You always set your alarm clock to wake you up one hour before you have to leave. But if you have to leave before 7 am, you give yourself 15 extra minutes to get ready since you know you’ll be groggy.

What’s the input for the process described above? What’s its output?

What time would you have to leave if the clock woke you up at 5:40am? 5:50am?

Your alarm wakes you up. Based on the time you see on the clock, how do you know when you’ll have to leave?

You’ve lost your calculator, and your diabolical arch nemesis has decided to exploit this opportunity. He’ll make calculations for you — for a price! Multiplication costs $\$1$, whereas exponentiation costs $\$6$. Unfortunately, you desperately need to compute ${7^{10}}$, and you’ve only got $\$5$. What will you do?

You input any word, and the AlphaBlast algorithm outputs the same word but with all the ‘a’s replaced with ‘z’s. Is AlphaBlast reversible? Example: AlphaBlast(“mathematics”) = “mzthemztics”. Test it carefully and then explain your answer. (An algorithm is called reversible if it’s possible to create an inverse algorithm for it.)

For each of the following algorithms, assume that the input can be any number, and call it $X$. Read each description, and decide in each case whether the algorithm is reversible. If so, write the inverse.

The Foo algorithm either multiplies $X$ by 2 (if it’s a whole number) or doesn’t change it (if it’s not a whole number).

The Bar algorithm either multiplies $X$ by 2 (if it’s NOT a whole number) or doesn’t change it (if it IS a whole number).

The FractionsAreFun algorithm takes $X$, adds 10, and divides that answer by 4, and outputs the result.

The NoReallyTheyAre algorithm divides $X$ by 20 then adds 1 and outputs the result.

The Flip algorithm outputs the value of 100/$X$, unless $X$ is zero, in which case the output is just zero.

The Flop algorithm divides 20 by $X$, then adds 1 and outputs the result.

The Square algorithm simply outputs the value of ${X^2}$.

Here’s a variation on the Number Devil game: the Number Devil is allowed to change its secret number after each time you guess. However, it can’t change it in a way that makes any of its previous answers untrue. (So, if you guessed “5” the first time and the answer was “higher”, it could not change its answer to “2”, but it $could$ change it to “7”.)

Explain why binary search algorithm’s worst-case guarantee is still the same.

Describe an algorithm the Number Devil could use to force a worst-case outcome every time, as long as you didn’t get lucky and guess right the first time.

Here’s a tougher variation of the game you played in problem 8. The setup is the same: your partner shuffles the cards and lines them up, and you can’t look at them. But now, instead of just finding the largest one, your goal is to sort all the cards into descending order (i.e., line them up from largest to smallest). Your basic operations are (i) to ask your partner to compare two cards (just like last time), and (ii) to swap the positions of any two cards.

Write an algorithm for this game. That is, create an algorithm that will sort a list of 5 numbers. After writing the algorithm, go ahead and try it out with a group member. Does it work?

Look back at your pancake-flipping algorithm from problem 34. If you extended your algorithm to stacks of $N$ pancakes, how many flips, maximum, could you guarantee it taking — that is, what’s the worst it could do?

Here’s a different version of the pancake problem. This time, you have three plates. Plate #1 has a stack of 5 pancakes, in order from the largest one on the bottom to the smallest on top. This time, though, you can only use the spatula to shift one pancake at a time to another plate. At no time can any larger pancake be on top of a smaller pancake. How many moves does it take to get the entire stack of pancakes from Plate #1 to Plate #3?

Don’t use a calculator for this problem.

Simplify $\frac{{20{a^4}{b^7}}}{{30{a^3}{b^{ - 2}}}}$

Reduce the fraction $\frac{x^3 +2x}{x^2 + x}$

Solve for $x$: ${x^{\frac{2}{3}}} = 64$

Solve for $x$: ${x^2} + 8x + 9 = 0$

Solve for $x$: $3{x^2} + 7x + 3 = 0$

If you double the size of a list in which you’re doing a binary search, will it (in the worst-case scenario) take you about twice as long to do the search?

The shape below is called a “tromino.”

Each of the following “checkerboards” has had one square removed. For each board, find a way to cover all of the remaining squares with trominoes.

You have only a compass and an unmarked straightedge. Come up with an algorithm for finding a point equidistant from three given points. In doing so, you will have proven that it is always possible to find such a point.

Alice is in the top right square of a giant chessboard, and the white rabbit she’s chasing is in the bottom left square. Each turn, Alice gets to move one square up, down, left, or right. Then, after that, the white rabbit gets to move (up, down, left, or right). Alice catches the rabbit when she’s able to move into the square he’s sitting in. Assuming that both Alice and the white rabbit are strategically quite savvy, predict the outcome.

The Floor operation takes any number whatsoever as an input. ${\mathop{\rm Floor}\nolimits} (x)$ simply takes $x$ and returns the greatest integer that’s less than or equal to $x$. In other words, the Floor function always “rounds down.” Examples: ${\mathop{\rm Floor}\nolimits} (2.001) = 2$, $\mathrm{Floor}(\pi ) = 3$, and $\mathrm{Floor}(5.9999)=5$.

The following algorithm, which uses Floor and some basic arithmetic operations, only takes positive integers for its input. What does it do?

${\mathop{\rm Mystery}\nolimits} (n) = 5\left( {\frac{n}{5} - {\mathop{\rm Floor}\nolimits} \left( {\frac{n}{5}} \right)} \right)$

The RightShift and LeftShift operations work for any non-negative integers with five or fewer digits:

${\mathop{\rm RightShift}\nolimits} (N,d)$ chops off the rightmost $d$ digits of $N$ and outputs the result. Example: ${\mathop{\rm RightShift}\nolimits} (3214,2) = 32$.

${\mathop{\rm Left}\nolimits} {\mathop{\rm Shift}\nolimits} (N,d)$ moves each digit $d$ places to the left, and fills in 0’s in the empty spaces; then, it chops off all but the last five digits. Examples: ${\mathop{\rm Left}\nolimits} {\mathop{\rm Shift}\nolimits} (75,3) = 75000$, and ${\mathop{\rm Left}\nolimits} {\mathop{\rm Shift}\nolimits} (3214,2) = 21400$.

Starting with the number 6789, how can you use only the LeftShift and RightShift operations to end up with the number 7?

Generalize your work from part a to create the PickADigit algorithm, where ${\mathop{\rm PickADigit}\nolimits} (N,d)$ outputs the $d$th digit of a positive integer $N$.

What does the following algorithm do, assuming that $N$ is a positive, five-digit integer?

You input a whole number to a certain algorithm. If it’s even, the algorithm doesn’t change the number. If it’s odd, the algorithm turns the number backwards — for example, 57 becomes 75. Is this algorithm reversible?

Come up with a non-trivial numerical algorithm that reverses itself. In other words, come up with an algorithm Boing such that, for every $N$,
$\text{(Boing(Boing}(N))= N$.

Imagine your friend gave you a very long table of $all$ the possible inputs to her algorithm and their corresponding outputs. Even if she didn’t explain the process her algorithm used, describe using a single sentence how you could figure out whether or not the algorithm was reversible.

Your basic operations for the following are addition, subtraction, multiplication, division, and testing which of two numbers is the biggest.

Write an algorithm for figuring out whether or not a positive integer $N$ is prime. Pick some three-digit numbers and try it out. (By the way, computer systems actually use algorithms like these to encrypt your private data.)

Suppose it takes you 3 seconds to use your calculator to do each of the basic operations above. How long would it take you to check whether or not 1,000,003 is prime? How about 1,000,005?

You have 4 identical-looking coins. Two of them are heavy and two of them are light (the two heavy ones weigh the same, and the two light ones weigh the same).

Call the coins A, B, C, and D. Your goal is to find out which two are heavy and which two are light. Your only instrument is a balance scale — you put some coins on the two sides of the scale, and it tells you which side is heavier (or that the two sides weigh the same).

Describe an algorithm for finding the light coins in as few steps as possible.

When you type a text message into a cell phone using multi-tap typing, you are essentially encoding English words into a long sequence of numbers (and pauses). For example, to type in the word “EIGHT”, I type: 3 3 4 4 4 [pause] 4 [pause] 4 4 8. Use this picture of a cell phone’s keypad to help you:

I typed 7 7 7 7 , 7 3 3 , 3 3 2 2 2 4 4. (Here the commas represent pauses). What word did I type?

Write an algorithm for “decoding” a string of numbers and pauses into a word.

You and a fellow explorer need to be able to communicate two numbers to each other — coordinates for your location, which will always be positive whole numbers, but can be small or large. The only means of communication you have is a carrier pigeon. Due to its tiny brain, the pigeon can only remember one $whole$ number (though it can be as big as you want).

Somehow, you need to fit the two numbers together into one number, but you need to make sure that the process is reversible, so that your companion can figure out what numbers you’re sending him.

Here’s the first process you try. Say your two numbers are 67 and 302. Then you just put them together to make 67302, and send that number along with the pigeon.

Why is this non-reversible?

The next strategy you develop fails as well. What you tried to do was separate the numbers with “000”. So, 67 and 302 gets written as 67000302.

This strategy will usually work, but not for every pair of numbers. When is this strategy non-reversible?

Create your own strategy, and show that it works — that it is reversible.

Look back at the sorting algorithm you wrote for problem 39.

Are there certain initial arrangements of the cards that make your algorithm finish in fewer steps?

Using your algorithm, what’s the maximum number of comparisons you have to make, in the worst case, when you play the game with five cards?

If you used your sort algorithm from problem 39 to sort a list with $N$ items in it, how many comparisons would it take? See if you can express your answer as a simple algebraic formula.

Five pirates come across 100 bars of gold. The pirates have a pecking order, with #5 and #1 indicating the top and bottom-ranked pirates, respectively. Each pirate wants to maximize his or her share of the gold bars. There are no coalitions or collusions between them. They are all very analytical — they can think things through!

Their process for splitting the loot is somewhat democratic. It begins with the top-ranked pirate making a proposal on how to divvy up the loot. (Note that the gold bars cannot be broken up, glued together, or otherwise changed; there are 100 bars of gold, period.) Each pirate gets one vote, up or down on the entire proposal. Remember, each pirate votes based on his or her own pocket and there are no side agreements of any sort. If the proposal gets 50% or more votes, it wins, and that’s that. On the other hand, if it fails to muster 50%, then the pirate who made the proposal is thrown overboard, and the process continues with the next pirate down the hierarchy.

What is the maximum number of bars that the most powerful pirate can get and what allocation to each pirate will ensure him the 50% vote that he needs?

(from http://www.eogogics.com/talkgogics/ezine/tech-talk/pirates1)

Create an algorithm to find the median of a list of numbers. Your only basic operation allowed is to compare two numbers to see which is larger. Assume the list has an odd number of items in it.

Write an algorithm that finds the second largest number in a list of 16 numbers, using only comparison as the basic operation. It’s possible to guarantee a correct answer in fewer than 29 comparisons — can your algorithm do this?

Sherlock and Watson create the following code to encrypt their messages:

First, turn each letter into a number (A is 1, B is 2, etc). Then for each number, multiply it by 7 and add 1. Finally, if any numbers have 3 digits, just ignore the hundreds digit — so 141 just gets written as 41, and so on.

Encrypt the word NO according to their code.

Sherlock sends Watson the message 36 34 22 8 13 36. Decode it.

Even though the code "erases” hundreds digits, it’s still perfectly reversible — you can always decode any message and know exactly what was written. Explain why it’s reversible.

You create an especially tricky locker code. Here’s what you do to encode the 3-number combination: The new first number will be the sum of all three numbers in the original.

The new second number will be the sum of the first two numbers in the original.

The new third number will be the sum of the last two numbers in the original.

Encode 12 – 8 – 1. Do you see how you could find the 12, 8, and 1 from the coded version?

Decode 17 – 14 – 5.

You receive a coded combination: A – B – C (A, B, and C represent the numbers you receive). Write instructions or equations for decoding it.

This is a famous problem that’s challenged many mathematicians. You have 12 coins that look alike. 11 are genuine and each have the same weight. 1 is counterfeit, and is $either$ too heavy or too light.

Write an algorithm for how to use a balance scale to find the fake coin, $and$ determine whether it’s too heavy or too light, in as few steps as possible. (It can be done with three weighings! See if you can figure out how.)

Hint: it will probably help a lot to make a chart to keep track of the different possibilities.

If ${\log _5}\left( x \right) = 5$ and ${\log _5}\left( y \right) = 7$, then what is the value of ${\log _5}\left( {\frac{y}{x}} \right)$?

If ${\log _3}\left( x \right) = 1.776$ and ${\log _3}\left( y \right) = 2.018$, then what is the value of ${\log _3}\left( {\frac{y}{x}} \right)$? (By the way: how old is the United States of America?)

Based on what you did in problems 28 and 29, complete the following generalization: “If ${\log _b}\left( x \right) = C$ and ${\log _b}\left( y \right) = D$, then ${\log _b}\left( {xy} \right) = $….” Now prove it using algebra.

Revise the generalization you wrote in the previous problem into an identity that relates ${\log _b}\left( {x \cdot y} \right)$ to ${\log _b}\left( x \right)$ and ${\log _b}\left( y \right)$. (Remember that an identity is just an equation that is always true, for all values of the variables involved.) This identity is going to come in handy, so write it down somewhere where it will be easy to refer to it.

Given that ${\log _2}\left( 5 \right) = 2.322$ and ${\log _2}\left( 3 \right) = 1.585$, what is the value of ${\log _2}\left( {15} \right)$?

State and prove an identity that relates ${\log _b}\left( {\frac{x}{y}} \right)$ to ${\log _b}\left( x \right)$ and ${\log _b}\left( y \right)$.

If ${\log _3}\left( {10} \right) = 2.096$, then what is the value of ${\log _3}\left( {30} \right)$? How about ${\log _3}\left( {100} \right)$?

Don’t use a calculator for this problem.

Simplify $16^\frac{1}{2} \cdot 2^3$.

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

Solve for x: $4x^3 + 2x^2 = 0$.

Simplify $\frac{{\sqrt {125} }}{{\sqrt {50} }}$.

If $\frac{1}{{{x^2}}} - \frac{1}{x} = 1$, find ${x^2} + x + 1$.

What is the value of ${\log _2}\left( {{4^{1005}}} \right)$?

For each of the following equations, use the log function of your calculator (and possibly some algebra) to approximate $x$ to the nearest thousandth.

${10^x} = 461.2$

$3 \cdot {10^x} = 5$

${10^{x + 2}} = 50$

Mr. Golthramis was in a bad mood one day, and decided to take it out on his students by giving them this problem: “Write down, using no exponents, a number $x$ such that $\log_{10}(x) > 1000$.” The students, having recently learned the definition of log, were outraged, and refused to do the homework. Why?

Verify with your calculator that ${\log _3}\left( {12} \right) = 2.2619$. Now, find the values of $\log_3 (36)$ , ${\log _3}\left( 4 \right)$, and $\log_3 (108)$.

According to the incomplete table below, ${\log _5}\left( {10} \right) = 1.4307$. Is this (approximately) correct? Correct it if not, and then complete the rest of the table.

Suppose we visualize some number $L$ as a tree diagram with $L$ leaves, and another number $M$ as a tree with $M$ leaves. Using these as building blocks, how could you build a tree to visualize the quantity $L \cdot M$? Can you use this to illustrate the multiplication and division identities of logarithms you proved? (It might help to use specific numbers first: try $L = 27$ and $M = 9$.)

Consider a balanced binary tree with a depth of $D$.

True or false: the total number of nodes in the tree will be $1 + 2 + {2^2} + ... + {2^D}$.

Find a simple formula for the total number of nodes in the tree. Answer in terms of $D$, and without using “…”. (Hint: try several small examples, and look for a pattern.)

If a balanced binary tree has 1023 nodes total, then how many of the nodes are leaves?

Have one of your group members make up a binary tree and show it to you. Your job is to add nodes to the tree until you have doubled the total number of nodes, while keeping the tree as “shallow” as possible. Try this a few times with trees of different sizes and shapes. If the original tree has $N$ nodes, then how many additional levels of depth do you need?

If you build a binary tree by adding the numbers 5, 2, 3, 7, 11, 17, and 13, in that order, the resulting tree would not be balanced. (Check this.) Instead of this, suppose you put these seven numbers into a hat and randomly chose one number at a time to add to the tree. What is the probability that your tree would be balanced?

Prove that each of the following are true:

${\log _b}\left( {{x^2}} \right) = 2 \cdot {\log _b}\left( x \right)$

${\log _b}({x^3}) = 3 \cdot {\log _b}(x)$

${\log _b}\left( {{x^4}} \right) = 4 \cdot {\log _b}\left( x \right)$

${\log _b}\left( {{x^5}} \right) = 5 \cdot {\log _b}\left( x \right)$

Use the rule you just proved in the previous problem to solve each of the following for $x$, and give your answer to the nearest thousandth. (Hint: $A = C$ if and only if ${\log _b}A = {\log _b}C$.)

${1.02^x} = 2$

${2^x} = 1000000$

${1.5^{x - 3}} = 21$

What is ${\log _{10}}\left( {{{\log }_{10}}\left( {{\mathrm{one \, googol }}} \right)} \right)$? What is ${\log _{10}}\left( {{{\log }_{10}}\left( {{{\log }_{10}}\left( {{\mathrm{one \, googolplex}}} \right)} \right)} \right)$?

(If you don’t know what a “googol” or “googolplex” is, Google them.)

Professor Arlo G. Smith poses the following questions:

What is the ratio of ${2^9}$ to ${2^5}$? What is the ratio of ${2^m}$ to ${2^n}$?

What is the ratio of $\log {2^9}$ to $\log {2^5}$? What is the ratio of $\log {2^m}$ to $\log {2^n}$?

What is the ratio of $100000000$ to $1000000$? What is the ratio of $\log 100000000$ to $\log 1000000$?

If:${\log _b}\left( X \right) = 3 \cdot {\log _b}\left( 2 \right) + 2 \cdot {\log _b}\left( 3 \right) + {\log _b}\left( 5 \right)$ then what is $X$?

Rewrite the expression ${\log _5}7 + {\log _5}15 - {\log _5}3$ as a single logarithm in the form ${\log _b}a$.

Not all calculators have a built-in way to calculate ${\log _2}\left( {47} \right)$. However, using the definition and properties of logarithms, there is a way to figure this out with any scientific calculator. Here’s a hint: re-write ${\log _2}\left( {47} \right) = x$ in exponential form, and then solve for $x$.

Can the ${\log _{10}}$ of an irrational number minus the ${\log _{10}}$ of a different irrational number ever equal a positive integer? Explain.

Solve each of the following equations, giving your answer to the nearest thousandth.

${1500 \cdot {1.13^t} = 4000}$

$3 \cdot {5^{x + 5}} = 8$

${\log _3}x - 2{\log _3}7 = 1$

${5^{x - 1}} = {2^{3x - 1}}$

$7 \cdot {x^3} = 1492$

A binary tree has 6000 nodes in it. What are its minimum and maximum possible depths?

Recall the Number Devil guessing game from the previous lesson, where the Number Devil picks a number between 1 and 15 and you have to guess it. When you guess, the Number Devil will tell you whether you’re correct, too high, or too low. If you use binary search, how many guesses will you need in the worst case? What if the Number Devil chose a number between 1 and 1 million?

That pesky bad coin you saw in the previous lesson is back. (Maybe the Number Devil brought it along.) You have 27 identical-looking coins. The bad coin is too light, and the rest all weigh exactly the same.

Using a balance scale, how many weighings do you need to find the bad coin?

How many weighings would it take to find one bad coin among 2187 coins? How about 1 million coins?

If you look up logarithms on the Web, you’re likely to see the following sentence, which is meant to be a helpful reminder when you’re learning what logarithms are:

“A logarithm is an exponent.”

What do you think people mean by this?

If $a_1,a_2, a_3, \ldots$ is a geometric sequence, then what kind of sequence is $\log_b(a_1), \log_b(a_2), log_b(a_3), \ldots$?

Recall that the Floor operation from the previous lesson takes a positive number $x$ and chops off anything after the decimal point. Similarly, the Ceiling function essentially “rounds up”: e.g., ${\mathop{\rm Ceiling}\nolimits} \left( 3 \right) = 3$, and ${\mathop{\rm Ceiling}\nolimits} \left( {2.0001} \right) = 3$. If $N$ is a positive integer, what does ${\mathop{\rm Ceiling}\nolimits} \left( {{{\log }_{10}}\left( N \right)} \right)$ tell you about $N$?

If you double a three-digit number 100 times, how many digits will the resulting number have?

Prove that squaring a number at most doubles the number of digits in the number.

Prove that ${\log _b}\left( {\frac{1}{x}} \right) = - {\log _b}\left( x \right)$.

Generalize what you did in problem 59 into an identity about logarithms.

What is ${\log _{10}}\left( 2 \right) \cdot {\log _2}\left( {10} \right)$?
How about ${\log _{10}}\left( 3 \right) \cdot {\log _3}\left( {10} \right)$?
And ${\log _7}\left( 3 \right) \cdot {\log _3}\left( 7 \right)$?

In a balanced ternary tree, every non-leaf node has three children, and all the leaves have the same depth. How many nodes are in a balanced ternary tree with a depth of 10?

Sweden and Croatia are locked in an escalating arms race. Every year, one of the countries builds some bombs and missiles, and then inevitably, the next year, the other country builds even more to compensate. In each country, the Department of Defense has the following strategy:

For every new bomb that the enemy built last year, we’ll build 2 new bombs. For every new missile that the enemy built last year, we’ll build 2 new missiles, AND a new bomb.

Neither country can afford the expense of this pointless arms race, and each one claims that the other country started the whole situation by building the first few weapons.

In 2008, Sweden built a total of 64 new bombs and 16 new missiles. What will happen in 2009? How about 2010?

Those 64 bombs and 16 missiles were constructed in response to the weapons that Croatia built in 2007. What were those weapons?

Continue working back in time to determine who must have started the situation. What year did it start, and what weapons were built?

In problem 12:

If the length of each side of the original triangle is 1 unit, write a recursion equation for ${P_n}$, the perimeter of the shape after $n$ stages. Will the perimeter ever reach 1,000,000 units?

Suppose that the area of the original triangle was 1 cm$^2$, what would be the area of the shape after 1 stage? After 2 stages? After 3 stages?

Write a recursive equation for ${A_n}$, the area of the shape after $n$ stages. Could the area get as large as 5 cm$^2$?

2000 people are exposed to a particularly contagious cold virus. Every day, $\frac{1}{4}$ of the people (rounded to the nearest person) that were healthy the day before, have now caught the virus. ${I_t}$ is the total number of people infected, after $t$ days.

If ${I_1} = 0$ (no one is infected on the first day), find ${I_t}$ up to $t = 5$.

If ${I_1} = 40$, do the same.

Write a recursive equation for ${I_t}$. (In your formula, you may use $\mathrm{Round}(x, 0)$, which is the operation on your calculator that rounds any number $x$ to the nearest integer.)

What happens eventually to these 2000 people?

Suppose that a less contagious cold virus infects 1/20 of the healthy population on a given day.

Find out how many days it takes for a population of 2000 to go from 0 sick people to half of the population being sick.

If you double the population to 4000, how many days do you think it will take for half of the population to get sick? Do the calculations to see if you are right.

You have 10 pet lizards and each gets fed 3 crickets at the end of every week. You buy 80 crickets, thinking that as the crickets reproduce at a rate of 50% per week, the population will sustain itself even though some crickets will be fed to the lizards. But the lizards reproduce at a rate of 10% per week. After the lizards and crickets both reproduce, then the lizards get fed.

At the end of the first week, how large is each population?

Draw a chart to track the populations over the next few weeks. Each week, round to the nearest whole number in your answers.

Write recursive equations for ${L_t}$ and ${C_t}$, the number of lizards and the number of crickets after t weeks respectively. Your equation for ${C_t}$, the number of crickets, will have to depend on how many crickets there were last week AND how many lizards there were this week.

Do you think you’ll always have enough crickets to feed the lizards, even though the lizard population will continue to grow?

A well-fed bacteria colony in a Petri dish grows as follows:

Find a pattern, and fill in the blanks.

Write a recursive equation for ${B_n}$, the number of bacteria after $n$ hours.

Check whether the pattern you discovered in problem 17 satisfies the equation ${B_n} = 60 * {3^{n - 1}}$. Find the size of the colony after 24 hours, by using either your recursive formula or the explicit formula.

Let the sequence $\{ {R_n}\} $ be generated by the quadratic equation ${R_n} = {n^2} + 1$.

Create a table for $\{ {R_n}\} $.

Write a recursive equation for $\{ {R_n}\} $.

Repeat problem 19 for the:

Quadratic equation, ${R_n} = 2{n^2} - 3n + 1$.

Exponential equation ${R_n = 3^n}$.

Exponential equation ${R_n} = 3 \cdot {2^n}$.

For each chart below find a recursive equation and an explicit equation.

You come across the following sequence in a textbook.

Find an explicit formula for ${T_n}$. (It might help to first carefully make a graph with $n$ as your horizontal axis and ${T_n}$ as your vertical axis.)

Find ${T_{100}}$.

A sequence follows this recursive rule: ${A_n} = {A_{n - 1}} - {A_{n - 2}}$.

Pick several different starting values for the sequence, and find the next ten or so values. What always happens?

How can you be sure of your conclusion in part a?

Write a recursive equation to describe the pattern you see in the Fibonacci sequence below.

You take out a loan from a bank. At the end of each year, you pay them \$200, and then they charge 3% interest on the amount that you still owe. For example, if you borrowed \$1200, you would owe \$1,030 after 1 year.

Find out how many years it would take to pay off your loan of $1200. (Once you hit a negative number, just assume the loan is now paid off.)

In part a, did the amount of money owed decrease faster and faster as time went on, or slower and slower? Why?

Write an equation ${M_t}$ for the amount of money you owe after t years.

What happens if instead you borrow $5000? Track the loan over at least ten years.

What happens if you borrow $10,000? Track the loan over at least ten years.

Find an initial loan amount that is an equilibrium solution.

One of the simplest models for population growth is based on the idea that the growth rate of the population is proportional to the current size of the population. So, for example, we might expect that, each year, the number of frogs in a pond will increase by 6% every year.

Write a recursive equation for this model of growth.

Does this equation have any equilibria?

The population model you looked at in the previous problem has a major limitation: it doesn’t take into account the fact that the frogs in the pond have a limited number of resources (food, lily pad real estate, etc.). The following equation, called the logistic model of population growth, attempts to take this into account:

${F_{n + 1}} = {F_n} + 1.1{F_n} \cdot \left( {\frac{{40 - {F_n}}}{{40}}} \right)$

The idea here is that, the pond has a “carrying capacity” of 40 frogs — i.e., it has enough resources for 40 frogs to live healthily. So, while the population’s growth rate is proportional to the current population size, it is also proportional to how close the population is to the carrying capacity.

Suppose that there were 15 frogs in the pond. According to the equation, how much would the population change the next year? How about if there were 35 frogs in the pond?

Does it seem plausible that this system (unlike the one in the previous problem) would have an equilibrium solution other than 0?

Starting with an initial population of 2 frogs, use the equation above to track the frog population over several years. What happens in the long run?

Predict what would happen if you repeated part c, but with an initial population of 60 frogs. Then try it and see.

Connect the midpoints of the sides of a triangle. Now connect the midpoints of the sides of your new triangle. If you could repeat this operation indefinitely it seems as though you would end up with a point. Where in the original triangle would that point be located?

Three kids sitting in a circle play a card game in which they are given a number of cards and asked on each turn to pass half of their cards to the kid sitting to their right. If they have an odd number of cards they must round up — for example, a kid who has 5 cards must pass 3. How is this game going to turn out?

Find the lowest value ever reached by the sequence $\{ {S_n}\} $, and explain why you know it’s the lowest value, given that:

${S_1} = 8000$
${S_n} = {S_{n - 1}} \cdot \frac{n}{{10}}$

A bank’s loan policy is that they calculate interest monthly: .8% per month. ${M_n}$ will represent how much money you owe the bank after $n$ months.

Suppose you start by borrowing $\$1500$ — so $M_0 = 1500$. How much do you owe after 1, 2, and 3 months?

Write a recursive equation for ${M_n}$.

If you choose to pay off the loan a little bit at a time, by paying $100 every month, calculate how much you’ll owe after 1, 2, and 3 months — assume that the \$100 is paid before interest is calculated for the month.

Write a recursive equation for ${M_n}$, under the assumptions of part c.

Now suppose the bank, hoping to make more money, decides to calculate interest every month before accepting your \$100 payment for the month. How much more will you owe at the end of 1, 2, and 3 months?

Write a recursive equation for ${M_n}$, under the assumptions of part e.

A sequence follows this recursive rule: ${F_n} = 2{F_{n - 1}}^2$.

Try starting with a few different values for ${F_1}$, and see what happens over the next 10 or so values of $n$. For which starting values do the values of the sequence increase, and for which do they decrease?

Without trying the numbers, say what will happen for the following starting values, based on what you noticed in part a — will the sequence’s values increase, decrease, or stay the same? The starting values are: 1, 1/10, ½, .99.

State and justify the “rule” you were using to make decisions in part b.

The recursive rule for a sequence $\{ {T_n}\} $ is that each term equals 30 more than the (positive) square root of the previous term.

If $T_1 = T_2$, find the number they must equal — call it $C$.

If ${T_1} > C$, what will happen to the value of ${T_n}$ as $n$ increases? How sure are you?

What if ${T_1} < C$? Are you sure?

Investigate what happens with the rule ${T_n} = \frac{{{{({T_{n - 1}})}^2}}}{n}$ when you use different starting values. You should find at least 3 different types of behavior.

For each of the sequences below an explicit equation is given. Start at $n = 1$ and figure out what happens to each sequence eventually — as $n$ gets larger and larger.

${A_n} = 1 + \frac{1}{n}$

${B_n} = {(1 + \frac{1}{n})^2}$

${C_n} = {(1 + \frac{1}{n})^3}$

${D_n} = {(1 + \frac{1}{n})^{10000}}$

Based on your answers to problem 35, what would you expect to happen to the following sequence eventually? After your guess, go ahead and check carefully on your calculator.

${E_n} = {(1 + \frac{1}{n})^n}$

You might have been a bit surprised by the answer you got for problem 36. This number happens to be quite an important number in mathematics — so important in fact that it has a special name and your calculator has a special button dedicated to it. The number is e. Check it out on your calculator.

Two animal populations are interdependent, because the predators (P) feed on the prey (Y). Here are recursive equations for the sizes of each population, in terms of time $t$ (in years):

${Y_t} = (1.03 - .001{P_{t - 1}}){Y_{t - 1}}$${P_t} = (.99 + .00002{Y_{t - 1}}){P_{t - 1}}$

If the starting population of predators is 0 and the starting population of prey is 100 (at time $t = 0$), what will happen to the prey population over time? Why does this make sense in context of the situation?

If on the other hand $Y_0 = 0$ but $P_0 = 50$, what will happen to the predator population over time? Why does this make sense?

If the starting populations are $Y_0 = 450$ and $P_0 = 45$, track what happens in the first ten years.

What should the starting populations be, for both populations to remain stable? Use what you saw in part c, or look at the equations themselves to figure it out.

You invest in a stock through a broker. ${M_t}$ is how much money you have after $t$ years. Every year, the value of your stock increases by 9%, but your stockbroker also charges a fee at the end of each year — $\$100$ the 1st year, $\$200$ the 2nd year, $\$300$ the third year, and so on.

If you invest $\$3000$ initially, how much money do you have each year over the next five years? What’s the trend?

Write a recursive equation for $\{ {M_t}\} $.

How much money would you have to start with, so that after 1 year you have exactly the same amount of money?

If you invested more at the beginning, would your money always grow, or would it still eventually decrease?

Don’t use a calculator for this problem.

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

Simplify $\frac{\sqrt{6} \sqrt{15}}{\sqrt{10}}$

Simplify ${\left( {\frac{{2{x^2}{y^{ - 3}}}}{{3{x^{ - 1}}{y^4}}}} \right)^{ - 2}}$

Add $\frac{x}{5} + \frac{4}{x}$

Simplify $\frac{{\sqrt {48} }}{8}$

Write a concluding statement for the following. If one is not a croot, then one is not a lurl. If one is a bleep, then one is a gurm. If one is a pift, then one is a lurl. If one is a croot, then one is not a gurm.

Therefore, if . . . . . . . then . . . . . . .

(Make sure your conclusion makes use of all of the postulates.)

What conclusion can you draw from the following?

If the triangle is acute, then the problem will be easy. If you are in Tony’s class, then the problem will not be easy. If the triangle is acute, then you are in Tony’s class.

Back to problem 13.

Prove that there exist 3 and only 3 lines. (You may use the theorem you proved in part b as well as postulates 4 and 1, and proof by contradiction.)

Try to draw a model that is different from your model in part a of problem 13.

Determine the consistency of each of the following systems.

In this system, the following 3 axioms hold. Squirrels and trees are the undefined terms. Postulate 1. There are exactly 3 squirrels. Postulate 2. Every squirrel climbs at least 2 trees. Postulate 3. No tree is climbed by more than 2 squirrels.

In this system, the following 4 axioms hold. Hearts, spades and trump(s) are the undefined terms. Postulate 1. There is at least one heart. Postulate 2. For each pair of spades, there is one and only one heart that trumps them. Postulate 3. Each heart trumps at least 2 spades. Postulate 4. Given any heart, there is at least one spade it does not trump.

In this system, the following 5 postulates hold. Lines and points are the undefined terms.

Postulate 1. Each pair of lines has at least one point in common. Postulate 2. Each pair of lines has no more than one point in common. Postulate 3. Every point is on at least 2 lines. Postulate 4. Every point is not on more than 2 lines. Postulate 5. There exist exactly 4 lines.

Is the system consistent?

Prove that every pair of lines has exactly one point in common.

Prove that every point belongs to exactly 2 lines.

Prove that there are exactly 6 points.

Prove that every line contains exactly 3 points.

Are there parallel lines in this system?

Here’s a chance for you to create your own geometry.

Invent an axiom system and a model for it.

Prove a theorem in your geometry.

Now add an additional axiom that would make the system inconsistent.

Use postulates to describe a geometry that would fit the model below.

Three large containers are filled with marbles. One is filled with red marbles, one with blue, and one with a mixture of blue and red marbles. The problem is that the containers have all been mislabeled, plus none of the containers has the correct label. Describe how you would correctly label the containers looking at the least number of marbles.

And how about this quite famous one. Someone is looking at a picture and says: “Brothers and sisters have I none, but this man’s father is my father’s son.” Who is in the picture?

In this system, the following 6 postulates hold. Lines and points are the undefined terms.

Postulate 1. Given any two points, there is at least one line that contains them. Postulate 2. Given any two points, there is at most one line that contains them. Postulate 3. Given any two lines, there is at least one point that is contained in both lines. Postulate 4. Given any line, there are at least three points that are contained in the line. Postulate 5. Given any line, there is at least one point that is not on the line. Postulate 6. There is at least one line.

Are there parallel lines in this system?

Prove that there exists at least one point.

Prove that any 2 points are on exactly one line.

Prove that 2 lines have exactly one point in common.

Prove that if P is any point, there is at least one line that does not contain P.

Prove that every point lies on at least 3 lines.

As we did in problem 15, let’s make problem 30 a little less abstract.

Replace undefined terms “points” and “lines” by new undefined terms (you use “students” and “committees” again if you like), and rewrite the postulates, changing the language a bit to accommodate the new terms.

Draw a chart that would represent the postulates. How many “points” did you end up with? How many “lines”?

Would the addition of another “point” or “line” violate any of the postulates?

Is the following diagram a model for the system in problem 30? If you think not, explain.

Decide whether or not the following system is consistent:

Postulate 1. The town of Fairhaven consists exclusively of married couples and their children. Postulate 2. There are more adults than children. Postulate 3. Every boy has a sister. Postulate 4. There are more boys than girls. Postulate 5. There are no childless couples.

A system is considered $dependent$ if at least one of the axioms follows from the other axioms; otherwise it is considered $independent$. To show that a system is dependent, it is only necessary to show that one axiom follows from the others. However to show that a system is independent, one must show that no one axiom follows from the others.

Construct two systems, one independent and one dependent.

Don’t use a calculator for this problem.

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

Subtract $\frac{{16}}{x} - \frac{{4 + x}}{{{x^2}}}$

Simplify $\sqrt {75} - \sqrt {24} + \sqrt {18} $ z

Solve ${x^2} + 3x = 2$