When you give $\unicode{x00a9}$ two numbers, it gives you the third number in the addition/subtraction pattern. For example, $\unicode{x00a9}\left\{ {21,23} \right\} = 25$ (going up by $2$’s), and $©\left\{ {95,90} \right\} = 85$ (going down by $5$’s).

Find $\unicode{x00a9} \left\{ {6,11} \right\}$ and $\unicode{x00a9} \left\{ {11,6} \right\}$.

Find $\unicode{x00a9}\left\{ {11,\unicode{x00a9}\left\{ {6,11} \right\}} \right\}$.

Write an equation for $\unicode{x00a9}\left\{ {m,n} \right\}$.

When you give $\unicode{x24}$ two numbers, it gives you the third number in the multiplication/ division pattern. For example, $\unicode{x24}\{3,15\} = 75$, and $\unicode{x24}\{48,24\} = 12.$

Find $\unicode{x24}\{2,3\}$, $\unicode{x24}\{12,13\}$, and $\unicode{x24}\{102,103\}.$

What is $\unicode{x24}\{2009,1\}$?

If $\unicode{x24}\{x,2\} = \unicode{x24}\{3,1\}$, then what is $x$?

The symbol $¤$ takes a single number, squares it, and then subtracts 4.

What is $¤(6)?$

Can you ever get a negative answer for $¤(x)?$ Why or why not?

Find an $x$ so that $¤(x)$ is divisible by 5.

Let the symbol $\unicode{xa5}$ mean: Add up the two numbers, then take that answer and subtract it from the product of the two numbers. What is $5\unicode{xa5}8$? $8 \unicode{xa5} 5$?

Look back at the problem above. Do you think the same thing would happen for any pair of numbers, if you used the same symbol? Explain your answer.

Fergie claims that each of the following symbols has the commutative property. Examine each of his claims.

$x \Downarrow y$ means add 1 to $y$, multiply that answer by $x$, and then subtract $x$.

Take two whole numbers $x$ and $y$. To do $x \% y$, you divide $x$ by 2, round down if it’s not a whole number, and then multiply by $y$.

To calculate $x\unicode{xa3} y$, imagine that you walk $x$ miles east and then $y$ miles northeast. $x\unicode{xa3}y$ is how far away you end up from your starting point.

$\unicode{xbf}$ works by adding up the two numbers, multiplying that by the first number, and then adding the square of the second number.

$\unicode{xa2}$ takes two numbers. You reverse the first number (for instance, 513 becomes 315); one-digit numbers stay the same), then add the reversed number to the second number, and finally add up the digits of your answer.

Let $x \unicode{x2605} y = x^2-y^2$. For example, $4 \unicode{x2605} 3 = 16-9 = 7$. True or false: $x \unicode{x2605} y$ always equals the sum of the two numbers — for example, $4 \unicode{x2605} 3 = 7$ which equals $4 + 3$. If it’s true, justify your claim. If it’s false, try to find out what kinds of numbers do make the claim work.

When you give $\heartsuit$ two numbers, it finds the sum of the two numbers, then multiplies the result by the first number. Finally it subtracts the square of the first number. Flinch claims that $\heartsuit$ is commutative. Is he correct?

Here’s how you might prove Flinch’s claim in problem 19 for $any$ two starting numbers.

Let’s call your first number $m$ and your second number $n$. Write down and simplify as much as you can the expression for $m \heartsuit n$.

Write down and simplify as much as you can the expression for $n \heartsuit m$.

Is the rule below commutative?

The rule $\unicode{x223c}$ adds up the two numbers, doubles the answer, multiplies the answer by the first number, then adds the square of the second number and subtracts the square of the first number.

Let $x \clubsuit y = \frac{1}{{1,000,000,000}} \thinspace {x^y}$.

Is there a value of $y$ such that $10 \clubsuit y > 1$?

Is there a value of $y$ such that $1.001 \clubsuit y > 1?$

The symbol “&”, applied to $any$ two numbers (not only whole numbers), means that you take the first number and then add the product of the numbers.

When you calculate $x$&$y$, you get 120. What could $x$ and $y$ be? Give several different answers.

When you calculate $x$&$y$, you get 120. Write an equation that expresses this fact, then solve for $y$ in terms of $x$ (meaning, write an equation $y = $…. with only $x$’s in the equation).

The command “CircleArea” is a rule that finds the area of the circle with the given radius. For example, ${\rm{CircleArea(3)}} = 9\pi $.

Find CircleArea(4).

Write the equation for CircleArea($x$).

Can you find CircleArea(-4)? Why or why not?

It’s January 1st, and you are counting the days until your birthday. Let $m$ be the month (as a number between 1 and 12) and $d$ the day (between 1 and 31) of your birthday, and pretend that there are exactly 31 days in each month of the year.

How many days are there until January 25? Until April 10?

For January 25th, $m = 1$ and $d = 25$, and for April 10th, $m = 4$ and $d = 10$. By looking at what you did in part a, explain how you can use the numbers $m$ and $d$ to count the days from January first to until any day of the year.

Let the symbol $♥$ represent this count. Write an algebraic rule for calculating $m♥d.$ Test your rule with an example.

Now, count how many days are from your birthday to New Year’s Eve (December 31st). Represent this with the symbol $\unicode{x2648}$. Again, pretend there are 31 days in each month.

Write an equation for in terms of $m$ and $d$, and use an example to show that your equation works. (You might want to try explaining it in words or in an example first.)

“max($a$,$b$)” takes any two numbers and gives you the larger of the two. The symbol $\partial $ is defined by $\partial (a,b)$= max($a$,$b$) – min($a$,$b$). Is $\partial $ commutative?

We say that the counting numbers (i.e., 1, 2, 3, ...) are “closed under addition” because any time you add two counting numbers, you get another counting number. Decide whether or not the counting numbers are closed under each of the following operations. In each case where the answer is no, try to find a group of numbers that $is$ closed under that operation.

* (multiplication)

- (subtraction)

/ (division)

The symbol ∂, from problem 27

The rule $\unicode{x2190}$ adds twelve to a number and divides the sum by four. What number $x$ can you input into $\unicode{x2190}(x)$ to get an answer of 5? An answer of -3?

To do the rule $\forall$, add 5 to the first number and add 1 to the second number, then multiply those two answers.

What’s $3 \forall 1$?

What’s $x \forall y$?

Your friend tells you that she needs to find numbers $x$ and $y$ so that $x \forall y$ gets her an answer of $A$— a whole number that she does not reveal. In terms of $A$, tell her what to plug in for $x$ and $y$. Make sure that your strategy would always get her the answer she wants.

What values of $x$ and $y$ give you an odd answer? Prove that your description is true and complete. (Make sure you know what it means to prove it’s complete!)

What values of $x$ and $y$ would give you an answer of zero?

Create 3 different rules that give an answer of 21 when you plug in 2 and 8.

Let $f$ be a rule that acts on a single number.

Create a rule for $f$ so that $f(2)=12$ and $f(11)=75$.

Now, create a new rule $g$ such that $g(5)=26$ and $g(10)=46$.

Let $x \triangle y = x^2+2xy$. ($x$ and $y$ have to be integers.)

If you plug in 6 for $x$, find a number you could plug in for $y$ to get an answer of zero.

Using part a as an example, describe a general strategy for choosing $x$ and $y$ to get an answer of zero, without making $x$ zero. Explain why your strategy works.

Suppose you use the same number for $x$ and $y$— call this number $N$. (So, you’re doing $N \triangle N$.) What is your answer, in terms of $N$? Simplify as much as possible.

Suppose $x \triangle y = 20$. Solve for $y$ in terms of $x$.

Describe a strategy to get any odd number that you want. Give an example, and also show why your strategy will always work (either give a thorough explanation, or use algebra to prove that it works).

Tinker to find a rule for $x \# y$ that gets the following answer: $5 \# 1 = 24$. Then try to write a rule that gives $5 \# 1 = 24$ and $4 \# 2 = 14$.

Create a rule $\alpha$ that works with two numbers, so that $\alpha(1,1) = 5$ and $\alpha(2,3) = 10$. Give your answer as an equation in terms of $a$ and $b$. $\alpha (a,b) = $ __________________________.

Let $\otimes \{a,b\}$ be the two digit number where the tens digit equals $a$ and the units digit equals $b$. For example, $\otimes \{9,3\} = 93$.

In terms of $f$, what do you get when you do $\otimes \{5,f\}$? Write your answer as an equation. (Remember, writing $5f$ doesn’t work, because it means $5 \cdot f$).

What do you get when you do $\otimes \{a,4\}$?

When you calculate $\otimes \{a,4\} - \otimes \{4,a\}$, what do you get in terms of $a$? Simplify as much as possible.

$\otimes \{6,x\} - \otimes \{x,3\} = 12$. Find $x$. Show your work algebraically.

Now, let’s redefine $\otimes \{a,b\}$. Let $\otimes \{a,b\}$ be the three-digit number where the hundreds and units digits are $a$, and the tens digit is $b.$

In terms of $b$, what is $\otimes \{b,2\}?$ (Again, $b2b$ won’t work).

In terms of $a$ and $b$, what do you get when you calculate $\otimes \{a,b\} - \otimes \{b,a\}$? Simplify as much as possible and check your answer with an example.

For any two positive numbers $a$ and $b$, let $a \bot b$ be the perimeter of the rectangle with length $a$ and width $b$.

Is $\bot$ commutative?

Is $\bot$ associative? In other words, does $(a\bot b) \bot c = a\bot(b\bot c)?$

A positive whole number is called a “staircase number” if the digits of the number go up from left to right. For example, 1389 works but not 1549.

The rule $x \nabla y$ works by gluing $x$ and $y$ together. For example, $63 \nabla 998$ gives an answer of 63998. If $x$ and $y$ are both staircase numbers, and $x$ is bigger than $y$, is it always true that $x \nabla y$ is bigger than $y \nabla x$ ? If you think it’s always true, explain why. If not, explain what kind of input would make it true.

Suppose that, for any two positive numbers $a$ and $b$, $a$■$b$ represents the area (ignoring units) of the rectangle with length $a$ and width $b$.

Is ■ associative?

In problem 23, you were asked to find some pairs of numbers $x$ and $y$ so that $x$&$y = 120$. (Recall that $x$&$y$ takes the first number and adds the product of the numbers).

Do you think you could find numbers $x$ and $y$ to get any number you wanted? Try a few possibilities and explain what you find.

Develop a rule for choosing $x$ and $y$ to get the number that you want, which we will call $N$. (Hint: Try starting out by picking a number for $x$, and then figuring out what $y$ would have to be. You might have to try different numbers for $x$ to find a solid strategy.)

Don’t use a calculator for this problem.

Find $\frac{3}{4} + \frac{5}{6}$

Find $2 \div \frac{1}{2}$

Simplify $3x - 2(x + 1)$

Write $.\overline 6 $ as a fraction.

Find $2 \frac{1}{4} +5 \frac{7}{8}$

Let $x \square y = xy - x - y$. Develop a strategy to pick $x$ and $y$ so that you can get any number you want.

For a number $x$, $f(x)$ subtracts twice $x$ from 100.

What is $f$(10)?

What is $f$(-4)?

Write an expression for $f(x)$.

Write an expression for $f$(3$x$).

Let Let $x \square y = xy - x - y$. Prove or disprove: if $x$ and $y$ are both larger than $2$, then $x \square y$ gives a positive answer. Make sure your explanation is thorough.

Write an equation for a rule $a \lozenge b$ , so that the answer is odd only when both $a$ and $b$ are even.

Let $a{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{\% }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} b = {a^2}b - {b^2}a$. What would have to be true about $a$ and $b$ for $a{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{\% }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} b$ to be positive?

Let $a \ ☺ \ b = ab + b^2$. What would have to be true about $a$ and $b$ for $a \ ☺ \ b$ to be negative?

Let $x \ominus y = xy + y/2$ , where $x$ and $y$ are whole numbers.

Describe what values of $x$ and $y$ would get you a negative answer.

Describe what values of $x$ and $y$ would get you an even whole number answer. (Careful! Your answer won’t be just in terms of odds and evens. You’ll have to tinker to see what type of numbers work. Think through it step by step and explain your reasoning).

What could $x$ and $y$ be to get 21? Give at least 3 different options.

Find a strategy to get any whole number you want, called $N$. You can explain your strategy in algebraic terms (“To get an answer of $N$ let $x$ equal … and let $y$ equal …”) or in words. Show an example to demonstrate that your strategy works.

For a number $x$ that might not be a whole number, $\Phi \left( x \right)$ represents the integral part of $x$— for example $\Phi \left( {6.51} \right)$ is 6 and $\Phi \left( {10} \right)$ is 10. Using the $\Phi $ notation, write a rule @ that gives an answer of 0 if $x$ is a whole number, and gives a non-zero answer if $x$ is not a whole number. For example @(20) should equal 0 and @(20.3) should give an answer not equal to zero.

The rule $\lfloor b \rfloor$ is called the “greatest integer function” — it outputs the largest integer that is not above $b$. For example, $\lfloor 9.21 \rfloor = 9$, $\lfloor {12} \rfloor = 12$, $\lfloor .92 \rfloor = 0$, and so on.

Write an equation for the rule $\left\{ b \right\}$, which rounds $b$ down to the hundreds — for example $\left\{ {302} \right\} = 300$, $\left\{ {599} \right\} = 500$, and $\left\{ 2 \right\} = 0$. For your equation to calculate $\left\{ b \right\}$, you can use any standard operations, and you should use the operation $\lfloor b \rfloor$.

To understand the rule behind the symbol $\diamondsuit $, you need to draw a picture (graph paper will help). To draw the picture for $5\diamondsuit 3$, pick a point to start and then draw a line stretching 5 units to the right. From there, draw a line stretching 3 units up. Then draw a line stretching 3 units to the right, and then a line stretching 5 units up. Finally, draw a line back to your starting point.

Let $5\diamondsuit 3$ be the area of the shape you just drew. Calculate this number exactly.

Pick 2 new numbers (the 1st number should still be bigger). Draw the picture and calculate the answer that $\diamondsuit $ would give for your numbers.

Draw a diagram to help you find an algebraic rule for $x {\kern 1pt} \diamondsuit {\kern 1pt} y$ (again, you can assume that $x$ is a bigger number than $y$). Check your rule by making sure it would give you the right answer for $5\diamondsuit 3$ and for your numbers in part b.

Will your rule still work if the two numbers are the same, such as $4\diamondsuit 4$? Explain why it will or will not always work.

Will your rule still work if the first number is smaller, such as $3\diamondsuit 7$? Explain why it will or will not always work.

(From Finkbeiner and Lindstrom) The Board of Directors of XYZ Corporation has 16 members and 6 committees of 5 persons each as indicated by the following chart.

How many distinct meeting times are needed to conduct the committee meetings of the board?

Draw a graph in which the total degree — that is, the sum of the degrees of each node — is odd.

The graph we used to represent bordering countries in problem 4 was planar. Create a map of your own countries for which the corresponding graph is nonplanar.

A graph is complete when every node is connected to every other node by an edge. Here’s a complete graph with four nodes:

Draw:

A complete graph with five nodes.

A complete graph with six nodes.

A complete graph with three nodes.

How many edges will a complete graph with each of the following number of nodes have?

Five nodes

Six nodes

Three nodes

How would you calculate the number of edges of a complete graph with…

A hundred nodes?

n nodes?

Now that you know how the number of nodes and edges in a complete graph are related, investigate this relationship further.

Are the number of nodes and the number of edges in a complete graph related directly? That is, if you double the number of nodes, does the number of edges double, too?

Are the number of nodes and edges in a complete graph related linearly? Or does the number of edges go up “faster” or “slower” than in a linear relationship?

What’s the minimum number of nodes needed for a graph with 1000 edges, if at most one edge can connect two nodes, and no edge can connect a node to itself?

If a complete graph has 500 nodes, can you estimate the number of edges quickly? Is it more likely to be roughly 12500, 125000, or 1250000 edges?

In the complete graph with four nodes, pictured earlier, the graph is drawn so that some of the edges crossed. However, just because it is drawn that way in that particular diagram does not mean the graph is not planar.

Is the complete graph with four nodes planar?

Is the complete graph with five nodes planar?

How about complete graphs with more than five nodes?

The symbol “Deg,” as in the degree of a node, represents a function. What kinds of objects does Deg take as input? What kinds of objects does it output? By the way, the set of allowable inputs to a function is called its domain, and the set of allowable outputs is called its range.

In this “Clue” game board, draw a graph in which you show which rooms can be reached from the others by the roll of a 6 on a die. Include the secret passages that you can take no matter what you roll from the Lounge to the Conservatory and from the Kitchen to the Study. If you roll a number higher than the number of spaces you need, you can still go in the room. By the way, is the Clue graph planar?

A group of six sophomores is going down to the Lower School to read to a group of six first-graders. The teachers asked for a list of preferences, both from the first-graders and the sophomores, as to who would read to whom.

Is there a way to form partners that will make everybody happy?

(From Epp) A traveler in Europe wants to visit each of six cities shown on this map exactly once, starting and ending in Brussels.
The distance (in kilometers) between each pair of cities is given in the table. Suggest a sequence of cities she can visit that minimizes the distance traveled.

Wiki-Racing! The game of wiki-racing involves starting on a randomly generated Wikipedia page and taking Wikipedia links to navigate through until you reach a designated ending page. For example, two players on two different computers might agree to end on the page for “Fruit Bat.” The first player to arrive at the Fruit Bat Wikipedia page using only the links on Wikipedia pages would win.

Below is a list of webpages in our miniature version of wiki-racing. Each webpage links to some, but not necessarily all, of the others.

How could you represent this situation using a graph?

What is the fewest number of clicks it will take to go from the Toy Story page to the Toy Story 3 page? From the Toy Story page to the Buzz Lightyear page?

Oh, no — your parents just announced that they’re dropping by your apartment for a surprise visit. But your place is a mess. Fortunately, you’ve got some elves — as many as you need — to help you clean the place, but even an elf can’t wash lights with darks. Here’s what you and the elves have got to do, and the time it takes to do each thing:

• Wash your load of light-colored laundry
  (30 mins)
• Wash your load of dark-colored laundry
  (30 mins, and remember you’ve only got one washer)
• Dry your lights (45 mins, and you have to wash the lights first)
• Dry your darks (45 mins, you have to wash the darks first, and you’ve only got one dryer)
• Scrape off the dishes in your sink (10 mins)
• Load and run the dishwasher
  (60 mins, and you have to scrape first)
• Pick your stuff up off of the floor
  (20 minutes)
• Vacuum (10 minutes, and you have to pick up your stuff first)

Find a good graph-theory way to represent this problem. How long will it take you to put your apartment in order?

What is the minimum number of elves you need to do everything in that amount of time?

You have a ${3 \times 3 }$ chessboard. The black knights are in the top two corners, and the white knights are in the bottom two corners. No other pieces are on the board. (Recall that the only moves knights are allowed to make is “2 up or down, 1 left or right,” or “1 up or down, 2 left or right,” but that they can “jump” pieces.)

How many moves will it take to switch the position of the knights, so that the black knights are now in the bottom two corners and the white knights are in the top two corners?

How many moves will it take just to switch the two rightmost knights?

Below is a map of the town of Königsberg, Prussia, as it existed in the 18th Century. A popular puzzle at the time was to cross all seven bridges without ever going back over a bridge you’ve crossed, and wind up at your starting point. Solve the puzzle yourself. (The mathematician Leonhard Euler invented graph theory by analyzing this problem.)

Image from solipsys.com.uk/new/KoenigsbergImages.html

For each figure, say whether you can trace it with your pencil without going over any segment twice, ending at your starting point.

A graph that can be traced in the manner of the previous problem is said to have an Euler Circuit, named after the mathematician who invented graph theory. Propose a method for determining if a given graph has an Euler circuit or not. (Are any of the concepts you’ve learned in this lesson relevant?) Make up new examples to test, if necessary.

Which complete graphs have Euler circuits?

The Irish mathematician William Rowan Hamilton invented a game in the 1850s called the Icosian Game. In it, you had to visit a list of cities connected by roads, as represented by the graph below.

(image from mathworld.wolfram.com/ IcosianGame.html)

The challenge was to start in one city and follow the roads to visit all the cities, winding up at your starting city without ever having visited a city twice. Solve the Icosian Game.

If you read the previous problem carefully, you won’t be surprised to learn that a circuit of the type you found — one where you need to visit each node exactly once and return to your starting node — is called a Hamiltonian circuit. Find Hamiltonian circuits for each graph below.

Do you suppose that all graphs have
Hamiltonian circuits?

Don’t use a calculator for this problem.

Factor ${x^2} + 8x + 16$

Factor ${x^2} - 16$

Solve for $x$: ${x^2} - 5x + 6 = 0$

Subtract $4 \frac{7}{5} - 2 \frac{1}{3}$

If $x^2+y^2=30$ and ${(x + y)^2} = 484$ , what is $xy$ ?

The diagram below shows a blueprint of three different utility companies and three houses. You want to add to the blueprint lines running from each utility company to each house, but the lines cannot cross one another. Suggest a way to do this or argue that it is impossible.

Draw a complete bipartite graph between two groups of two nodes. This graph is abbreviated $K_{2,2}$. Now draw $K_{2,4}$. Using this notation, what graph were you working with in Problem 43?

How many colors are needed to color the graph $K_{2,4}$? How about $K_{m,n}$?

Find a formula for the total degree
of $K_{m,n}$.

In the three-utilities puzzle (see problem 43), suppose now that lines are allowed to cross, and so the lines are actually run that way. If you accidentally cut one of the pipes without knowing which one it is, what’s the probability that the gas in the leftmost house will go out? How about if there are $m$ houses and $n$ utilities, instead?

Any graph that could represent “who beat whom” in a situation where each team plays every other team once is called a tournament. Which of the graphs below are tournaments?

The Game of Sprouts: Mark a few dots on a piece of paper. A move consists of two actions: connect two different dots with a single edge then place a dot at around the middle of this edge. You can only connect two dots if each has a degree of less than 3, and you may not repeat an edge or cross edges. The person who has no move is the loser.

If you start with only two dots, then what is the maximum number of moves the game can last?

At the end of the two-dot sprouts game, what will be the total degree of the diagram?

At the end of the $n$-dot sprouts game, what will be the total degree of the diagram?

How many moves will the $n$-dot sprouts game take?

The following is a floor plan of a house. Is it possible to enter the house in room A, travel through every interior doorway of the house exactly once, and exit out of room E?

Earlier we learned that if two graphs are isomorphic, each node has a "partner" of the same degree. Is it also true that when nodes can be matched into partners having the same degree, the two graphs must be isomorphic?

Propose a criterion by which you’d be able to tell if a graph has an Euler path. Does it matter which nodes are your starting and ending points? Remember to examine a similar problem by making up specific cases for yourself.

In problem 32 and the tournament
problem (see problem 48), you drew a directed graph — a graph where the edges are one-way. The question of finding Euler circuits is more complicated on directed graphs. Do some experiments and propose criteria for whether a directed graph has an Euler circuit.

Does every tournament have a Hamiltonian path? What does it mean for the team rankings if there is more than one path through a tournament?

Come up with your own system to determine the winner of a tournament. It may help to create some of your own examples of tournaments.

The following graph represents streets and corners; the portion of a street between two corners is a “block”. The neighborhood association decides to place streetlights at some of the various corners so that the neighborhood never gets too dark at night. They want there to be enough streetlights so that every home is on a block that has a streetlight on at least one of its corners. Assume that there are homes between each pair of adjacent corners. Find the smallest number of street lights needed.

Use a graph to represent and solve the following maze.

The “doughnut” shape below is called a torus. You know you can’t draw a complete graph with five nodes on a flat surface. But can you draw it on the surface of a torus?

If the blueprint for the houses and utilities (see problem 43) were printed on paper shaped like a torus, could you run the wiring with no overlaps?

How many prime numbers under 10000 have digits that add up to 9?

If the sum of two prime numbers is 999, what is their product?
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Explain why the product of any three consecutive integers has to be divisible by 6.

The dimensions of a rectangular box (in cm) are all positive integers, and the volume of the box is 2002 cm$^3$. What is the least possible sum of the three dimensions?

(Calculators allowed.) What is the smallest positive integer that is NOT a factor of $1 \times 2 \times 3 \times 4 \times 5 \times 6 \times ... \times 18 \times 19 \times 20?$
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Does ${6^6} + {6^6} + {6^6} + {6^6} + {6^6} + {6^6}$ equal ${36^6}$? ${6^{36}}$? ${6^7}$? ${36^{36}}$? Or something else in the form ${a^b}$? Explain.
(Appeared on AMC-12 competition, 1992)

Express $\frac{{{2^1} + {2^0} + {2^{ - 1}}}}{{{2^{ - 2}} + {2^{ - 3}} + {2^{ - 4}}}}$ as a single, simple fraction. (Appeared on AMC-12 competition, 1987)

When 270 is divided by the odd number X, the answer is a prime number. What is X?
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What is the simplest expression for $\frac{{{2^{40}}}}{{{4^{20}}}}$?
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What is the simplified value of $\frac{{{{4444}^4}}}{{{{2222}^4}}}$?
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If you double $x$, by what factor does the expression $\frac{{{x^7}{x^{ - 2}}}}{{{x^3}}}$ grow?

A composite number is a whole number that is greater than 1 and not prime — that is, it has factors other than itself and 1.

What is the smallest composite number that is not divisible by 2 or 3?

What is the smallest composite number that is not divisible by 2, 3, or 4?

What is the smallest composite number that is not divisible by 2, 3, 4, or 5?

(Calculators allowed.) Now determine the smallest composite number that is not divisible by 2, 3, 4, 5, 6, 7, or any of the numbers up to and including 100.

Order these, from smallest to largest, without using a calculator: ${1000^{\frac{1}{{1000}}}}$, $\big(\frac{1}{1000}\big)^{1000}$ , ${1000^{ - 1000}}$, $\big(\frac{1}{1000}\big)^{\frac{1}{1000}}$, $\big(\frac{1}{1000}\big)^{-1000}$

You are going to randomly choose three integers (repetition allowed) from 0 to 5, and call them $k$, $m$, and $n$. What’s the probability that ${2^k}{3^m}{5^n}$ is NOT divisible by 5?

What is the only number $x$ which satisfies $\sqrt {1992} = 1992\sqrt x $?
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In circle P, the length of $\overline {AB} $ is $\sqrt {50} $. Find the area of the shaded region.

Find EF exactly.

Triangle ABC has a right angle at C. If sin B $= \frac{2}{3}$ what is tan B, expressed as a fraction?

Why must $\sqrt {17} $ be a non-terminating decimal? Put another way, why can’t a terminating decimal multiplied by itself = 17?

What is the positive number $x$ for which $x = \sqrt[3]y$ and $\sqrt y = 8$?
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If ${x^{64}} = 64$, what is the exact value of ${x^{32}}$?
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In simplest form, what is the numerical value of $\sqrt {1985} \,\,\sqrt[3]{{1985}}\,\,\sqrt[6]{{1985}}$?
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Which kind of number does not appear to have a 10th root? Which kind of number has an integer as its 10th root?

If $\sqrt[5]{x} = 4$ , what is the value of $\sqrt x $?
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What does $\sqrt 7 \sqrt 3 \sqrt 2 \sqrt 5 \sqrt {10} \sqrt {21} $ equal? Think before calculating!!

Rewrite $\big(\frac{1}{4}\big)^{-\frac{1}{4}}$ as the root (i.e. square root, cube root, and so on—you choose !) of an integer that is NOT 4.

What is the value of $x$ which satisfies $\sqrt[3]{{x\sqrt x }} = 4$?
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If $x \ge 0$, then $\sqrt {x\sqrt {x\sqrt x } } $ is equivalent to ${x^N}$. What is $N$?

The function $s$ takes a number and outputs the sum of its “proper” divisors, meaning that 1 is considered to be a divisor but the number itself is not.

Find $s(8)$.

A perfect number is a number $n$ for which $s(n)=n$. How many perfect numbers are between 2 and 10?

An abundant number is a number n for which $s(n)>n$. Find the first abundant number.

A pair of numbers is said to be amicable when the proper divisors of each number add up to the other number.

Verify that 220 and 284 are an amicable pair.

Write the definition of amicable numbers using the $s(n)$ notation.

Which is bigger, ${2^{3000}}$ or ${3^{2000}}$? Prove your answer.
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Write $\frac{{{{15}^{30}}}}{{{{45}^{15}}}}$ in the form ${a^b}$, if possible. If it is not possible, explain why not.
(Appeared on AMC-12 competition, 1993)

If ${3^x} = 5$, what is the value of ${3^{\left( {2x + 3} \right)}}$?
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What is the integer $n$ for which ${5^n} + {5^n} + {5^n} + {5^n} + {5^n} = {5^{25}}$?
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Take any 6-digit number that repeats 3 digits twice in the same order, like 596596$. It will always be divisible by 7, 11, and 13. Why? (Calculators allowed.)

How many distinct pairs of positive integers $\left( {m,n} \right)$ satisfy ${m^n} = {2^{20}}$?
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How many positive integers less than 50 have an odd number of positive integer divisors?
(Appeared on AHSME 41 competition)

If $x > y > 0$, then express $\frac{{{x^y}{y^x}}}{{{y^y}{x^x}}}$ using only a single exponent.
(Appeared on AMC-12 competition, 1992)

Earlier, you learned how to calculate a number such as ${8^{\frac{2}{3}}}$.

In fact, there are two different plausible ways of evaluating a number like ${8^{\frac{2}{3}}}$: as $\big(8^\frac{1}{3}\big)^2$ or as ${\left( {{8^2}} \right)^{\frac{1}{3}}}$. Are these two numbers in fact equal? Explain why or why not.

Try calculating both ${27^{\frac{4}{3}}}$ and ${32^{\frac{6}{5}}}$ in each of the ways described in part a. Which is easier, and why?

Multiplying and dividing numbers and/or expressions that have roots and/or exponents in them becomes considerably easier if all the roots are converted to exponents. Also often helpful in harder problems is to try to give all the exponents the same base. For example, the easiest way to simplify ${2^5} \cdot {8^4}$ is to think of “8” as “${2^3}$”:

Simplify the following:

${4^3} \cdot {64^5}$

${2^{ - 4}} \cdot {32^2}$

$\frac{{{3^5} \cdot {{18}^4}}}{{16}}$

${7^{\frac{5}{6}}} \cdot {49^3}$

$\frac{{{{81}^{\frac{3}{8}}}}}{{{{27}^{\frac{1}{2}}}}}$

(Calculators allowed.) In addition to the greatest common divisor, another interesting numerical concept is called the least common multiple.

What is the smallest positive integer that is a multiple of 30 and also a multiple of 42? This number is called the “least common multiple” and is written as lcm(30, 42). (For example, lcm(6, 8) = 24, as 24 is the smallest number that is both a multiple of 6 and a multiple of 8.)

If your answer to part a involved testing a large amount of numbers, try using the prime factorizations of the numbers to help you see how to find the lcm efficiently.

Find lcm(84, 126) efficiently. How can you be sure you have found the smallest possible number that works?

Find lcm(${2^{16}} \cdot {3^4}$, ${2^{13}} \cdot {3^8}$). Looking at the original prime factorizations of each number, what do you notice about the prime factorization of your answer?

Create a procedure that allows one to efficiently determine the lcm of two numbers. Once again, check with 3 examples you devise, with at least one of the examples similar to part d.

Take any two positive integers $x$ and $y$. (Calculators allowed.)

Find their gcd and lcm.

Multiply the gcd and lcm together.

Now multiply the original numbers $x$ and $y$ together. What do you notice?

Try parts a through c with 5 different pairs of positive integers $x$ and $y$.

What is going on here? Explain carefully in terms of the prime factorizations of $x$ and $y$. (Hint: Go back to questions 13 and 92e and look at how you were able to find the gcd and the lcm efficiently, and compare.)

Sometime around 5th grade, you were taught to add fractions $\frac{a}{b}$ and $\frac{c}{d} \;$ by finding a common denominator. Assume $b$ and $d$ are positive.

Explain whether you should be finding the gcd or the lcm of $b$ and $d$ if you want to find their smallest possible common denominator.

By the way, why do you need a “common” denominator to add fractions anyway? Why not just add them using different denominators?

If $x = \sqrt {2000} $ and $y = \sqrt {2001} $, what is the simplified numerical value of ${(x + y)^2} + {(x - y)^2}$?
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What on earth could ${4^{\sqrt 2 }}$ mean? Could you determine approximately how big it would have to be?

Let’s learn about how simplest radical form directly relates to prime factorization. (Calculators allowed.)

Put $\sqrt {99} $ and $\sqrt {884} $ into simplest radical form.

Explain, for an arbitrary integer $n$, how you would go about putting $\sqrt n $ into simplest radical form.

Now try simplifying $\sqrt {907} $. What makes you confident that you are definitely in simplest radical form?

Explain why you don’t have to test any numbers above 31 when checking to see if 907 is prime.

Now try simplifying $\sqrt {529} $, $\sqrt {551} $, and $\sqrt {557} $.

Given what you learned in parts c through e, can your answer in part b be improved or clarified? Are you confident your method will efficiently put $\sqrt n $ into simplest radical form? Explain.

Try your method on $\sqrt {343} $, $\sqrt {379} $, $\sqrt {403} $, $\sqrt {765} $, $\sqrt {1517} $, $\sqrt {1373} $, and $\sqrt {1763} $.

Let’s take a look at a proof the Greek mathematician Euclid came up with over 2000 years ago to show that there are an infinite number of primes. Euclid did this by assuming that there were a finite number of primes, and then showed that assuming that is the case turns out to be self-contradictory. Follow along to see his “moves”!

If there were a finite number of primes, we could call the largest one $“P”$. Euclid then asked us to consider $“Q”$, the number that has a prime factorization that includes each of the primes exactly once. Explain clearly how one could calculate $“Q”$.

Euclid then made his biggest “move”. Remembering that all the primes divide evenly into $“Q”$, Euclid asks us to consider the number $Q + 1$. Which of the primes do you think would go evenly into $Q + 1$ as well? Why?

What can you conclude about $Q + 1$, now that you have established that it has no prime factors?

Why does your conclusion contradict what you assumed at the start of the problem?

Given that $(x + 2)(x + b) = {x^2} + cx + 6$ for all values of $x$, what must $c$ be?
(Appeared on AHSME 1988 competition, #5)

If $Q = 2(3\pi - 7)$, how many times bigger than $Q$ is $8(12\pi - 28)$?

A tree broke at a point $\frac{1}{4}$ the distance up the trunk, and when it fell the top of the tree was 60 feet from its base (so a triangle is formed — the upright $\frac{1}{4}$ of a tree, the diagonal rest of the tree, and the ground). How tall was the tree?

If $x = {y^2}$, what is the value of ${y^{1996}} - {x^{998}} + 499$?
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$f$ is a function that squares a number, then adds 16. $g$ is a function that adds 4 to a number, then squares the result. Anthony says that $f$ and $g$ are really the same function. Is he right?

Prove that the square of an even number is always even. Then prove that the square of an odd number is always odd. (Hint: Every even number is a multiple of 2; every odd number is one more than a multiple of 2.)

Prove that the square of an odd number minus the square of another odd number is always a multiple of 4.

Take two numbers: the first should be a multiple of 12, and the second 3 more than the first. Prove that the second number squared minus the first number squared will never be divisible by 6.

If $\frac{1}{{x - 1}} - \frac{1}{{x + 1}} = \frac{C}{{{x^2} - 1}}$, what number must $C$ be?

A box whose dimensions are 4 ft, 4 ft and 4 ft. is packed with cylindrical cans that are 2 ft high with a diameter of 6 in. When the box is fully packed with cans, how much space is wasted in the box? Prove that your answer is the same as $16(4 - \pi )$ ft$^3$.

If $a + b = 0$, but $a \ne 0$, what is the value of $\frac{{{a^{2007}}}}{{{b^{2007}}}}$? Of $\frac{{{a^{2008}}}}{{{b^{2008}}}}$?
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The length (in cm) of the diagonal connecting opposite corners of a cube is the same as the volume of the cube (in
cm$^3$). What is the surface area of the cube?

Three adjacent faces of a rectangular box have areas 20, 30, and 40 cm$^2$. What is the exact volume of the box?

Make up a problem like Problem 2 and try it out on your classmate. Make sure to prove that it works before doing it!

This trick involves two numbers — your age, and the amount of change in your pocket (expressed in pennies — a number from 1 to 99).

Double your age.

Add 5.

Multiply by 50.

Add the amount of change in your pocket (and if you don’t have change, just think of a number between 1 and 99).

Subtract the number of days in a non-leap year.

Add 115.

Divide by 100.

Look to the left and to the right of the decimal point. Cute, no? How does this one work?

Here’s a trick for you to finish. Take two positive numbers, $x$ and $y$.

Multiply their sum by their difference.

Now multiply your answer by the sum of their squares.

Add $y$ raised to the 4th power.

Take the square root of your result.

How could someone always figure out what “$x$” was if they had the answer to Part d?

Pick two numbers.

Multiply the first by three more than the second — call it “Sonny”.

Multiply the first by three less than the second — call it “Cher”.

Subtract Cher from Sonny, and then divide by 6. What do you notice?

Now add Cher to Sonny, and then divide by double the first number. What have you found?

Explain the mysteries of Sonny and Cher. (It might help to think of the two numbers you picked at the start as $x$ and $y$.)

John from Cincinnati claims that he has a snappy way to calculate ${37^2} - {33^2}$ in his head — just multiply the sum of the two numbers by their difference (i.e. $70 \cdot 4 = 280$).

Does John’s method actually work? Try it on a few pairs of numbers.

If two numbers that you pick are $x$ and $y$, John’s method claims that there is an easier way to calculate $x^2- y^2.$ In terms of $x$ and $y$, what is John saying is the easier way?

Prove that John’s method always works, or that it doesn’t always work.

Calculate $104 \cdot 96$ in your head! (Hint: Work backwards!)

Distribute $(2w - 12z)(2w + 12z)$ and $(x^3+y^7)(x^3-y^7)$; what do you notice?

Jake made up the following number trick, which he was sure would dazzle all his friends. “Pick a number without telling me. Multiply it by 100, and subtract that from 2500. Now add the square of the original number, and tell me the answer.” Jake knew that all he had to do was call the final result $Y$, use his calculator to compute $50 - \sqrt Y $, and he’d have their original number back.

Confirm that Jake’s method works for a few numbers. Then explain why it works.

For a while, Jake’s trick worked wonderfully. But the other day, his friend Rachel said that Jake had failed to read her mind! Her number had been 99, but Jake had guessed 1. How could he have been so hideously far off?

Phil from Duluth has another calculating trick. He says that he can calculate $33 \cdot 27$ in his head by taking the square of the sum of the two numbers and subtracting the square of their difference, and then dividing the answer by 4: $\frac{{{{(33 + 27)}^2} - {{(33 - 27)}^2}}}{4}$. This is quicker, Phil points out, because the numbers are easier—

is a lot easier to calculate in one’s head than $33 \cdot 27$.

Is Phil correct that this method always gives the right answer? Will this always work with any two numbers? Prove your answer.

Is Phil correct that this method is always quicker? Give some examples to back up your position.

A terminating decimal is a decimal like $.3764$, which ends. A repeating decimal is one that repeats forever, like $.862862862... .$

How can one express any terminating decimal as a fraction? Show that you can do so with $.3674$ and $.1864597.$

It isn’t obvious whether repeating decimals can be expressed as fractions or not. Still, if $.862862862...$ were to be equal to a fraction, it would have to be close to $\frac{{862}}{{1000}}$. See if you can find a fraction which appears to do better.

Will the idea of part b work with any repeating decimal? Try it out on $.9166591665...$ and $.333... .$

Let $x =.333... .$ By multiplying both sides of this equation by 10, one gets a new equation. By combining equations, prove why $.333... = \frac{1}{3}$.

Now prove that $.862862862...$ is a fraction as well by a similar method to part d. Will this method let you convert any repeating decimal into a fraction? Does it work on $.9166591665...,$ for example?

Don’t use a calculator for this problem.

Factor $x^2-x+12$.

Solve ${x^2} + 4x + 2 = 0$ by completing the square.

Divide $5.\bar 3 \div 8$.

Simplify $\sqrt{50a^4b^3}$.

If $(x - 2)(x + 2) = 5$, find ${x^4} - 4{x^2}$.

Ask a friend to think of a three-digit number in which the digits are all the same (such as 777). Tell them that you don’t want to know the number, but that you would like them to add up the digits, multiply by 37 and tell you the answer. Why would they think you are being a wiseacre? Can you prove it always “works”?
(Hint: $777 = 700 + 70 + 7 = 7\left( {100 + 10 + 1} \right)$.)

If I have a larger number divisible by 3 and subtract a smaller number divisible by 3, will the answer necessarily be divisible by 3? Try some examples. Explain why what you have found makes sense.

If I have a larger number and subtract a smaller number divisible by 3 and get an answer that is divisible by 3 — is the larger number necessarily divisible by 3? Again, try some examples, and explain why your conclusions are justified.

After having completed and digested the ideas of Problems 44 and 45, we are now ready to understand why the divisibility test for 3 works. Remember that the divisibility test for 3 states that if the sum of the digits of a number is divisible by 3, then the number itself is as well. (You will prove below that it works for a three digit number, but your proof can be adapted easily to any number of digits.)

We can write 741 in the following way: $700 + 40 + 1$, and similarly we can write 6518 as $6000 + 500 + 10 + 8$. How could you write any number “ABC” in the same way, where A is the hundreds digit, B is the tens digit, and C is the units digit?

Consider 568, which is not divisible by 3. Notice that when you subtract the sum of its digits (5 + 6 + 8 =19) from itself, the answer you get, 549, is divisible by 3. Does this always work for any 3-digit number?

Use algebra to prove the result you found in part b. Your answer to part a. will be helpful here.

If $\left( {{\rm{A + B + C}}} \right)$ is also divisible by 3, by using what you learned in Problem 45 and parts b and c above, what can you now say about the number “ABC"? Explain!

If $\left( {{\rm{A + B + C}}} \right)$ is not divisible by 3, what, if anything, can we say now about “ABC”? Explain carefully.

Pick a 3-digit number. If one subtracts the “reverse” of the number from the number itself, the answer is always a multiple of 9. For example $852 - 258 = 594$, which is a multiple of 9 (and negative numbers “work” as well — e.g. If we had picked $258, \ 258 - 852 = -594$). Explain why this is so. (Remember that any three digit number “ABC” can be written as $100{\rm{A}} + 10{\rm{B}} + {\rm{C}}$ ; how do you think you could write its reverse, “CBA”?) Will this work for 4 or 5 digit numbers as well?

Go to the following web page and have fun!: http://digicc.com/fido/ What is going on here? Can you come up with an explanation of how it is done?

Look at the following YouTube video on the method of “Vedic Multiplication”: www.youtube.com/watch?v=gwaAAEYIW_8 Why does this method work? Will it always work?

Explain what the following “Proof without Words” is equivalent to algebraically:

Explain what the following (slightly harder) “Proof without Words” is equivalent to algebraically:

Is the sum of 3 consecutive integers always divisible by 3? Is the sum of 4 consecutive integers always divisible by 4? Is the sum of 5 consecutive integers always divisible by 5?

When is the sum of $n$ consecutive integers divisible by $n$? Try different values of $n$ to get a feel for the problem, and then see if you can give good reasons for the pattern you see.

Prove that the square of an odd number is always one more than a multiple of 8. (Hint: Try factoring part of the expression you came up with to represent the square of an odd number, and also note that with any two successive integers, one of them must be odd and one of them must be even.)

Which primes can be expressed as the difference of two squares? Why? Can any be written as the difference of two squares in two different ways? Why or why not?

Here’s a quick way of multiplying any two 2-digit numbers, like 52 and 58, that have the same tens digit and have units digits that add up to 10.

1. Multiply the tens digit by the next largest integer and call it “$M$” (here, $M = 5 \cdot 6 = 30$).
2. Multiply the units digits of the two numbers together and call it “$N$” (here, $N = 2 \cdot 8 = 16$).
3. Write the digits of $M$ followed immediately by the digits of $N$ (here, 3016). This gives you the right answer!

Four whole numbers, when added three at a time, give the following sums: 188, 199, 212, and 220. Find the four numbers.

How many colors would it take to color each one of the graphs below? Explain your answer for each graph.

What if there were “$n$” nodes on the circle (instead of the 5 nodes and 6 nodes in the graphs of part a) — then how many colors would it take to color? Answer clearly and completely.

Chris draws a graph that has an Euler Circuit. Alonso adds an edge to Chris’s graph, and now, although the graph no longer has an Euler Circuit, Alonso says it has an Euler Path. Can Alonso possibly be right? Explain.

From Crisler and Froelich: Discrete Mathematics Through Applications. The following is a list of chemicals and the chemicals with which each cannot be stored.

Draw a graph to represent the situation.

How many different storage facilities are necessary to keep all 7 chemicals?

Uler says she just drew a graph with 1001 nodes, and exactly 500 of the nodes are of even degree. Do you believe her? Explain.

How many colors does it take to color a complete graph with n nodes? Explain.

A graph has 6 nodes with degrees 3, 3, 4, 5, 3, and 2. Is it possible for the graph to have an Euler path?

For each of the following say whether it’s true or false. If false, produce a counterexample or say precisely why.

Sum of degrees of any graph is even.

Every graph has an even number of nodes of odd degree.

If a graph has an Euler path then it must have exactly two vertices/nodes of odd degree.

Every graph of 4 or fewer nodes is planar.

All the nodes of a graph could be of odd degree.

A planar graph could be isomorphic to a non-planar graph.

All complete graphs with four nodes are isomorphic to each other.

For some graphs that have Euler circuits, it is actually still possible to walk around those graphs in such a way that you get stuck before you’ve crossed every edge. Draw one such graph, and show (a) how to walk around it so that you cross every edge exactly once, and (b) how to walk around it so that you get stuck before you get to every edge.

Simplify $\big(\frac{120x^{-8}y^7}{y^{-2}x^{-4}z^3 \cdot 15}\big)^{-2}$ making sure there are only positive exponents in your simplified answer.

Write the fraction in as simplified a form as possible, without using a calculator:
$\big(\frac{105/117}{165/363}\big)^{-3}$

If $x$ and $y$ are > 0, express $\frac{{{x^y}{y^x}}}{{{y^y}{x^x}}}$ , using only a single exponent.

Find the prime factorization of 533610. Use a calculator to help! Now, find the prime factorization of 20! (20 factorial), without a calculator.

How many of the first 200 positive integers are divisible by all of the numbers 2, 3, 4 and 5?

Suppose $p$ and $q$ are prime numbers.

What are all the factors of the number $pq$ ?

List all the factors of ${p^{10}}$ .

How about ${p^2}{q^2}$ ?

Write each of the following numbers in the form ${2^a}{3^b}{5^c}{7^d}$ , where $a$, $b$, $c$, $d$ can be whatever you want.

2100

$\frac{1}{{21}}$

2

1

2.1

Write the GCD of ${2^a}{3^b}{5^c}{7^d}$ and ${2^{a + 1}}{3^{b - 2}}{5^{c + 4}}{7^d}$ , assuming that $a$, $b$, $c$, $d$ have to be integers larger than 1.

Is $\frac{{50^{24}}}{{20^{12}}}$ , simplified, an integer or a
fraction? Explain.

Simplify the following expressions, and make sure there are no negative exponents:

$\frac{{36{{(x{y^4})}^{ - 2}}}}{{40{x^5}{y^{10}}}}$

$\big( \frac{(x^2y^{-3})^2}{x^{-2}y^5} \big)^{ - 1}$

. $\big(\frac{16^{-4}x^{-4}y^{-2}}{128^{-2}x^{-3}y^{-3}}\big)^3$

If $m \ne 0$ , then $\frac{{{{({m^4})}^4}}}{{{m^4}}} = $ …?

If ${4^{6x - 9}} = 64$ then what is the value
of $x$?

You ask your friend to do the following calculations:

Take a number

Add 1

Multiply by 3

Multiply your answer in (c) by one less than the original number …

When your friend tells you the answer in d, what’s the fastest process you can use to figure out what the original number was?

Another number trick:

Pick a number.

Double your number.

Square the result in “b”

Subtract 4. Call this answer “Fred.”

Now take your original number and add one.

Multiply this number by one less than the original number. This is “Lilli”.

What is $\frac{\rm{Fred}}{\rm{Lilli}}$ ? Why does this work?

You ask your friend to think of a number, then

• Multiply by 6

• Add 9

• Divide by 3

• Subtract 3.

Your friend tells you the answer. What do you need to do to get the original number back so that you can convince your friend you are a mind-reader? Show some algebra to explain why.

Which equation represents the following statement?
Twice the difference between a certain number and its square root is 15 more than twice the number.

$2N - \sqrt{N} = 15 + N$

$2\left( {N - \sqrt N } \right) = 15 + 2N$

$2N - \sqrt N = 16 + N$ )

$2N - \sqrt N + 15 = N$

$2\left( {N - \sqrt N } \right) + 15 = N$

$2N - 2\sqrt N = 15 + 2N$

Simplify the following so that there are only positive exponents: $\big(\frac{168p^{-4}q^5r^{-6}}{42pq^{-2}r^3}\big)^{-2}$

Simplify this until it is a single fraction that is reduced as much as is possible:
$\big(\frac{180 \cdot 5}{4 \cdot 196}\big) \cdot \big(\frac {448 \cdot 49}{1800 \cdot 15}\big)^2$

Simplify by multiplying out and collecting like terms.
(By the way, these specific expressions tend to show up very often in algebra.)

${(x + y)^2}$

${(x - y)^2}$

$(x + y)(x - y)$

$(1 + x)(1 - x)$

$(x + 1)(x - 1)$

Simplify by multiplying out and collecting like terms:

$4b(b - 9) - b(2 - b)$

$({z^2}y + {z^2} - x)(z - x)$

$(m + n)({m^2} - {n^2})$

$-5x(x+3y)^2$

$(5t + 2)(25{t^2} - 10t + 4)$

${({z^4} - 3)^2}$

$(2a{b^2} - 5{a^2}b)(21{a^5}{b^4} - 14{a^4}{b^5} + 7a{b^3} - b)$

${(2a + 3b)^3}$

$1 - (1 + [1 - (1 + y)])$

If ${x^2} + 7x + 8 = (x + 3)(x + 4) + p$ is always true, then $p = $ …

Reduce each fraction as much as you can (which may be not at all).

$\frac{{14{a^2} - 18a}}{{2a}}$

$\frac{{{x^2}y - {x^7}{y^2}}}{{{x^2}y}}$

$\frac{{ - 12m(m - 13)}}{3}$

$\frac{{4x + 2y - 3}}{2}$

$\frac{{8{x^3}{y^5} - 6{x^4}{y^3}}}{{{x^2}{y^3}}}$

$\frac{{24{r^3}{s^{ - 2}} - 36{r^{ - 1}}{s^5}}}{{{r^{ - 2}}{s^{ - 3}}}}$

. $\frac{{6{x^3} - 12x{y^{\frac{2}{3}}}}}{{{x^{\frac{1}{2}}}y}}$

$\frac{{6{f^4}g - 5{f^2}{g^3}}}{{fg}}$

$\frac{{\left( {\frac{{2x + 2}}{x}} \right)}}{{\left( {\frac{x}{{x + 1}}} \right)}}$

$\frac{{{{({n^2} - 6n + 14)}^6}}}{{{{({n^2} - 6n + 14)}^4}}}$

Add the fractions:

$\frac{8}{p} + \frac{q}{2}$

$\frac{1}{x} + \frac{3}{{{x^2}}}$

$\frac{{2y + 3}}{6} - \frac{{y + 3}}{4}$

$\frac{3}{{ab}} - \frac{4}{{{b^2}}}$

$\frac{4}{{2x - 1}} - \frac{3}{{2x}}$

$\frac{2}{{x - 1}} + \frac{2}{{x + 1}}$

$\frac{{{w^2} + 2w + 1}}{w} - \frac{{{{(w - 1)}^2}}}{w}$

What is the result when $3 - 2x$ is subtracted from the sum of $x - 3$ and $5 - x$ ?
(From Andres et al, Preparing for the SAT Mathematics. Amsco, 2005.)

If $3x - 7 = 5$ , then $9x - 21$ =…?

If ${x^2} + {y^2} = 37$ and $xy = 24$ , what is the value of ${(x - y)^2}$ ?

Prove that if one positive integer is 3 more than another, the difference in their squares must be a multiple of 3, but cannot be a multiple of 6.

Write each in the form ${a^b}$ , where $a$ and $b$ are integers not equal to 1:

${3^4}{9^5}{27^6}$

$4^5 15^{10}$

How many ${2^3}$ terms are there on the left side of the equation?

Solve for $x$:

$3{x^5} = 96$

$\frac{2}{3} \; x^3 = 22$

${x^{\frac{4}{5}}} = 37$

$6{x^{- \frac{{1}}{6}}} = 66$

$217{p^{\frac{2}{9}}} = 53$

$5\sqrt {5x - 1} = 7\sqrt {2x + 5} $

$\sqrt{3x+3} = x + \frac{3}{2}$

Rewrite in the form ${a^b}{c^d}$ , where $a$, $b$, $c$, and $d$ are each integers not equal to 1 and less than 10:
$27^{\frac{4}{3}} \cdot 216^{\frac{2}{3}} \cdot 32^{\frac{3}{5}}$

Simplify the following expressions, making sure there are no negative exponents:

$\frac{{{x^{ - \frac{2}{3}}}}}{{{x^{\frac{4}{3}}}}}$

$\frac{{{x^2}{y^{\frac{1}{4}}}}}{{{y^{ - \frac{1}{2}}}{x^{\frac{5}{3}}}}}$

$\big(\frac{16^{\frac{3}{4}} x^\frac{4}{3} y^{-2}} {121^{-\frac{1}{2}} x^{-\frac{2}{3}} y^{-3}}\big) ^3$

How many distinguishable 6 letter words can be formed from the letters of “twists”, where each letter can only be used once per word? What if the t’s and s’s were given different colors, so they could be distinguished from each other as well?

How many ways can you step right 4 times and left 7 times in a sequence of 11 steps? Put another way, how many ways can you arrange 4 R’s and 7 L’s in a straight line, where the R’s and L’s are indistinguishable from each other?

Suppose you’re going to toss a coin 8 times and record the sequence of heads and tails. How many different sequences are possible?

Josephine flips a coin 9 times. What is the probability that she’ll get exactly 6 heads? 4 heads? 3 heads?

How many ways are there to form a committee of 5 people from a class of 80? (There are at least two different methods to solve this problem)

In going packing for a trip, you and your spouse need to pack 4 ties and 6 dresses. You own 7 ties and 10 dresses. How many different ways can you select the ties and dresses for your trip?

How many ways can you select 4 letters from the word “fishbowled”, if the order of the letters selected is unimportant? What if the order is important?

If you start in the lower left corner of an 8x8 chess board, and for each “step” you can move either one square to the right or one square up, how many different ways are there to reach the upper right corner of the board?

In how many ways can five girls be chosen from a class of 20 if Sarah Dreadful has to be chosen (so that she won’t throw a fit)?

Julius goes to a yard sale and sees 8 books on sale. He thinks that he would like to buy either 2, 3, or 4 books. How many different possibilities are there for what books he could buy at the yard sale?

Ten people meet at a party, and each pair of people shake hands. How many handshakes are there?

Three identical prizes are to be given to three lucky people in a crowd of 100. In how many ways can this be done? What if the prizes were not identical?

You plan on forming a Sophomore Prom committee consisting of 8 sophomores. The Sophomore Class has 45 boys and 35 girls in it. How many distinct committees can be formed if

there are no restrictions placed on the committee other than there must be 8 sophomores on the committee?

there are to be 4 girls and 4 boys on the committee?

the committee cannot have Sarah Dreadful on it?

A boat has 3 red, 3 blue, and 2 yellow flags with which to signal other boats. All 8 flags are flown in various sequences (1 flag at a time) to denote different messages. How many such sequences are possible?

Five boys and five girls stand in a line. How many different arrangements are
possible:

If boys and girls can be intermixed (or not) in the line?

If all the boys stand at the back of the line?

Henry’s Hamburger Heaven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburgers can be ordered?

A “necklace” is a circular string with several beads on it. Two necklaces are the same if they are just rotations, in the plane, of each other. How many different necklaces can be made with 10 different beads?

How many 6-digit numbers have at least one even digit?

There are five books on a shelf. How many ways are there to arrange some or all of them in a stack? The stack may consist of one book.

The map of a town is depicted below. Ian lives at point I; the library is at point L; and the grocery store is at point G. Ian can only walk south or east. The town is many times bigger than the map.

How many different ways are there for Ian to walk from his house to the library?

How many different ways are there for Ian to walk from the library to the grocery store?

If Ian wants to walk from his house to the grocery store by way of the library, how many different ways can he do this?

Suppose Ian uses a coin flip to decide whether he’s going to go one block south or one block east. After he walks a block, he then tosses the coin again and follows the previous guidelines regarding his movement. If it takes Ian one minute to walk one block, then what is the probability that he will make it from his house to the grocery store in 13 minutes? Assume it takes him no time to flip the coin and interpret the meaning of the outcome.

One student has 6 novels and another has 7 novels. How many ways are there for the first student to exchange 3 novels with 3 novels owned by the second student?

How many 5 digit numbers have factors of 4 or 5?

The equation below shows a series of 1’s summing to 14.

$1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 14$

How many ways can you choose 3 of the plus signs to circle?

Clearly explain how your answer to Part a also answers the question below.

How many solutions are there of the equation $x + y + z + w = 14$ in positive integers? Note: the solution $\left( {1,2,8,3} \right)$ is different from the solution $\left( {2,3,1,8} \right)$.

What is the probability of getting 5 heads out of 10 tosses of a fair coin?

Suppose you draw at random 5 cards from a standard 52-card deck.

How many possible different 5-card hands are there?

How many ways can you pull 5 cards such that you get exactly one ace?

How many ways can you pull 5 cards such that you get exactly two aces?

How many ways can you pull 5 cards such that you get exactly two aces and three kings?

What’s the probability of pulling 4-of-a-kind?

Suppose you are standing at zero on the real number line. You toss a fair coin and move 1 unit to the left if you get heads. You move one unit to the right if you get tails. You do this 17 more times for a total of 18 coin tosses.

After 18 tosses what are the possible numbers you might be at on the real number line?

What is the probability that you will be standing at 0 after 18 tosses? At -2? at -3?

For fun on a Friday night you and a friend are going to flip a fair coin 10 times. Let H represent the outcome that a flip shows heads and T represent tails. Assume that ${\rm{Prob}}\left( {\rm{H}} \right) = {\rm{Prob}}\left( {\rm{T}} \right) = 0.5$ .

You flip the coin 10 times and get the sequence HTHHHTTHHT. Your friend does likewise and gets HHHHHHHTTT. Which of these two sequences was more likely to occur? Justify your response.

What is the probability that you will get at least two heads in 10 tosses of the coin?

The numbers 1, 2, 3, …, 25 are placed in random order. What’s the probability that the numbers 1, 2, 3 are next to each other?

a. Suppose that 14 people are arranged randomly in a line and two of these people are Emma and Eric. What’s the probability that there are 3 people standing between Emma and Eric?

Suppose that the 14 people in Part a are arranged in a circle. What’s the probability that there are 3 people standing between Emma and Eric?

Suppose you will roll a standard, fair six-sided die until you get a 2. What’s the probability that you will stop rolling after the first roll? What’s the probability that you will stop after the second roll? What’s the probability that you will stop after n rolls?

Eight first graders, 4 girls and 4 boys, arrange themselves at random around a merry-go-round. What is the probability that boys and girls will be seated alternately?

Rick writes letters to 8 different people and addresses 8 envelopes with the people’s addresses. He randomly puts the letters in the envelopes. What is the probability that he gets exactly 6 letters in the correct envelopes? What about 7 letters?

There are 40 people waiting to be selected for a 12-person jury. Of the forty, 20 are African-American, 15 are Hispanic, and 5 are Caucasian.

How many different 12-person juries (called “jury panels”) can be picked from this group of 40? Show your work.

How many jury panels can be picked that have only African-Americans? Show your work.

Show that the probability that an all African-American panel would be picked just by chance is approximately .0000225. Show your work.

If you were a judge and the two lawyers working a case in your court picked an all African-American jury panel (or a jury panel with no African Americans on it) from the pool of 40 given in the beginning of this problem, would you believe that this panel was created without consideration of race? Explain, making direct reference to the probability in Part c.

Suppose Tony’s class is trying to form two committees — the Social Committee and the Fund Raising Committee — from the fourteen students in his class. How many possible arrangements are there of the fourteen students into these two committees, if each student is allowed to choose one committee or neither to be on? Show your work.