Related
I am interested in simulating the phenomenon of "regression to the mean". Say a 0-1 vector V of length N is "gifted" if the number of 1s in V is greater than N/2 + 5*sqrt(N).
I want Maple to evaluate a string of M 0-1 lists, each of length N, to determine whether they are gifted.
Then, given that list V[i] is gifted, I want to evaluate the probability that list V[i+1] is gifted.
So far my code is failing in a strange way. So far all the code is supposed to do is create the list of sums (called 'total') and the list 'g' which carries a 0 if total[i] <= N/2 + 5sqrt(N), and a 1 otherwise.
Here is the code:
RS:=proc(N) local ra,i:
ra:=rand(0..1):
[seq(ra(),i=1..N)]:
end:
Gift:=proc(N,M) local total, i, g :
total:=[seq(add(RS(N)),i=1..M)]:
g:=[seq(0,i=1..M)]:
for i from 1 to M do
if total[i] > (N/2 + 5*(N^(1/2))) then
g[i]:=1
fi:
od:
print(total, g)
end:
The trouble is, Maple responds, when I try Gift(100,20),
"Error, (in Gift) cannot determine if this expression is true or false: 5*100^(1/2) < -2"
or, when I try Gift(10000,20), "Error, (in Gift) cannot determine if this expression is true or false: 5*10000^(1/2) < -103."
Where are these negative numbers coming from? And why can't Maple tell whether 5(10000)^{1/2} < -103 or not?
The negative quantities are simply the part of the inequality that results when the portion with the radical is moved to one side and the purely rational portion is moved to the other.
Use an appropriate mechanism for the resolution of the conditional test. For example,
if is( total[i] > (N/2 + 5*N^(1/2)) ) then
...etc
or, say,
temp := evalf(N/2 + 5*N^(1/2));
for i from 1 to M do
if total[i] > temp then
...etc
From the Maple online help:
Important: The evalb command does not simplify expressions. It may return false for a relation that is true. In such a case, apply a simplification to the relation before using evalb.
...
You must convert symbolic arguments to floating-point values when using the evalb command for inequalities that use <, <=, >, or >=.
In this particular example, Maple chokes when trying to determine if the symbolic square root is less than -2, though it tried its best to simplify before quitting.
One fix is to apply evalf to inequalities. Rather than, say, evalb(x < y), you would write evalb(evalf(x < y)).
As to why Maple can't handle these inequalities, I don't know.
I'm working on a homework problem that asks me this:
Tiven a finite set of numbers, and a target number, find if the set can be used to calculate the target number using basic math operations (add, sub, mult, div) and using each number in the set exactly once (so I need to exhaust the set). This has to be done with recursion.
So, for example, if I have the set
{1, 2, 3, 4}
and target 10, then I could get to it by using
((3 * 4) - 2)/1 = 10.
I'm trying to phrase the algorithm in pseudo-code, but so far haven't gotten too far. I'm thinking graphs are the way to go, but would definitely appreciate help on this. thanks.
This isn't meant to be the fastest solution, but rather an instructive one.
It recursively generates all equations in postfix notation
It also provides a translation from postfix to infix notation
There is no actual arithmetic calculation done, so you have to implement that on your own
Be careful about division by zero
With 4 operands, 4 possible operators, it generates all 7680 = 5 * 4! * 4^3
possible expressions.
5 is Catalan(3). Catalan(N) is the number of ways to paranthesize N+1 operands.
4! because the 4 operands are permutable
4^3 because the 3 operators each have 4 choice
This definitely does not scale well, as the number of expressions for N operands is [1, 8, 192, 7680, 430080, 30965760, 2724986880, ...].
In general, if you have n+1 operands, and must insert n operators chosen from k possibilities, then there are (2n)!/n! k^n possible equations.
Good luck!
import java.util.*;
public class Expressions {
static String operators = "+-/*";
static String translate(String postfix) {
Stack<String> expr = new Stack<String>();
Scanner sc = new Scanner(postfix);
while (sc.hasNext()) {
String t = sc.next();
if (operators.indexOf(t) == -1) {
expr.push(t);
} else {
expr.push("(" + expr.pop() + t + expr.pop() + ")");
}
}
return expr.pop();
}
static void brute(Integer[] numbers, int stackHeight, String eq) {
if (stackHeight >= 2) {
for (char op : operators.toCharArray()) {
brute(numbers, stackHeight - 1, eq + " " + op);
}
}
boolean allUsedUp = true;
for (int i = 0; i < numbers.length; i++) {
if (numbers[i] != null) {
allUsedUp = false;
Integer n = numbers[i];
numbers[i] = null;
brute(numbers, stackHeight + 1, eq + " " + n);
numbers[i] = n;
}
}
if (allUsedUp && stackHeight == 1) {
System.out.println(eq + " === " + translate(eq));
}
}
static void expression(Integer... numbers) {
brute(numbers, 0, "");
}
public static void main(String args[]) {
expression(1, 2, 3, 4);
}
}
Before thinking about how to solve the problem (like with graphs), it really helps to just look at the problem. If you find yourself stuck and can't seem to come up with any pseudo-code, then most likely there is something that is holding you back; Some other question or concern that hasn't been addressed yet. An example 'sticky' question in this case might be, "What exactly is recursive about this problem?"
Before you read the next paragraph, try to answer this question first. If you knew what was recursive about the problem, then writing a recursive method to solve it might not be very difficult.
You want to know if some expression that uses a set of numbers (each number used only once) gives you a target value. There are four binary operations, each with an inverse. So, in other words, you want to know if the first number operated with some expression of the other numbers gives you the target. Well, in other words, you want to know if some expression of the 'other' numbers is [...]. If not, then using the first operation with the first number doesn't really give you what you need, so try the other ops. If they don't work, then maybe it just wasn't meant to be.
Edit: I thought of this for an infix expression of four operators without parenthesis, since a comment on the original question said that parenthesis were added for the sake of an example (for clarity?) and the use of parenthesis was not explicitly stated.
Well, you didn't mention efficiency so I'm going to post a really brute force solution and let you optimize it if you want to. Since you can have parantheses, it's easy to brute force it using Reverse Polish Notation:
First of all, if your set has n numbers, you must use exactly n - 1 operators. So your solution will be given by a sequence of 2n - 1 symbols from {{your given set}, {*, /, +, -}}
st = a stack of length 2n - 1
n = numbers in your set
a = your set, to which you add *, /, +, -
v[i] = 1 if the NUMBER i has been used before, 0 otherwise
void go(int k)
{
if ( k > 2n - 1 )
{
// eval st as described on Wikipedia.
// Careful though, it might not be valid, so you'll have to check that it is
// if it evals to your target value great, you can build your target from the given numbers. Otherwise, go on.
return;
}
for ( each symbol x in a )
if ( x isn't a number or x is a number but v[x] isn't 1 )
{
st[k] = x;
if ( x is a number )
v[x] = 1;
go(k + 1);
}
}
Generally speaking, when you need to do something recursively it helps to start from the "bottom" and think your way up.
Consider: You have a set S of n numbers {a,b,c,...}, and a set of four operations {+,-,*,/}. Let's call your recursive function that operates on the set F(S)
If n is 1, then F(S) will just be that number.
If n is 2, F(S) can be eight things:
pick your left-hand number from S (2 choices)
then pick an operation to apply (4 choices)
your right-hand number will be whatever is left in the set
Now, you can generalize from the n=2 case:
Pick a number x from S to be the left-hand operand (n choices)
Pick an operation to apply
your right hand number will be F(S-x)
I'll let you take it from here. :)
edit: Mark poses a valid criticism; the above method won't get absolutely everything. To fix that problem, you need to think about it in a slightly different way:
At each step, you first pick an operation (4 choices), and then
partition S into two sets, for the left and right hand operands,
and recursively apply F to both partitions
Finding all partitions of a set into 2 parts isn't trivial itself, though.
Your best clue about how to approach this problem is the fact that your teacher/professor wants you to use recursion. That is, this isn't a math problem - it is a search problem.
Not to give too much away (it is homework after all), but you have to spawn a call to the recursive function using an operator, a number and a list containing the remaining numbers. The recursive function will extract a number from the list and, using the operation passed in, combine it with the number passed in (which is your running total). Take the running total and call yourself again with the remaining items on the list (you'll have to iterate the list within the call but the sequence of calls is depth-first). Do this once for each of the four operators unless Success has been achieved by a previous leg of the search.
I updated this to use a list instead of a stack
When the result of the operation is your target number and your list is empty, then you have successfully found the set of operations (those that traced the path to the successful leaf) - set the Success flag and unwind. Note that the operators aren't on a list nor are they in the call: the function itself always iterates over all four. Your mechanism for "unwinding" the operator sequence from the successful leaf to get the sequence is to return the current operator and number prepended to the value returned by recursive call (only one of which will be successful since you stop at success - that, obviously, is the one to use). If none are successful, then what you return isn't important anyhow.
Update This is much harder when you have to consider expressions like the one that Daniel posted. You have combinatorics on the numbers and the groupings (numbers due to the fact that / and - are order sensitive even without grouping and grouping because it changes precedence). Then, of course, you also have the combinatorics of the operations. It is harder to manage the differences between (4 + 3) * 2 and 4 + (3 * 2) because grouping doesn't recurse like operators or numbers (which you can just iterate over in a breadth-first manner while making your (depth-first) recursive calls).
Here's some Python code to get you started: it just prints all the possible expressions, without worrying too much about redundancy. You'd need to modify it to evaluate expressions and compare to the target number, rather than simply printing them.
The basic idea is: given a set S of numbers, partition S into two subsets left and right in all possible ways (where we don't care about the order or the elements in left and right), such that left and right are both nonempty. Now for each of these partitions, find all ways of combining the elements in left (recursively!), and similarly for right, and combine the two resulting values with all possible operators. The recursion bottoms out when a set has just one element, in which case there's only one value possible.
Even if you don't know Python, the expressions function should be reasonably easy to follow; the splittings function contains some Python oddities, but all it does is to find all the partitions of the list l into left and right pieces.
def splittings(l):
n = len(l)
for i in xrange(2**n):
left = [e for b, e in enumerate(l) if i & 2**b]
right = [e for b, e in enumerate(l) if not i & 2**b]
yield left, right
def expressions(l):
if len(l) == 1:
yield l[0]
else:
for left, right in splittings(l):
if not left or not right:
continue
for el in expressions(left):
for er in expressions(right):
for operator in '+-*/':
yield '(' + el + operator + er + ')'
for x in expressions('1234'):
print x
pusedo code:
Works(list, target)
for n in list
tmp=list.remove(n)
return Works(tmp,target+n) or Works(tmp,target-n) or Works(tmp, n-target) or ...
then you just have to put the base case in. I think I gave away to much.
UPDATE:
Combinatorics and unranking was eventually what I needed.
The links below helped alot:
http://msdn.microsoft.com/en-us/library/aa289166(v=vs.71).aspx
http://www.codeproject.com/Articles/21335/Combinations-in-C-Part-2
The Problem
Given a list of N symbols say {0,1,2,3,4...}
And NCr combinations of these
eg. NC3 will generate:
0 1 2
0 1 3
0 1 4
...
...
1 2 3
1 2 4
etc...
For the ith combination (i = [1 .. NCr]) I want to determine Whether a symbol (s) is part of it.
Func(N, r, i, s) = True/False or 0/1
eg. Continuing from above
The 1st combination contains 0 1 2 but not 3
F(N,3,1,"0") = TRUE
F(N,3,1,"1") = TRUE
F(N,3,1,"2") = TRUE
F(N,3,1,"3") = FALSE
Current approaches and tibits that might help or be related.
Relation to matrices
For r = 2 eg. 4C2 the combinations are the upper (or lower) half of a 2D matrix
1,2 1,3 1,4
----2,3 2,4
--------3,4
For r = 3 its the corner of a 3D matrix or cube
for r = 4 Its the "corner" of a 4D matrix and so on.
Another relation
Ideally the solution would be of a form something like the answer to this:
Calculate Combination based on position
The nth combination in the list of combinations of length r (with repitition allowed), the ith symbol can be calculated
Using integer division and remainder:
n/r^i % r = (0 for 0th symbol, 1 for 1st symbol....etc)
eg for the 6th comb of 3 symbols the 0th 1st and 2nd symbols are:
i = 0 => 6 / 3^0 % 3 = 0
i = 1 => 6 / 3^1 % 3 = 2
i = 2 => 6 / 3^2 % 3 = 0
The 6th comb would then be 0 2 0
I need something similar but with repition not allowed.
Thank you for following this question this far :]
Kevin.
I believe your problem is that of unranking combinations or subsets.
I will give you an implementation in Mathematica, from the package Combinatorica, but the Google link above is probably a better place to start, unless you are familiar with the semantics.
UnrankKSubset::usage = "UnrankKSubset[m, k, l] gives the mth k-subset of set l, listed in lexicographic order."
UnrankKSubset[m_Integer, 1, s_List] := {s[[m + 1]]}
UnrankKSubset[0, k_Integer, s_List] := Take[s, k]
UnrankKSubset[m_Integer, k_Integer, s_List] :=
Block[{i = 1, n = Length[s], x1, u, $RecursionLimit = Infinity},
u = Binomial[n, k];
While[Binomial[i, k] < u - m, i++];
x1 = n - (i - 1);
Prepend[UnrankKSubset[m - u + Binomial[i, k], k-1, Drop[s, x1]], s[[x1]]]
]
Usage is like:
UnrankKSubset[5, 3, {0, 1, 2, 3, 4}]
{0, 3, 4}
Yielding the 6th (indexing from 0) length-3 combination of set {0, 1, 2, 3, 4}.
There's a very efficient algorithm for this problem, which is also contained in the recently published:Knuth, The Art of Computer Programming, Volume 4A (section 7.2.1.3).
Since you don't care about the order in which the combinations are generated, let's use the lexicographic order of the combinations where each combination is listed in descending order. Thus for r=3, the first 11 combinations of 3 symbols would be: 210, 310, 320, 321, 410, 420, 421, 430, 431, 432, 510. The advantage of this ordering is that the enumeration is independent of n; indeed it is an enumeration over all combinations of 3 symbols from {0, 1, 2, …}.
There is a standard method to directly generate the ith combination given i, so to test whether a symbol s is part of the ith combination, you can simply generate it and check.
Method
How many combinations of r symbols start with a particular symbol s? Well, the remaining r-1 positions must come from the s symbols 0, 1, 2, …, s-1, so it's (s choose r-1), where (s choose r-1) or C(s,r-1) is the binomial coefficient denoting the number of ways of choosing r-1 objects from s objects. As this is true for all s, the first symbol of the ith combination is the smallest s such that
∑k=0s(k choose r-1) ≥ i.
Once you know the first symbol, the problem reduces to finding the (i - ∑k=0s-1(k choose r-1))-th combination of r-1 symbols, where we've subtracted those combinations that start with a symbol less than s.
Code
Python code (you can write C(n,r) more efficiently, but this is fast enough for us):
#!/usr/bin/env python
tC = {}
def C(n,r):
if tC.has_key((n,r)): return tC[(n,r)]
if r>n-r: r=n-r
if r<0: return 0
if r==0: return 1
tC[(n,r)] = C(n-1,r) + C(n-1,r-1)
return tC[(n,r)]
def combination(r, k):
'''Finds the kth combination of r letters.'''
if r==0: return []
sum = 0
s = 0
while True:
if sum + C(s,r-1) < k:
sum += C(s,r-1)
s += 1
else:
return [s] + combination(r-1, k-sum)
def Func(N, r, i, s): return s in combination(r, i)
for i in range(1, 20): print combination(3, i)
print combination(500, 10000000000000000000000000000000000000000000000000000000000000000)
Note how fast this is: it finds the 10000000000000000000000000000000000000000000000000000000000000000th combination of 500 letters (it starts with 542) in less than 0.5 seconds.
I have written a class to handle common functions for working with the binomial coefficient, which is the type of problem that your problem falls under. It performs the following tasks:
Outputs all the K-indexes in a nice format for any N choose K to a file. The K-indexes can be substituted with more descriptive strings or letters. This method makes solving this type of problem quite trivial.
Converts the K-indexes to the proper index of an entry in the sorted binomial coefficient table. This technique is much faster than older published techniques that rely on iteration. It does this by using a mathematical property inherent in Pascal's Triangle. My paper talks about this. I believe I am the first to discover and publish this technique, but I could be wrong.
Converts the index in a sorted binomial coefficient table to the corresponding K-indexes.
Uses Mark Dominus method to calculate the binomial coefficient, which is much less likely to overflow and works with larger numbers.
The class is written in .NET C# and provides a way to manage the objects related to the problem (if any) by using a generic list. The constructor of this class takes a bool value called InitTable that when true will create a generic list to hold the objects to be managed. If this value is false, then it will not create the table. The table does not need to be created in order to perform the 4 above methods. Accessor methods are provided to access the table.
There is an associated test class which shows how to use the class and its methods. It has been extensively tested with 2 cases and there are no known bugs.
To read about this class and download the code, see Tablizing The Binomial Coeffieicent.
This class can easily be applied to your problem. If you have the rank (or index) to the binomial coefficient table, then simply call the class method that returns the K-indexes in an array. Then, loop through that returned array to see if any of the K-index values match the value you have. Pretty straight forward...
This logic programming is really making a lap dance on my imperative programming skills. This is homework, so please just don't drop me the answer. This is what I have:
fibo(N,1) :-
N < 2,
!.
fibo(N,R) :-
N1 is N-1,
N2 is N-2,
fibo(N1,R1),
fibo(N2,R2),
R is R1+R2.
I'm suppose to make another function that looks like this; fib(N,Value,LastValue).
N is the n'th number, and value is the return value. I don't understand how I can rewrite this using accumulation. And since it counts backwards I don't see how it can "know" a last value before it calculates anything. :s Any input is appreciated.
I could post here the solution, but since that this is homework, it would be counter-productive. Instead, here's a lead:
The problem with the version of Fibonacci that you listed is that it is inefficient. Each call to fibo/2 causes another two calls, but some of these calls calculate the values of the same Fibonacci numbers. For example, in pseudo-code:
(a) fibo(4) -> fibo(3), fibo(2)
(b) fibo(3) -> fibo(2), fibo(1)
(c) fibo(2) -> fibo(1), fibo(0) % called from (a)
(d) fibo(2) -> fibo(1), fibo(0) % called from (b), redundant
To overcome this deficiency, you were asked to rephrase Fibonacci in terms of returning not just the last value, but the last two values, so that each call to fib/3 will cause only a single recursive call (hence calculate the Fibonacci series in linear time). You'll need to change the base cases to:
fib(1,1,0).
fib(2,1,1).
I'll leave the recursive case to you.
For the impatient
Here is the recursive case as well:
fib(N, Val, Last) :-
N > 2,
N1 is N - 1,
fib(N1, Last, Last1), % single call with two output arguments,
% instead of two calls with one output argument
Val is Last + Last1.
See the related discussion:
Generalizing Fibonacci sequence with SICStus Prolog
and consider ony's very good solution using finite domain constraints from there.
Perhaps using tail recursion is a good option
edit:
Instead of breaking fib(6) into fib(5)+fib(4) you might try something like fib(6) = fib(6,0,0) the first parameter is the count of steps, when it reaches 0 you stop, the second parameter the last value you calculated, and the third parameter is the value to calculate which is equal to the sum of current second and third parameters (with the exception of the first step, in which 0 + 0 will be 1)
So to calculate you set the second parameter at each call and acumulate in the third so fib(6,0,0) => fib(5,0,1) => fib(4,1,1) => fib(3,1,2) => fib(2,2,3) => fib(1,3,5) => fib(0,5,8) then you return 8
In that method you dont actually have to save in the stack the adress return, avoiding stack overflow
Remember that there is another way to calculate the Fibonacci sequence: starting from the base case and moving up.
Right now, to calculate fib(n), you add fib(n-1) and fib(n-2). Instead, flip that around and calculate fib(0) and fib(1) based on the definition of the Fibonacci sequence, and build up from that.
You almost already have it. Just rewrite:
fibo(N, Value) :-
N1 is N-1, N2 is N-2,
fibo(N1, LastValue),fibo(N2, SecondToLastValue),
Value is LastValue + SecondToLastValue.
in terms of
fibo2(N, Value, LastValue):- ...
I don't understand how I can rewrite
this using accumulation
Just don't, this is not needed (although it's possible to do so).
I'm working on a homework problem that asks me this:
Tiven a finite set of numbers, and a target number, find if the set can be used to calculate the target number using basic math operations (add, sub, mult, div) and using each number in the set exactly once (so I need to exhaust the set). This has to be done with recursion.
So, for example, if I have the set
{1, 2, 3, 4}
and target 10, then I could get to it by using
((3 * 4) - 2)/1 = 10.
I'm trying to phrase the algorithm in pseudo-code, but so far haven't gotten too far. I'm thinking graphs are the way to go, but would definitely appreciate help on this. thanks.
This isn't meant to be the fastest solution, but rather an instructive one.
It recursively generates all equations in postfix notation
It also provides a translation from postfix to infix notation
There is no actual arithmetic calculation done, so you have to implement that on your own
Be careful about division by zero
With 4 operands, 4 possible operators, it generates all 7680 = 5 * 4! * 4^3
possible expressions.
5 is Catalan(3). Catalan(N) is the number of ways to paranthesize N+1 operands.
4! because the 4 operands are permutable
4^3 because the 3 operators each have 4 choice
This definitely does not scale well, as the number of expressions for N operands is [1, 8, 192, 7680, 430080, 30965760, 2724986880, ...].
In general, if you have n+1 operands, and must insert n operators chosen from k possibilities, then there are (2n)!/n! k^n possible equations.
Good luck!
import java.util.*;
public class Expressions {
static String operators = "+-/*";
static String translate(String postfix) {
Stack<String> expr = new Stack<String>();
Scanner sc = new Scanner(postfix);
while (sc.hasNext()) {
String t = sc.next();
if (operators.indexOf(t) == -1) {
expr.push(t);
} else {
expr.push("(" + expr.pop() + t + expr.pop() + ")");
}
}
return expr.pop();
}
static void brute(Integer[] numbers, int stackHeight, String eq) {
if (stackHeight >= 2) {
for (char op : operators.toCharArray()) {
brute(numbers, stackHeight - 1, eq + " " + op);
}
}
boolean allUsedUp = true;
for (int i = 0; i < numbers.length; i++) {
if (numbers[i] != null) {
allUsedUp = false;
Integer n = numbers[i];
numbers[i] = null;
brute(numbers, stackHeight + 1, eq + " " + n);
numbers[i] = n;
}
}
if (allUsedUp && stackHeight == 1) {
System.out.println(eq + " === " + translate(eq));
}
}
static void expression(Integer... numbers) {
brute(numbers, 0, "");
}
public static void main(String args[]) {
expression(1, 2, 3, 4);
}
}
Before thinking about how to solve the problem (like with graphs), it really helps to just look at the problem. If you find yourself stuck and can't seem to come up with any pseudo-code, then most likely there is something that is holding you back; Some other question or concern that hasn't been addressed yet. An example 'sticky' question in this case might be, "What exactly is recursive about this problem?"
Before you read the next paragraph, try to answer this question first. If you knew what was recursive about the problem, then writing a recursive method to solve it might not be very difficult.
You want to know if some expression that uses a set of numbers (each number used only once) gives you a target value. There are four binary operations, each with an inverse. So, in other words, you want to know if the first number operated with some expression of the other numbers gives you the target. Well, in other words, you want to know if some expression of the 'other' numbers is [...]. If not, then using the first operation with the first number doesn't really give you what you need, so try the other ops. If they don't work, then maybe it just wasn't meant to be.
Edit: I thought of this for an infix expression of four operators without parenthesis, since a comment on the original question said that parenthesis were added for the sake of an example (for clarity?) and the use of parenthesis was not explicitly stated.
Well, you didn't mention efficiency so I'm going to post a really brute force solution and let you optimize it if you want to. Since you can have parantheses, it's easy to brute force it using Reverse Polish Notation:
First of all, if your set has n numbers, you must use exactly n - 1 operators. So your solution will be given by a sequence of 2n - 1 symbols from {{your given set}, {*, /, +, -}}
st = a stack of length 2n - 1
n = numbers in your set
a = your set, to which you add *, /, +, -
v[i] = 1 if the NUMBER i has been used before, 0 otherwise
void go(int k)
{
if ( k > 2n - 1 )
{
// eval st as described on Wikipedia.
// Careful though, it might not be valid, so you'll have to check that it is
// if it evals to your target value great, you can build your target from the given numbers. Otherwise, go on.
return;
}
for ( each symbol x in a )
if ( x isn't a number or x is a number but v[x] isn't 1 )
{
st[k] = x;
if ( x is a number )
v[x] = 1;
go(k + 1);
}
}
Generally speaking, when you need to do something recursively it helps to start from the "bottom" and think your way up.
Consider: You have a set S of n numbers {a,b,c,...}, and a set of four operations {+,-,*,/}. Let's call your recursive function that operates on the set F(S)
If n is 1, then F(S) will just be that number.
If n is 2, F(S) can be eight things:
pick your left-hand number from S (2 choices)
then pick an operation to apply (4 choices)
your right-hand number will be whatever is left in the set
Now, you can generalize from the n=2 case:
Pick a number x from S to be the left-hand operand (n choices)
Pick an operation to apply
your right hand number will be F(S-x)
I'll let you take it from here. :)
edit: Mark poses a valid criticism; the above method won't get absolutely everything. To fix that problem, you need to think about it in a slightly different way:
At each step, you first pick an operation (4 choices), and then
partition S into two sets, for the left and right hand operands,
and recursively apply F to both partitions
Finding all partitions of a set into 2 parts isn't trivial itself, though.
Your best clue about how to approach this problem is the fact that your teacher/professor wants you to use recursion. That is, this isn't a math problem - it is a search problem.
Not to give too much away (it is homework after all), but you have to spawn a call to the recursive function using an operator, a number and a list containing the remaining numbers. The recursive function will extract a number from the list and, using the operation passed in, combine it with the number passed in (which is your running total). Take the running total and call yourself again with the remaining items on the list (you'll have to iterate the list within the call but the sequence of calls is depth-first). Do this once for each of the four operators unless Success has been achieved by a previous leg of the search.
I updated this to use a list instead of a stack
When the result of the operation is your target number and your list is empty, then you have successfully found the set of operations (those that traced the path to the successful leaf) - set the Success flag and unwind. Note that the operators aren't on a list nor are they in the call: the function itself always iterates over all four. Your mechanism for "unwinding" the operator sequence from the successful leaf to get the sequence is to return the current operator and number prepended to the value returned by recursive call (only one of which will be successful since you stop at success - that, obviously, is the one to use). If none are successful, then what you return isn't important anyhow.
Update This is much harder when you have to consider expressions like the one that Daniel posted. You have combinatorics on the numbers and the groupings (numbers due to the fact that / and - are order sensitive even without grouping and grouping because it changes precedence). Then, of course, you also have the combinatorics of the operations. It is harder to manage the differences between (4 + 3) * 2 and 4 + (3 * 2) because grouping doesn't recurse like operators or numbers (which you can just iterate over in a breadth-first manner while making your (depth-first) recursive calls).
Here's some Python code to get you started: it just prints all the possible expressions, without worrying too much about redundancy. You'd need to modify it to evaluate expressions and compare to the target number, rather than simply printing them.
The basic idea is: given a set S of numbers, partition S into two subsets left and right in all possible ways (where we don't care about the order or the elements in left and right), such that left and right are both nonempty. Now for each of these partitions, find all ways of combining the elements in left (recursively!), and similarly for right, and combine the two resulting values with all possible operators. The recursion bottoms out when a set has just one element, in which case there's only one value possible.
Even if you don't know Python, the expressions function should be reasonably easy to follow; the splittings function contains some Python oddities, but all it does is to find all the partitions of the list l into left and right pieces.
def splittings(l):
n = len(l)
for i in xrange(2**n):
left = [e for b, e in enumerate(l) if i & 2**b]
right = [e for b, e in enumerate(l) if not i & 2**b]
yield left, right
def expressions(l):
if len(l) == 1:
yield l[0]
else:
for left, right in splittings(l):
if not left or not right:
continue
for el in expressions(left):
for er in expressions(right):
for operator in '+-*/':
yield '(' + el + operator + er + ')'
for x in expressions('1234'):
print x
pusedo code:
Works(list, target)
for n in list
tmp=list.remove(n)
return Works(tmp,target+n) or Works(tmp,target-n) or Works(tmp, n-target) or ...
then you just have to put the base case in. I think I gave away to much.