## The Riddle Bridge

Taking that internship in a remote mountain lab might not have been the best idea. Pulling that lever with the skull symbol just to see what it did probably wasn’t so smart either. But now is not the time for regrets because you need to get away from these mutant zombies…fast. Can you use math to get you and your friends over the bridge before the zombies arrive? Alex Gendler shows how.

## The Burning Island Puzzle

A man is stranded on an island covered in forest.

One day, when the wind is blowing from the west, lightning strikes the west end of the island and sets fire to the forest. The fire is very violent, burning everything in its path, and without intervention the fire will burn the whole island, killing the man in the process.

There are cliffs around the island, so he cannot jump off.

How can the man survive the fire? (There are no buckets or any other means to put out the fire).

## The Black And White Hats Problem

Cannibals ambush a safari in the jungle and capture three men. The cannibals give the men a single chance to escape uneaten.

The captives are lined up in order of height, and are tied to stakes. The man in the rear can see the backs of his two friends, the man in the middle can see the back of the man in front, and the man in front cannot see anyone. The cannibals show the men five hats. Three of the hats are black and two of the hats are white.

Blindfolds are then placed over each man’s eyes and a hat is placed on each man’s head. The two hats left over are hidden. The blindfolds are then removed and it is said to the men that if one of them can guess what color hat he is wearing they can all leave unharmed.

The man in the rear who can see both of his friends’ hats but not his own says, “I don’t know”. The middle man who can see the hat of the man in front, but not his own says, “I don’t know”. The front man who cannot see ANYBODY’S hat says, “I know!”

How did he know the color of his hat and what color was it?

## The 5 Pirates Puzzle

5 pirates of different ages have a treasure of 100 gold coins.

On their ship, they decide to split the coins using this scheme:

The oldest pirate proposes how to share the coins, and ALL pirates (including the oldest) vote for or against it.

If 50% or more of the pirates vote for it, then the coins will be shared that way. Otherwise, the pirate proposing the scheme will be thrown overboard, and the process is repeated with the pirates that remain.

As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard.

Assuming that all 5 pirates are intelligent, rational, greedy, and do not wish to die, (and are rather good at math for pirates) what will happen?

## Redesigned SAT: What You Need To Know In March of 2014, College Board President, David Coleman, announced plans for an overhaul of the SAT. The new SAT, which will be officially introduced this coming March, aims to realign the test with the work done in high schools and more accurately assess what student should be learning before college.

Much of the discussion surrounding the new SAT has centered on how the changes will impact colleges, high schools, and test prep providers. But what the students? What type of impact will the new exam have on test takers? Here is where future SAT test-takers should focus:

1) Pick Up A Book

The College Board has decided to eliminate the abstruse and outdated language from the SAT. They test will no feature words that students are more likely to encounter in college and the workplace. Many words still look familiar, but their meaning will depend on context, and answering questions correctly will require strong skills in reading comprehension and interpretation.

The new SAT will contain a large number of passages from the Declaration of Independence, the Federalist Papers, and other Founding Documents. Therefore, taking advanced coursework in U.S. history could really help improve your standardized test score. Consider enrolling in an honors or AP U.S. History course. The skills you learn in these classes may help you negotiate the exam’s new writing section, which requires that students analyze a passage and explain how its author uses evidence and reason to build an argument.

3) Statistics

The redesigned math section has a primary focus on quantitative reasoning and real world applications. Test takers will be asked to analyze rates, interpret graphs, synthesize mathematical information, and make sense of patterns within a particular dataset. All of these skills are considered to be essential in today’s big data environment and should be developed during a statistics course.

4) Sample Free Test-Prep

The College Board has partnered with Khan Academy, a leading online learning platform, to provide free test preparation services. This is a great way to get a better idea of how the test will actually read. However, you still may benefit from fee-based options for some more in-depth training as to what will be on the test.

## What It Takes To Be A Successful Tutor Effective tutoring is all about producing results. From a tutor’s perspective, it’s about witnessing your student achieve proposed academic goals and make way for more, in-depth learning. From a student’s point of view it’s about feeling confident with newly learned information and putting it to good use. As simple as this may all sound, effective learning is not something that happens overnight. Effective tutoring should be a direct reflection of a solid tutor-student relationship that’s been built over time, where both parties have a reached a point where they each feel fulfilled by their progress and excited for the academic challenges that are bound to come their way. Here are three key points for tutors to become more effective:

## Prepare

You can never be prepared enough, but there more you are the better. Because tutoring can take place outside of a formal educational environment, it is very important that you shape your own. Make sure to set a schedule with your students, preferably a consistent schedule ahead of time. Establishing a learning rhythm will be key for your students’ academic improvement. Also, try to carve out what each session will be about. Of course there will always be questions that may arise not pertaining to the day’s lesson, so it’s good to leave some extra room for that. But for the most part, focusing each session on either a specific chapter, theme, etc. will also help establish a beneficial learning rhythm for your students.

The best way to build a good rapport with your students is taking the time to know them. Every student is different. Figuring out your student’s personality will help you understand their learning process and how you should angle your sessions with them. For example, does your student have a sense of humor? Do they need your undivided attention? Are they shy? Does the student feel confident about the subject matter you’re going over? The more empathetic you can be, the more the student will feel comfortable around you and will allow himself or herself to continue to learn from you.

## Set Expectations

It is very important that both you and your student know your roles and responsibilities in a tutoring session. Tutoring is not the same as teaching. The tutor is not there to teach or re-teach something that the student has already learned. Instead, tutoring should be more of a well-balanced exchange of questions and information. In other words, tutors are there to guide the students to improve their studying skills, while the students are there to discover things through independent questioning and develop self-confidence.

## Peano’s Postulates

The Natural Numbers,  = {1, 2, 3, …}, can be built from five basic axioms, as follows:

1.  There is a natural number 1; that is, 1 ∈ .

2.  If n ∈ , then there is another number called the successor of n and written as S(n) that is also a natural number; that is:  n ∈   S(n) ∈ .

3.  Two different numbers have different successors; that is:  m  n  S(m)  S(n).

4.  Every natural number but 1 is the successor of some (other) number; that is:

n ∈ , n  1 m ∈  S(m) = n.

5.  The Principle of Mathematical Induction:  All natural numbers are formed in this way—either 1, or a successor.  More formally,

If there is a subset S of natural numbers such that 1 ∈ S and n ∈ S  S(n) ∈ S, then S is the set of Natural Numbers, that is S = .

Define addition:  n + 1 = S(n).

Can prove addition follows commutative and associative laws.

Define subtraction to be S(n) – 1 = n.

## Algebraic Structures — Field

Mathematics does not deal so much in numbers, but in “mathematical structures.”  These are basically abstract concepts that fit together according to certain rules or patterns.

One abstract concept that is very important is that called a Set.  A set in mathematics is a very abstract concept, but unless we are involved in higher mathematics, a set is something very similar to what we call a “collection” in everyday English.  Since we are not involved in higher math in this course, we will consider a set to be a collection of some objects (often numbers, but it could be a collection of other things, such as functions, triangles, etc.).  Many books will define a set to be a collection.  This definition is not correct, so as a matter of honesty, we point to that it is not correct.  However, the differences between a set and a collection are so subtle and abstract, we need get involved, but will consider a set to be a collection.

Another important concept is that of an operation—it means a process where some members of a set are combined together to get something new, like mixing flour and water to get dough.  In mathematics, in dealing with numbers, the most common operations are addition, subtraction, multiplication, and division.  There are others, of course, but we need not be concerned with them.  In dealing with functions, the most important operations are there are these four and a FIFTH operation that is important, called composition.  We studied it in class–f◦g means f(g(x)).

An operation is called binary, if, by definition we must always combine exactly two things together.  All the above operations are binary.  Note we cannot combine three numbers with addition together, for example, we cannot add 2+3+4 all at once.  We must first add only two numbers together, get a temporary answer, and then we add the third number to this temporary answer.  This means the operation is binary—we can only deal with two at a time, although we can keep adding more numbers to our answer after we get it.

An operation is called unique, if, unlike tossing a pair of dice, for example, combining the same two inputs always produces the same output.

1. The sum of any two real numbers a and b is a real number written a+b

This is called the Law of Closure for addition of Real numbers.

2. For all real numbers a, b, c:  (a+b)+c=a+(b+c)

This is called the Associative Law for addition of Real numbers.

3. There is a one unique real number called zero, written 0, such that a+0 = a and 0+a = a  for all real a.  (Note both parts are needed because Rule 5 does not necessarily hold in all systems.)   That is to say, that there is one and only one real number that can be used for I to make the following two equations come true:  For any Real number, a, a+I = a and I+a = a  for all real a.  (And of course, I is 0).

This is called the Additive Identity Law for addition of Real numbers.

4. For each real number a there is a real number called –a such that a+(-a)=0 and

(-a)+a = 0.  (Note both parts are needed because Rule 5 does not necessarily hold in all systems.)

This is called the Inverse Law for addition of Real numbers.

5. a+b=b+a

This is called the Commutative Law for the addition of Real numbers.

Rules for multiplication:

1. The product of any two real numbers is a real number written ab or ab

2. For all real numbers a, b, c:  (ab) c=a (bc)

3. There is a real number called one, written 1, such that a1=a and 1a=a for all real a.  (Note both parts are needed because Rule 5 does not necessarily hold in all systems).

4. For each real number a0,  there is a real number called 1/a such that
a (1/a)=1 and (1/a) a= 1.   1/a is also written as a-1.  (Note both parts are needed because Rule 5 does not necessarily hold in all systems.)

5. ab=ba

Rule 11 connecting addition and multiplication:

a(b+c)=ab+ac

Rule 1 is called the Law of Closure.

Rule 2 is called the Associative Law.

Rule 3 is called the Identity Law.

Rule 4 is called the Inverse Law.

Rule 5 is called the Commutative Law.

Rule 11 is called the Distributive Law.  (Adding columns and rows—switching order.)

Note:

1.  Rule 2 does not make sense unless Rule 1 is first established.

2.  Rule 4 does not make sense unless Rule 3 is first established.

3.  If a system has just one binary operation and the first four laws, it is called a “Group.”  Since Rule 5 is not necessarily true in an arbitrary group, it is necessary to state the identity and inverse laws on both sides.

4.  If a system has just one binary operation and the first five laws, it is called a Commutative Group or an Abelian Group.

5.  If a system has rule 2 and 5, then the General Associative Law is true, which says you can change the order and the parentheses any way you want.

6.  If a system has two binary operations and all 11 laws, then it is called a field.

You should be able to state these rules in the abstract—If S is a set, and if a, b, and c are any arbitrary members of S, and if * is a unique binary operation defined for the members of S, then (commutative law, for example, would be a*b = b*a.

Now some results or consequences of these rules:

1.  If a+c=a+b then b=c.  (This will be true in any group that has + as an operation.)

2.  If ab=ac, then b=c, providing a0  (This will be true in any group that has multiplication as an operation.)

3.  These laws enable us, if we are in a field,  to solve any equation of the following forms:

a+ x=b,

x+a=b,

ax=b (provided a0),

xa=b (provided a0), and

ax + b = c

4.  a0=0  for any Real number a.

5.  ab = 0 means a = 0 or b = o or both.

6.  The zero principle:  In a field, ab = 0if and only ifa = 0 or b = 0 or both.

This is a consequence of #4 and #5 above.  We needed all 11 laws (except for the commutative laws) to prove #3 and #4, so we see the zero principle will always hold in a field, but not necessarily in a group.

5 – 7

7.  (a)  b = (ab)  (Henceforth, we will omit the  for multiply, or use a dot instead.)

All the above represent the basic algebraic properties of a field.  There are many fields besides the real numbers.  Some are finite  and some are infinite, such as , , and –and there are many other examples of both finite and infinite fields not mentioned.  Any system that is a group will look a lot like our number system, , the real numbers.  However, any field will look even more like .

To characterize the Real Numbers mathematically, we need to say it is an infinite field with the following additional properties—these additional properties, however, are generally not considered “algebraic” properties, but “geometric” properties:

It is infinite, ordered, Archimedian, and complete.  These three properties are explained briefly below, but you are not responsible for them.

Ordered means given any two different numbers, then one is greater than the other (or equivalently, one is smaller than the other).

Archimedian means that given any small number, s, regardless of how small, and a big number, b, regardless of how big, then it is possible to add enough s’s together so that eventually the sum  s + s + … + s > b.  In other words there is a number n, so that ns > b.  For example, is s is 1/100 and b is 1,000,000 we can keep adding 1/100 + 1/100 etc. until eventually the sum is bigger than 1,000,000.  Equivalently, if n is any number larger than 100,000,000 then ns > b.

## Proof that Square Root of Two is Irrational

First, let us review some basic facts:

1.  N or the set of natural numbers, is the set (collection) {1, 2, 3, …}

2.  Every natural number > 1 can be decomposed into the product of (other) numbers, often smaller numbers, and often in several ways.  For example, 60 = 3·20 or 60 = 6·10.  These other numbers, 3 and 20 or 6 and 10, are called divisors or factors of 60.  P, or the prime numbers, are those natural numbers > 1 that have no divisors other than the number itself and 1, such as 2, 3, 5, and 7.  Other natural numbers greater than 1 (those that have other divisors) are called composite numbers.

3.  The Fundamental Theorem of Arithmetic says that every natural number greater than 1 can be expressed uniquely as a product of primes, up to order.

4.  One consequence of this fundamental theorem is that if there is some number, call it m, and we square it to get m2, then m2 will have exactly the same prime divisors that m has, only the number of occurrences of each prime divisor will be double that of m.  Hence, m will have an even number of prime divisors in total.

5.  The Whole Numbers, or W ,are the numbers in the set {0, 1, 2, 3, …}.

6.  The Integers, or Z, are plus or minus the whole numbers, namely, the set of numbers

{… -3, -2, -1, 0, 1, 2, 3,   }

7.  It is clear that there is a “lot of space” between integers on the number line.  Even though the next integer after 0 is 1, there is room for many other numbers between these two on the number line.  We call this fact “the integers are discrete”, meaning we could draw a small circle (say of radius ¼) around each integer and there would be no other integers inside that circle.

8.  The Rational Numbers, or Q, are all fractions or quotients of the form A/B where A and B are both integers and B is not equal to 0.

9.  Note that if B is plus or minus 1, then A/B is an integer, so we see that all integers are considered rational numbers.

10.  Unlike the integers, the rational numbers are very closely packed on the number line.  The word we give for this fact is dense.  If we draw a circle of very small radius around any fraction on the number line, regardless of how small we make the radius of the circle, there will still be another fraction inside that circle.  For example, if we draw a circle of radius ¼ around the fraction ½, then 5/8 is still inside this circle.  In fact there are many fractions inside this circle.  Even if we make the radius very small, say 1/10 or 1/100 then since ½ = 5/10 = 10/20 = 50/100 we can add one to the numerator and get 11/20 or 51/100 and this new fraction will be inside the circle.  So any circle centered around each and every fraction will contain another fraction or rational number.  There is no “next” rational number.  The next integer after 0 is one, but what is the next fraction after ½?  11/20, 51/100, 251/500?  There is no next fraction, because they are dense.

11.  In spite of the fact that the fractions or rational numbers are dense on the number line, still there is room for more numbers.  There are numbers on the number line that are not fractions, for example, the square root of two or pi.  (For convenience we will write the square root of two as (sr2) in the future.)  All numbers on the number line are called real numbers.  Those numbers on the number line that are not fractions are called irrational numbers.  The Greeks called them irrational because they knew they were there but they could not see how they could fit in since the fractions were dense, so they called them irrational.

12.  We proved in class that if any fraction will be changed to a decimal then the decimal will either terminate, or, if it goes on forever, it will be an infinite repeating fraction.  We also showed the converse, namely that if we start with an infinite repeating decimal, we can change it back to a fraction.  These two facts together mean that the set of fractions or rational numbers is exactly the same as the set of decimals that either terminate or repeat, whereas the set of irrational numbers are those decimals that are infinite but they do not repeat.

This is the end of the review.  Now, we wish to prove that (sr2) is one of the irrationals.  To do we take several steps.  The first step is that we use an indirect proof.  We assume there are integers A and B such that (sr2) = A/B.  We then plan to show that this assumption leads to a contraction.  In mathematics, when an assumption leads to a contradiction, we know it is not true.  Therefore, if we get a contradiction, then our starting assumption that (sr2) = A/B will be false–that is, the square root of 2 cannot be expressed as a fraction or rational number; hence, it must be an irrational number, an infinite, non-repeating decimal.  This method of proof—where we assume the opposite of what we want to prove and show that this assumption results in a contradiction—this method, is called indirect proof.

So the first step is we assume that (sr2) = A/B

Second step:  Multiply both sides of the equation by the same number, B.  We can always multiply both sides of an equation by the same number, and we do so here to get rid of the fraction and use just whole numbers instead.  If we multiply both sides by B, we obtain:

B·(sr2) = B·(A/B)

Since the B’s on the Right-Hand Side (henceforth RHS) will cancel out, this gives us:

B(sr2) = A.

Third step:  Whenever we have an equation, we can square both sides.  For example, if x = y, then x2 = y2.  In general (LHS) (LHS) = (RHS) (RHS).  Let us do that here to get rid of the square root and have only whole numbers:

B(sr2)B(sr2)= A A.

Fourth step:  We can rearrange the terms on the left to be (sr2)(sr2)BB.  Since, by definition, the square root of two meets the condition (sr2)(sr2) = 2 and since BB = B2 (and since AA = A2), we can rewrite this equation to be:

2B2 = A2

Fifth step:  Let us use the Fundamental Theorem of Arithmetic, (review facts #3  and #4) and factor this number.  We can factor it by looking at either the right-hand side or the left-hand side.  The Fundamental Theorem says we must get the same result, but in fact, we will not.  Notice that the right-hand side must contain an even number of primes.  On the left side, we factor the number as 2 times B2, and the B2  contains an even number of prime factors.  But there is also the number 2 on the LHS, so the entire left-hand side has one more prime, namely two, for a total of however many primes there are inside of B2 (which is an even number) plus one more prime (for the number 2) giving a total of an odd number of primes.  Thus, we have shown the RHS has an even number of primes, and the LHS contains an odd number of primes.  This fact contradicts the Fundamental Theorem that every number can be factored into primes only one way.  Conclusion:  No such numbers A and B exist, that is, (sr2) must be irrational.

Note, a similar proof will show that (sr5) or any other square root that is not a whole number will be irrational.

What is Russell’s Paradox?  What is the solution to it?

Background (you will need to know to understand the answer):

Russell’s Paradox arises when sets get “too big” and sets contain sets.  It is possible for sets to contain sets, but not all such combinations are acceptable.  We will now explain how the idea of a “set containing a set” can be problematic.

To start, first notice that many sets, even if they are infinite in size, can generally be expressed in English sentences containing a few words.  For example, “The set of all even numbers” is a sentence describing an infinite set—the even numbers—and the description should be very clear to anyone, even someone in the second grade.  Yet this description only takes six words!  (Side point:  In general mathematicians like to be very terse, and the brevity of mathematical statements can confuse people who are not used to paying attention to the detail and exact meaning of every word—an absolute necessity in mathematics.)  In spite of the tendency towards terseness, there are some sets that might take many words to describe, for example, consider this set:  “Let K = the set of all people who (1) were born in New York, (2) have red hair, (3) moved to California after age 10, (4) drive a Ford, (5) did not graduate college, (6) like to follow football games on television, (7) like Big Macs, (8) own a computer, etc. “  If we were to list 100 different conditions, even using terse wording, the description would certainly be over 100 words long.  Of course, it could be there is no person on earth who meets all 100 conditions, especially if some of them are rare, such as (over 6 feet 5 finches tall).  While that one condition might be met by some people, the number of people meeting this condition might be relatively small, so it could be there is no one who meets all 100 conditions.  If this is the case, then K, the set described is the empty set, but we would not know that without a survey unless we created a ridiculous condition, such as the person must be over 10 feet tall.  In short, this set could very well not be the empty set, but the description could be very long—over 100 words.

Now let H be the set of all sets that can be described in an English sentence containing less than 100 words.  By definition, the elements of H are themselves sets.  Clearly, the set K described above is not inside of H, because its description is too long.  However, all other sets mentioned above are sets that are inside of H—as well as many other sets, such as the set of real numbers, the set of rational numbers, the set of all people on earth, etc.  These sets are infinite, but their descriptions are short, so they are all inside of H.  In fact, the set H is also inside of H because the description of H above contained 17 words (which is certainly less than 100).  Thus we have something like the following H ={H, N, Z, Q, S, T, ….}  H is in fact a member of H!  This concept is counter-intuitive, because if we think a set as a collection, a collection cannot obviously contain itself.  So while it is nice to think of a set as a collection, the analogy breaks down when we deal with more complicated examples, as is so in this case.

The Answer as it Should Appear on a Test:

Now, here is Russell’s Paradox.  All the sets we have above, or “standard” sets, such as whole numbers, natural numbers, integers, etc. we will call “ordinary” sets.  However, any set, such as H, which contains itself, we will call an “extra-ordinary” set.

Let Obe the set of all ordinary sets.  Question:  Is O  ordinary or extra-ordinary?  We will find we have a problem (contradiction) or paradox, either way we look at it.  If O  is ordinary, then since O  contains all ordinary sets (by its very definition) then it must contain itself (since it is ordinary)—and by containing itself it would then become extra-ordinary—a contradiction!  Again, if O is extra-ordinary, then O  must contain itself, by the definition of the word “extra-ordinary.”  But this fact contradicts the definition of O , since O is supposed to contain only ordinary sets, not an extra-ordinary ones!  So as a result, we cannot decide whether O is ordinary or extra-ordinary.

What is the solution to Russell’s Paradox?

The solution to this paradox, is to redefine or rethink what we mean by sets.  They are not simply any collections, but they must be limited so they do not grow too big.  There are several ways to achieve this re-definition—but most are too sophisticated to explain here.  We will explain, briefly (not in detail) the simplest method here.

We start by defining some set, which we call U, the universal set.  U could be the real numbers, natural numbers, or any other set you want it to be.  However, we cannot just create “new” sets at random.  They must be formed by certain rules—for example, by taking elements from U, by taking subsets from U, or by taking unions or intersections, or complements of these sets.  So while we can make an infinite number of sets, and we can even create a set of subsets, we can’t create “higher levels” of sets containing sets, so we stay clear of this paradox.

Questions:  Let S be the set of all sets.  Does S exist?

Answer:  No.  It is too big.  The solution to Russell’s Paradox sets limits as to how big a set can get—and S violates those limits.  We need to have some “upper” set such as the real numbers, and all other sets are subsets of this upper set or universal set.