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I'm pretty close to an exam where I have to answer some questions about Haskell and Prolog.
I would like to find a web like "learnyouahaskell" but about Prolog.
I don't know why logical languages are harder for me, they aren't logical for my knowledge =.=
Any recommends?
Thanks in advance.
I think that the equivalent of leanyouahaskell would be Learn Prolog Now.
also, check this question
It might also help to re-evaluate (or rather backtrack xD) your definition of logical;
for example, when you have
x=3
2x=x+1
you would normally conclude that there is no x that satisfies this system of equations, not that x is first 3 and then 1 :b
anyway, it's true that it's a bit hard to adjust; good luck with your exam!
also this is a very good website for understanding the basics with examples, diagrams and so on:
http://www.csupomona.edu/~jrfisher/www/prolog_tutorial/contents.html
AllDistinct(a1 , . . . , an )
if (n = 1)
return True
for i := n down to 2
begin
if (LinearSearch(a1 , . . . , ai−1 ; ai ) != 0)
return False
end
return True
Give a big-O bound on the running time of AllDistinct. For full credit, you must show
work or briefly explain your answer.
So the actual answer for this according to the solution for this problem is O(n^2). However, since BigO is the worst case running time, could I have answered O(n^100000) and gotten away with it? Theres no way they can take points off for that since its technically the correct answer right? Although the more useful O(n^2) is obvious in this algorithm, I ask because we might have a more difficult algorithm on the upcoming exam, and incase I cant figure out the 'tight' bound, I could make up some extremely large value and it should still be correct, right?
Yes, if a function is in O(n^2), it is also in O(n^1000).
Whether you'll get full (or any) credit for answering this way depends on the person grading your exam of course, so I can't tell you that (probably not though). But yes, it is technically correct.
If you do decide to go this way though, you should probably chose something like O(n^n) or O(Ackermann(n)), since for example exponential functions are not in O(n^1000).
Another problem is that you will probably be asked to proof the bound as well. This will be hard to do if you don't actually know the running time of the function. "n^n is really large, so the running time will probably be less than that" is not a proof. Though on the upside if you manage to correctly proof that the function is in O(n^n), you'll probably get at least partial credit.
That would be a trivial answer to the question. Although correct, it tells you nothing and is thus worthless. It's not about right or wrong, it's about bad and good. The better your answer, the more points you'll get for it. The question does not say you'll get credit for a terrible bad bound. Bad answers give bad marks?
(Asking for Big Theta would be a harder question. I would play nice :)
No.
It might be all clever and HA! I got you! but that's not the idea. (and you know that)
If the professor ask you for BigO of it you can answer whatever BigO you think but you must prove it as it say For full credit, you must show work or briefly explain your answer.
BigO is not useless. For problems it's easy to get Upper Bound (BigO) graeter; e.g.
Sorting problem: you have the simple bubble sort and you can proof that is n^2 (right?), so upper bound of Sorting problem is n^2 (because exists and algorithm that solve it in that time, but if you go on with maths, you see that the problem has a lower bound of log(n!) ). So n^2 was a good answer until you proof it's log(n!). There are many problems that we know just BigO but not the lower bound, so it's not useless.
If you can say that a program halts you can always compute is BigO with some math, but is not always easy (exists even ammortized complexity) but it's simpler than problems lowerBound. So BigO is not so important in algorithm, but it's not useless.
The important thing is that you understand what it means; then if you can get any BigO of that program you can write it on that exam paper that is a function from Student to number.. and good luck.
At a guess, you'd probably have to talk to the professor, and argue with him a bit to even get partial credit for an answer like that. Depending on how much he values theory vs. practicality, he might give you partial credit, or he might give no credit -- but I can hardly imagine a professor who'd give any credit without your explicitly pointing out how it's (semi-)correct, and some might not even then.
I was a prof. Profs make up exam questions, and those can have bugs. It's embarrasing when you have to throw out a question because it's got a bug and people can give trivial answers. In this case the bug is "a big-O bound". Making exam questions is tricky, because you don't want to err on the side of saying too much, like some kind of airtight lawyer statement, because that will confuse people even more.
After all, the reason for doing this is, hopefully, you'll learn something useful. If you see an ambiguous question like this, the prof will appreciate it if you say something like "I assume you mean a good big-O bound."
I am reading 'The Little Schemer' in an effort to better understand some of the core elements of programming (namely recursion) and to get more of an idea how to think like a programmer.
The book comes recommended as an entry-level book and the introduction states that all I need to know are English, numbers and counting (which I do).
I am kind of confused though as the first section and questions start off by asking "Is it true that this is an atom?"
Am I missing something? Am I supposed to know what an atom is? I am confused as I thought it was meant to be in more plain English.
Thanks in advance,
Tim
It can be a hard book to get into; it took me two tries separated by about a year. The way you to read it is that you are figuring out these concepts for yourself by listening in on a dialogue between two other people. The first question about a concept will lose you, but the hope is that you say, "Aha! I've figured out the concept they must be talking about" before the end of the questions on a given topic. By the end of the section you'll be answering the questions yourself before reading the answers in the book.
If you hit the end of a section and haven't gotten to that point, start over again but try to give the answers yourself without reading them. When you can supply the answers yourself, you've either figured out the concept in your own terms or memorized the answers in the book. Later sections will refer back to these concepts, though, and will reinforce your understanding.
Think of the student in the book as a proxy for you who seems to begin each section smarter than you, but who you outpace by the end of the section.
The book uses a sort of "constructivist" learning model. It asks you to figure things out before you know the formal definitions. The idea is to develop an intuition before formality (I believe, although that may not be the intention of the authors). You may find this annoying at first, but when you get to the higher-level concepts, you will find yourself understanding things way better than you would from reading R5RS, for example. Continuations had me completely baffled until I read all the way through this book. Stick with it and you'll get why the authors take this approach.
On the left of the page:
"Is it true that this is an atom?
atom"
On the right of the page, 2cm away:
"Yes, because atom is a string of
characters beginning with the letter
a".
And similar questions and answers about atoms in the same format for the remainder of the page. I don't think it takes a genius to work out what is going on here.
An atom in Scheme is like in english, something that you can't divide.
Here are some atoms:
'foo 'bar 'baz 123 '() '+
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Closed 10 years ago.
First of all, sorry, but my English is not very good.
I'm at the third semester at a programming course and I can't "get it".
I passed previous exams because I've studied three times the amount my colleagues did. And, now in my Data Structures class when the professor asks us to make list algorithms ( for example ), my colleagues just start to write the code while me ( or would be "I"? ), even after seeing it done, can't understand.
I find it so hard it's painful. It takes me 1 hour to understand simple algorithms.
So the question is: can anyone recommend me books or something, that will help me think more like a programmer? I mean, "normal" people, after a given problem, seem to immediately make the picture in their head and star to write the code.
you might want to take a philosophy class on intro to logic. it's exactly the same thing as you're doing with computers, but in a different context.
I think there are two ways to learn to write algorithms:
1. Try it yourself (and then review it with the "right" answer). Of course you should start with the simple ones.
2. Implement other's algorithms (to code). Going from algorithms to code and the other way around gives you a great "feeling" of how algorithms are created.
But the best advice I can give you (which you can use the two ways above to acquire) - try being very good at some algorithms.
If you could remember, write and really understand even just a few basic algorithms - you're on the right way.
Anyways, I think learning algorithms is mainly practice and less "abstract knowledge", so get ready to get your hands dirty..
Best of luck!!
I found that the most important skill for mastering data structures is to "visualize" them. If you have no clear picture of your data structure, everything stays fuzzy and vague. E.g. take a stack (single linked list). It may help to visualize it as a stack of plates or people in a polonaise (everyone has the hands on the shoulders of the person in front of him).
Or a double linked list: Imagine a row of people grabbing the belt of the left and right neighbor. Now you can imagine what you must do to remove one person: Its left neighbor needs to grab the belt of the right neighbor and the right neighbor the one of the left neighbor. Then you can "delete" that person (or the garbage collector does this for you in Java etc)
The second important skill is to understand recursion. You need always a base case, usually involving an empty list or empty node. When you follow the algorithm in the recursive method, check that it is correct, but don't follow recursive calls inside that method. That will drive you mad and lead to nothing. Just assume that recursive calls always "do the right thing". If your current method is correct and the base case as well, you are done. Once you understood this, you understand recursion.
If it helps, you might want to try books in your native language. If there is any gap in your understanding that comes from your English comprehension, that would add additional difficulty for you.
Second, an algorithm, is just a list of steps for achieving something. Try to focus on something simple like sorting algorithms, they are fun and easy to learn. Hope that helps.
I do not know how far this problem is concerned, but it might help.
Algorithms and data structures are way cleaner for me if I think of them just as informal ideas/boxes/shapes/forms/connections/... instead of abstract things.
That is, everytime I come across a new data structure I try to visualize it somehow in order to actually beeing able to "see" what the single operations do with the structure.
I had also sometimes the problem that I didn't actually know instantly how to solve something algorithmically, so I just went ahead and started drawing.
Let's take - as an example - the problem of converting a binary tree into a list. So, I draw a tree (of arbitrary size and holding arbitrary elements, but beeing a "good" example). Then I think how I would convert the tree into a list manually:
1
2 3
4 5 6 7
So I think: I want to have the result [4,2,5,1,6,3,7]. How can I do this step-by-step?
Ok, first of all, I find the leftmost element (4); how can I do this? Ok, I start just by the root and go left until there's nothing more. Then I remove this element and go on with the remaining tree:
1
2 3
. 5 6 7
Ok, now I would pick the 2. How can I reach the 2? Ok, I can either start again at the root and go left until there's nothing more or i just go back from the deleted node.
Then I go on and look for recurring patterns, how one can generalize steps etc.
The outcome is, it might often be helpful to have a visual representation of what the algorithm is doing and then you can implement it.
I think it is a good practice to use diagrams before implementing anything. You can make diagrams or write a pseudo code, they are helpful before you start to design or implement the actual code.
I had a similar problem when I was first introduced to programming languages. I missed a lot of lectures because it was my first year of college! For me there was no books or lecturers that I found that knew how to help you think like a programmer. I always found the people teaching didn't know how 'not' to think like a programmer anymore and as a result they assume you know the simple concepts. So finally by the end of my first year I had to cram majorly to catch up and and had to fill in the gaps myself...! This is how I think about programming problems now:
OBJECTS: For Object-Oriented programming objects are the key to the whole thing. If you think about what it is your program needs to be able to do then you can break up the program into smaller chunks. For example, if you imagine making a cup of tea, the objects you need to make the cup of tea are:
1 -> A cup
2 -> A tea bag
3 -> Water
4 -> A kettle
5 -> A spoon
6 -> Milk
7 -> Sugar
So now your program has 7 objects which will interact in some way to make a cup of tea. Objects are always declared as their own class and will have constructor methods which when called will create a copy (instantiation) of your object that can then be used in your program. All of the method that are inside your class will define what functionality your object can provide.
Kettle kettle = new Kettle();
kettle.boilWater();
So now that you have your objects you should think about your algorithm.
ALGORITHMS: In all programming languages an algorithm is basically just a list of steps that you take that will help you achieve your end goal. In our case our end goal is to make a cup of tea.
The steps you will take in your algorithm have to come one after the other in a logical fashion i.e. You cannot pour the milk into the kettle, or pour cold water into the cup and boil the sugar etc.
So our algorithm can be as follows:
Step 1: Pour water into Kettle
Step 2: Turn kettle on - to boil the water
Step 3: Put tea-bag into cup
Step 4: "IF" water is boiled -> pour into cup
"ELSE" wait until water has boiled
Step 5: Stir teabag with spoon
Step 6: Pour milk into cup
Step 7: Put sugar into cup
Step 8: Stir
There are always a few different ways you can arrange the steps in an algorithms that will still work but always remember to have a logical order or else you will make a mess!!
The same principle can be applied to even the most complex problems. The most important thing to do is try to break the problem down into the simplest steps and arrange the steps in a common sense way.
When it comes to more complex tasks it is obviously very important to know what tools you have available to you i.e. know what functionality APIs provide you with and be familiar with the syntax. But as people have mentioned to you already before practice makes perfect. It is the only way that you will begin to understand it and believe me you will get it eventually... One day it WILL all make sense to you it is just about thinking a certain way. Break everything down to small simple steps and then order the steps in a logical way. Do this and it will start to make sense to you. I PROMISE!!
If you want to jump in the deep end and are willing to work hard, there is "Structure and Interpretation of Computer Programs". It is out of print but available on the net.
It goes deep into conceptual thinking processes behind programming and converting them to code. It uses scheme, but the principles are applicable everywhere and it is great training for flexing the abstraction muscles.
Don't do it in one go... take your time with it, come back regularly to it, and have fun!
Try to think about how you approach everyday cognitive problems, and formalize the steps you take to solve them, on paper.
For example, if you've ever had to reorder a deck of 52 jumbled cards, you probably know Insertion sort or some variant of Merge Sort already. If you're asked to sort five 2-digit numbers in your head, you're probably using Selection Sort. If you've had to find a specific card from a jumbled deck, you probably used Linear Search. If you've ever seen the 'High Low Game' on 'The Price is Right', you probably know Binary Search. If you're asked to solve the 8-Queens problem as a 'puzzle', you will almost certainly use the classic backtracking technique.
Don't let jargon like 'loop invariant' or 'asymptotic time complexity' intimidate you straight away. Think about how an algorithm is dealing with the problem, and you will find yourself 'rediscovering' these terms yourself when you want to reason about its correctness or efficiency.
I'd suggest you could try out some of the websites which contain a number of algorithmic problems to solve. Usually you could find a number of very easy problems to start with, and forums to discuss them. Then it's up to you and how much time you spend on them. Just try to go into as much detail as possible and question yourself what exactly is it that you don't understand, then how could you figure that out (maybe someone already had figured that out and you just need to find the answer). For the easiest problems Google is more than enough to find the answers you're looking for.
And here is the list of websites you could try:
uva.onlinejudge.org - Huge list of problems related to different algorithms.
www.topcoder.com/tc - Live competitions and lots of tutorials to get started.
www.algorithmist.com - Constains a list of links collected over time by problem-solvers.
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Closed 10 years ago.
I'm about to start (with fellow programmers) a programming & algorithms club in my high school. The language of choice is C++ - sorry about that, I can't change this. We can assume students have little to no experience in the aforementioned topics.
What do you think are the most basic concepts I should focus on?
I know that teaching something that's already obvious to me isn't an easy task. I realize that the very first meeting should be given an extreme attention - to not scare students away - hence I ask you.
Edit: I noticed that probably the main difference between programmers and beginners is "programmer's way of thinking" - I mean, conceptualizing problems as, you know, algorithms. I know it's just a matter of practice, but do you know any kind of exercises/concepts/things that could stimulate development in this area?
Make programming fun!
Possible things to talk about would be Programming Competitions that either your club could hold itself or it could enter in locally. I compete in programming competitions at the University (ACM) level and I know for a fact that they have them at lower levels as well.
Those kind of events can really draw out some competitive spirit and bring the club members closer.
Things don't always have to be about programming either. Perhaps suggest having a LAN party where you play games, discuss programming, etc could be a good idea as well.
In terms of actual topics to go over that are programming/algorithm related, I would suggest as a group attempting some of these programming problems in this programming competition primer "Programming Challenges": Amazon Link
They start out with fairly basic programming problems and slowly progress into problems that require various Data Structures like:
Stacks
Queues
Dictionaries
Trees
Etc
Most of the problems are given in C++.
Eventually they progress into more advanced problems involving Graph Traversal and popular Graph algorithms (Dijkstra's, etc) , Combinatrics problems, etc. Each problem is fun and given in small "story" like format. Be warned though, some of these are very hard!
Edit:
Pizza and Soda never hurts either when it comes to getting people to show up for your club meetings. Our ACM club has pizza every meeting (once a month). Even though most of us would still show up it is a nice ice breaker. Especially for new clubs or members.
Breaking it Down
To me, what's unique about programming is the need to break down tasks into small enough steps for the computer. This varies by language, but the fact that you may have to write a "for loop" just to count to 100 takes getting used to.
The "top-down" approach may help with this concept. You start by creating a master function for your program, like filterItemsByCriteria();
You have no idea how that will work, so you break it down into further steps:
(Note: I don't know C++, so this is just a generic example)
filterItemsByCritera() {
makeCriteriaList();
lookAtItems();
removeNonMatchingItems();
}
Then you break each of those down further. Pretty soon you can define all the small steps it takes to make your criteria list, etc. When all of the little functions work, the big one will work.
It's kind of like the game kids play where they keep asking "why?" after everything you say, except you have to keep asking "how?"
Linked lists - a classic interview question, and for good reason.
I would try to work with a C subset, and not try to start with the OO stuff. That can be introduced after they understand some of the basics.
Greetings!
I think you are getting WAY ahead of yourself in forcing a specific language and working on specific topics and a curriculum.. It sounds like you (and some of the responders) are confusing "advising a programming club" with "leading a programming class". They are very different things.
I would get the group together, and the group should decide what exactly they want to get out of the club. In essence, make a "charter" for the club. Then (and only then) can you make determinations such as preferred language/platform, how often to meet, what will happen at the meetings, etc.
It may turn out that the best approach is a "survey", where different languages/platforms are explored. Or it may turn out that the best approach is a "topical"one, where there topic changes (like a book club) on a regular basis (this month is pointers, next month is sorting, the following is recursion, etc.) and then examples and discussions occur in various languages.
As an aside, I would consider a "language-agnostic" orientation for the club. Encourage the kids to explore different languages and platforms.
Good luck, and great work!
Well, it's a programming club, so it should be FUN! So I would say dive into some hand on experience right away. Start with explaining what a main() method is,then have students write a hello world program. Gradually improve the hello world program so it has functions and prints out user inputs.
I would say don't go into algorithm too fast for beginners, let them play with C++ first.
Someone mentioned above, "make programming fun". It is interesting today that people don't learn for the sake of learning. Most people want instant gratification.
Teach a bit of logic using Programming. This helps with(and is) problem solving. The classing one I have in my head are guessing games.
Have them make a program that guesses at a number between 0 and 100.
Have them make a black jack clone ... I have done this in basic :-(
Make paper instructions.
Explain the "Fried eggs" story. Ask the auditory what they would do to make themselves fried eggs. Make them note the step they think about. Probably you will receive less than 5 steps algorithm. Then explain them how many steps should be written down if we want to teach a computer to fry eggs. Something like:
1) Go to the Fridge
2) Open the fridge door
3) Search for eggs
4) If there are no eggs - go to the shop to buy eggs ( this is another function ;) )
5) If there are eggs - calculate how many do you need to fry
6) Close the fridge door
7) e.t.c. :)
Start with basics of C - syntax semantics e.t.c, and in parallel with that explain the very basic algorithms like bubble sort.
After the auditory is familiar with structured programming (this could take several weeks or months, depending how often you make the lessons), you can advance to C++ and OOP.
The content in Deitel&Deitel's C++ programming is a decent introduction, and the exercises proposed at the end of each chapter are nice toy problems.
Basically, you're talking about:
- control structures
- functions
- arrays
- pointers and strings
You might want to follow up with an introduction to the STL ("ok, now that we've done it the hard way... here's a simpler option")
Start out by making them understand a problem like for instance sorting. This is very basic and they should be able to relate quite fast. Once they see the problem then present them with the tools/solution to solve it.
I remember how it felt when I first was show an example of merge-sort. I could follow all the steps but what the hell was I for? Make then crave a solution to a problem and they will understand the tool and solution much better.
start out with a simple "hello world" program. This introduces fundamentals such as variables, writing to a stream and program flow.
Then add complexity from there (linked lists, file io, getting user input, etc).
The reason I say start with hello world is because the kid will get to see a running program really quick. It's nearly immediate feedback-as they will have written a running program right from the start.
IMO, Big-O is one of the more important concepts for beginning programmers to learn.
Have a debugging contest. Provide code samples that include a bug. Have a contest to see who can find the most or fastest.
There is an excellent book, How Not to Program in C++, that you could use to start with.
You always learn best from mistakes and I prefer to learn from some else's.
It will also let those with little experience learn by see code, even if the code only almost works.
In addition to the answers to this question, there are certain important topics to cover. Here's an example of how you could structure the lessons.
First Lesson: Terminology and Syntax
Terminology to cover: variable, operator, loop (iteration), method, reserved word, data type, class
Syntax to cover: assignment, operation, if/then/else, for loop, while loop, select, input/output
Second Lesson: Basic Algorithm Construction
Cover a few simple algorithms, involving some input, maybe a for or a while loop.
Third Lesson: More Advanced Algorithm Topics
This is for things like recursion, matrix manipulation, and higher-level mathematics. You don't have to get into too complex of topics, but introduce enough complexity to be useful in a real project.
Final Lesson: Group Project
Make a project that groups can get involved in doing.
These don't have to be single day lessons. You can spread the topics across multiple days.
Pseudocode should be a very first.
Edit: If they are total programming beginners then I would make the first half just about programming. Once you get to a level where talking about algorithms would make sense then pseudocode is really important to get under the nails.
Thanks for your replies!
And how would you teach them actual problem solving?
I know a bunch of students that know C++ syntax and a few basic algorithms, but they can't apply the knowledge they know when they solve real problems - they don't know the approach, the way to transcribe their thoughts into a set of strict steps. I do not talk about 'high-level' approaches like dynamic programming, greedy etc., but about basic algorithmic mindset.
I assume it's just because of the poor learning process they were going through. In other sciences - math, for example - they are really brilliant.
Just because you are familiar with algorithms does not mean you can implement them and just because you can program does not mean you can implement an algorithm.
Start simple with each topic (keep programming separate from designing algorithms). Once they have a handle on each, slowly start to bring the two concepts together.
Wow. C++ is one of the worst possible languages to start with, in terms of the amount of unrelated crap you need to get anything working (Java would be slightly worse, I guess).
When teaching beginners in a boilerplate-heavy environment, it's usual to start with "here's a simple C program. We'll discuss what all this crap at the top of the file is for later, but for now, concentrate on the lines between 'int main(void)' and the 'return' statement, which is where all the useful work is accomplished".
Once you're past that point, basic concepts to cover include the basic data structures (arrays, linked lists, trees, and dictionaries), and the basic algorithms (sorting, searching, etc).
Have your club learn how to actually program in any language by teaching the concepts of building software. Instead of running out an buying a dozen licenses for Visual Studio, have students use compilers, make systems, source files, objects and librarys in order to turn their C code into programs. I feel this is truly the beginning and actually empowers these kids to understand how to make software on any platform, without crutches that many educational institutions like to rely on.
As for the language of choice - congratulations - you'll find C++ is very rich in making you think of mathematical shortcuts and millions of ways to make your code perform even better (or to implement fancy patterns).
To the question: When I was beggining to program I would always try to break down one real life problem into several steps and then as I see similarity between tasks or data they transform I would always try to find a lazier, easier, meanier way to implement it.
Elegance came after when learning patterns and real algorithms.
Hank: Big O??? you mean tell beginning programmers that their code is of O(n^2) and yours is of n log n ??
I could see a few different ways to take this:
1) Basic programming building blocks. What are conditional statements, e.g. switch and if/else? What are repetition statements, e.g. for and while loops? How do we combine these to get a program to be the sequence of steps we want? You could take something as easy as adding up a grocery bill or converting temperatures or distances from metric to imperial or vice versa. What are basic variable types like a string, integer, or double? Also in here you could have Boolean Algebra for an advanced idea or possibly teach how to do arithmetic in base 2 or 16 which some people may find easy and others find hard.
2) Algorithmically what are similar building blocks. Sorting is a pretty simple topic that can be widely discussed and analysed to try to figure out how to make this faster than just swapping elements that seem out of order if you learn the Bubblesort which is the most brain dead way to do.
3) Compiling and run-time elements. What is a call stack? What is a heap? How is memory handled to run a program,e.g. the code pieces and data pieces? How do we open and manipulate files? What is compiling and linking? What are make files? Some of this is simple, but it can also be eye-opening just to see how things work which may be what the club covers most of the time.
These next 2 are somewhat more challenging but could be fun:
4) Discuss various ideas behind algorithms such as: 1) Divide and conquer, 2) Dynamic programming, 3) Brute force, 4) Creation of a data structure, 5) Reducing a problem to a similar one already solved for example Fibonacci numbers is a classic recursive problem to give beginning programmers, and 6) The idea of being, "greedy," like in a making change example if you were in a country where coin denominations where a,b, and c. You could also get into some graph theory examples like a minimum weight spanning tree if you want something somewhat exotic, or the travelling salesmen for something that can be easy to describe but a pain to solve.
5) Mathematical functions. How would you program a factorial, which is the product of all numbers from 1 to n? How would you compute the sums of various Arithmetic or Geometric Series? Or compute the number of Combinations or Permutations of r elements from a set of n? Given a set of points, approximate the polynomial that meets this requirement, e.g. in a 2-dimensional plane called x and y you could give 2 points and have people figure out what are the slope and y intercept if you have solved pairs of linear equations already.
6) Lists which can be implemented using linked lists and arrays. Which is better for various cases? How do you implement basic functions such as insert, delete, find, and sort?
7) Abstract Data Structures. What are stacks and queues? How do you build and test classes?
8) Pointers. This just leads to huge amounts of topics like how to allocate/de-allocate memory, what is a memory leak?
Those are my suggestions for various starting points. I think starting a discussion may lead to some interesting places if you can get a few people together that don't mind talking on the same subject week after week in some cases as sorting may be a huge topic to cover well if you want to get into the finer points of things.
You guys could build the TinyPIM project from "C++ Standard Library from Scratch" and then, when it's working, start designing your own extensions.