Fermi Questions

Fermi Questions was last a Division C event for the 2013 competition year.

What is a Fermi Question?

A Fermi question is one where a seemingly impossible-to-calculate answer is estimated. A famous example of a Fermi question is "How many licks does it take to get to the center of a tootsie roll pop?", where there is very little data to use and assumptions must be made. Fermi questions are named after Enrico Fermi, a physicist who is known for solving these types of questions.

In Science Olympiad, answers to Fermi questions are given in powers of ten. For example, an estimated answer to the above question of 400 licks is put in scientific notation as [math]\displaystyle{ 4\cdot 10^2 }[/math], and the exponent on the ten is used as the answer, yielding 2. If the estimate was 600 licks, or [math]\displaystyle{ 6\cdot 10^2 }[/math], then the answer would be 3, rounding up.

Points are usually given as follows:

• 5 points for the correct power of ten
• 3 points for one away from the correct power of ten
• 1 point for two away from the correct power of ten.

For example, if the correct answer to the number of licks to the center of a tootsie roll pop is 2, and the given answer is 2, five points are awarded. If the given answer is 3 or 1, 3 points are awarded, and if the given answer is 4 or 0, one point is awarded. All other answers would receive 0 points.

Sample Question

Here is a fermi question, worked out, with explanations.

How many pieces of paper could a package of pencil lead cover?

Determine the Facts

The first step in solving a fermi question is to determine what facts are necessary. For this case:

• How much paper a piece of lead can cover
• How large a piece of paper is
• How many pieces of lead are in a package

Now, the size of a piece of paper is 8.5 x 11 inches, which most people should know. The number of pieces of lead in a package could be known, but if not, would have to be assumed. However, the area a piece of lead can cover is not known, and must be assumed.

Making the Assumptions

The key to fermi questions is having good assumptions. Obviously, the best assumptions are the ones that aren't assumed, like the size of a piece of paper. However, more often than not, assumptions will be necessary.

The area a piece of lead can cover is based on the width and length of the line it can draw. Using common knowledge, we know that a piece of lead is either 0.5 mm or 0.7 mm. However, we choose to use 0.5 mm because 5 is an easier number to deal with than 7.

Now that we know the width of the line, we need to know the length of the line. In order to do this, you have to use common sense. If you say it can draw 1 meter, then that's too small. If you say it can draw 100 km, then that's too large. It may be 10 meters, but that seems a little short. It could be 1 km, but that seems a little long. 100 m still seems a little short, but close than 1000 m (1 km). Therefore, we will use 300 m, keeping in mind it could still be small.

So we take [math]\displaystyle{ 5 \cdot 10^{-4} }[/math] times [math]\displaystyle{ 1 \cdot 3\cdot 10^2 }[/math], and get [math]\displaystyle{ 15 \cdot 10^2 }[/math], or [math]\displaystyle{ 2 \cdot 10^{-1} }[/math] (going up because the 300 m could have been small) now, this is, in plain English, 2 tenths of a square meter, or 2000 square centimeters, per piece of lead.

Now, we must assume the number of pieces of lead in a package. The number of pieces of lead in a package vary greatly, so it's not very important to be accurate. 10 pieces seems low, and 1000 seems too high. 100 still seems a little high, so we choose 70 as the number of pieces of lead in a package.

[math]\displaystyle{ 2\cdot 10^3\text{ cm}^2 }[/math] times [math]\displaystyle{ 60= 1.4\cdot 10^5 }[/math], which we will change to [math]\displaystyle{ 2\cdot 10^5 }[/math] in order to simplify the math. This can be done because first, the number comes from assumptions that may not be right, second, because the final answer will only be an exponent, and third, because we will round down later to compensate.

The last step is the divide this by the number of square centimeters in a piece of paper. 8.5 and 11 are awkward numbers, so we simply change them to ten. This will remain somewhat accurate, because when we rounded down on one, we rounded up on the other. Note that this will always result in a larger answer.

Now we need to convert 1 square foot into square centimeters we can do this quite simply if we remember that foot long rulers can fit 30 cm on them. [math]\displaystyle{ 30\cdot 30=900 }[/math], so we say there are [math]\displaystyle{ 1000 cm^2 }[/math] in a piece of paper. This value is high, but if we remember that we're dividing, we'll realize this will cancel out the other rounding up (1.5 to 2) we did earlier.

So, [math]\displaystyle{ \frac{2\cdot 10^5}{1\cdot 10^2} }[/math] is [math]\displaystyle{ 2\cdot 10^3 }[/math] sheets of paper. Therefore, our answer is 3.

Making Calculations Quickly

Magnitude Notation

The above question was worked out sequentially with numbers in to allow for explanation. This works well for shorter problems, but with longer problems with longer numbers, this can take a while, and can become a problem with many questions and limited time and space. When doing such problems, it is better to note numbers in a format that is easier to work with. This format is called magnitude notation.

Magnitude notation works as follows:

• [math]\displaystyle{ 1\cdot 10^3 }[/math] would be E3
• [math]\displaystyle{ 4\cdot 10^3 }[/math] would be +E3
• [math]\displaystyle{ 7\cdot 10^3 }[/math] would be -E4

Basically, the number is put in exponential notation (like your calculator does) and rounded so there is only one digit.

Then, if the one digit is 0, 1 or 2, leave the exponent part (E3), but do not put a plus or minus. If the digit is an 8 or 9, add one to the exponent and do not put a plus or minus. If the digit is a 3 or 4, then you leave the exponent and put a plus. If the digit is a 6 or 7 add one to the exponent and put a minus. If 5 is rounded up, put add one to the exponent and put a minus. If five is rounded down, then put a plus.

Multiplication

Magnitude notation has a few important rules regarding multiplication.

• With magnitude notation, when two numbers with plus signs are multiplied, the pluses are removed and one is added to the resulting exponent. for example, +E5 times +E7 equals E13.
• When two numbers, one with a +, and one with a -, are multiplied, the signs are canceled without changing the exponent.
• When two numbers, both with minus signs, are multiplied, one is subtracted from the exponent and the signs removed.

With division, same signs cancel. Opposing signs are removed, adding one to the exponent of the value with the + sign.

Addition and Subtraction

Addition and subtraction have somewhat complicated rules.

• If the exponents are equal:
  • If the signs are both pluses, add one to the exponent and remove the signs.
  • If there are no signs, put a plus (exponents still equal).
  • If both signs are minus, remove the signs (exponents still equal).
• If one exponent is one number larger than the other:
  • Remove a minus sign if there is one on the larger one
  • Add a plus sign if there is no sign on the larger one
  • Increase the exponent on the larger one while removing the sign if there is a plus sign on the larger one.
• If the exponents are 2 or more apart, simply ignore the smaller number.

Although magnitude notation appears more complicated, it is faster and neater to work with, and thus easier to use with limited time and space.

Equations

In Fermi Questions, there is little need for accurate complex equations. Therefore, use simplified versions of the needed formulas. If a particular formula is unknown, make up an equation that makes sense. For example, instead of calculating values using a parabola, use the triangle formula instead, slightly increasing the base and height of the triangle to compensate for the curve. Likewise, for an ellipsoid, use a cylinder, decreasing the length to compensate for the curve.

Competition Tips

Teams can do very well in Fermi Questions by being as accurate as possible with the most questions. This can be done in certain ways.

Partner Pairing

In Fermi questions, the partner pairing is important. You generally want two people with different interests and personalities. The ideal pair would be the estimator and the number cruncher.

Estimator

The estimator should be a visual/kinetic (but more visual) learner with a good memory. They need to be able estimate the dimensions of, for example, a football stadium based on their memory of one. They need to be able to estimate the weights of objects they know the visual size of, but may or may not have held before. He (or she) should also know random facts, like the frequency of a cordless phone, or the number of Crayola colors. But above all, the estimator must be able to determine all of these estimations with relative accuracy.

Number Cruncher

The number cruncher is someone quick to perform the calculations. The number cruncher should know physical values and conversion factors, like how many pounds are in a kilogram, the speed of light, speed of sound, etc.

The number cruncher is less crucial to event success. The estimator can memorize the values, but the number cruncher cannot easily learn how to estimate.

Use Facts

Facts provide a good basis for estimations, with or without an estimator. Because even an estimator is not infallible, it is crucial to know unit conversions, as well as many basic facts about things ranging from stellar distances to volumes of the earth's oceans. Learn esoteric units, like BTU's, barns, outhouses, sheds, furlongs, and others, which helps when answers are requested to be in terms of an unusual unit. There have been questions were students were asked to put answers in terms of a poronkusema, so nothing is off limits.

Useful Facts

• Physics facts, like the speed of sound, or the wavelength of a certain color of light.
• Human body facts.
• State facts from the state in which the competition is taking place
• Facts about the U.S.A.
• Facts about the world.
• Esoteric units, such as those listed above.

Sometimes fermi questions simply ask for an obscure fact, such as the mass of a swallow and the mass of a coconut. As it is impossible to know everything, just guess. Answers are in powers of ten, and therefore, they can be off by a lot and still be right.

Other Tips

• You can be off by as much as 250% and still get points. Being off by 50% will still give full points for that problem.
• Guess away. There are no penalty for being wrong. Just don't guess blindly. Don't say 3 elephants weigh as much as the titanic.
• When you round something down, round something else up (or down if you're dividing).
• These are exponents. E3 plus +E7 is +E7 (this is magnitude notation)
• The competition is timed. Don't dwell on a single problem unless it's your last.
• Keep your calculations neat and simple. Often times, space is limited.
• Use common sense.