
a 1/4
pound of butter looks like - it is a stick about 1 inch x 1 inch x 5 inches. But
ask them what 100 g of butter looks like and they may throw up their hands in
defeat. The answer is that the stick is almost the same size since 100 g is
close to 1/4 pound.
5. What
subject matter is covered? Everything is fair game! If the item in question
has numbers associated with it, it might be used. On the past tests, questions
have been used from math, chemistry, physics, biology, geology, geography,
economics, swimming, basketball, running, census, food, waste generation,
Examples.
(abbreviation: F?s = Fermi Question solution) These can be done by the students
as practice; have them show all work and what assumptions they made in solving
the problems.
Note that both answers are the same. Budding Fermi Question experts should
remember that there are 3*107 seconds in a year - this will probably
save them time in solving another F?.
ii. How many miles are
there in a light-year?
Exact
solution: 186,000 m/s * 3.15*107 s/y = 1.86*3.15*1012
= 5.9*1012 m/y
FA 13
F?s: 2*105
m/s * 3*107 s/y = 6*1012
m/y
This quantity is a basic unit used in astronomy. As noted in problem i., knowing
that there are 3*107 seconds in a year has shortened the work
considerably.
iii. How many kilometers are
there in a light-year?
vi. How many pounds of rice were
consumed in the U.S. in the year 2001?
Assumptions:
20 pounds of rice eaten per year by a person, 3*108
people in the U.S.
F?s: 20 #/p *
3*108 p = 6*109 # = 1010
FA 10
This answer was checked using data from the U.S. Dept. of Agriculture; 5.2*109
lbs. If the students assume 2-10 or 200-1000 #/p, they would still get 3 points.
vii. What is the density of butter in
g/cc?
Assumptions:
1 pound of butter is a package 2 inch x 2 inch x 5 inches.
F?s: V = 2 in
* 2.5 cm/in * 2 in * 2.5 cm/in * 5 in * 2.5 cm/in = 5 * 5 * 12 = 300 cm3
Density = M/V
= 500 g / 300 cm3 = 1.5 g/cm3 ~ 1 = 100
FA 0
