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• Maths Question: number of different permutations

  - by KingCong

  This is more of a maths question than programming but I figure a lot of people here are pretty good at maths! :) My question is: Given a 9 x 9 grid (81 cells) that must contain the numbers 1 to 9 each exactly 9 times, how many different grids can be produced. The order of the numbers doesn't matter, for example the first row could contain nine 1's etc. This is related to Sudoku and we know the number of valid Sudoku grids is 6.67×10^21, so since my problem isn't constrained like Sudoku by having to have each of the 9 numbers in each row, column and box then the answer should be greater than 6.67×10^21. My first thought was that the answer is 81! however on further reflection this assume that the 81 number possible for each cell are different, distinct number. They are not, there are 81 possible numbers for each cell but only 9 possible different numbers. My next thought was then that each of the cells in the first row can be any number between 1 and 9. If by chance the first row happened to be all the same number, say all 1s, then each cell in the second row could only have 8 possibilites, 2-9. If this continued down until the last row then number of different permutations could be calculated by 9^2 * 8^2 * 7^2 ..... * 1^2. However this doesn't work if each row doesn't contain 9 of the same number. It's been quite a while since I studied this stuff and I can't think of a way to work it out, I'd appreciate any help anyone can offer.

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• Ruby Programming Techniques: simple yet not so simple object manipulation

  - by Shyam

  Hi, I want to create an object, let's say a Pie. class Pie def initialize(name, flavor) @name = name @flavor = flavor end end But a Pie can be divided in 8 pieces, a half or just a whole Pie. For the sake of argument, I would like to know how I could give each Pie object a price per 1/8, 1/4 or per whole. I could do this by doing: class Pie def initialize(name, flavor, price_all, price_half, price_piece) @name = name @flavor = flavor @price_all = price_all @price_half = price_half @price_piece = price_piece end end But now, if I would create fifteen Pie objects, and I would take out randomly some pieces somewhere by using a method such as getPieceOfPie(pie_name) How would I be able to generate the value of all the available pies that are whole and the remaining pieces? Eventually using a method such as: myCurrentInventoryHas(pie_name) # output: 2 whole strawberry pies and 7 pieces. I know, I am a Ruby nuby. Thank you for your answers, comments and help!

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• Pohlig–Hellman algorithm for computing discrete logarithms

  - by drelihan

  Hi Folks, I'm working on coding the Pohlig-Hellman Algorithm but I am having problem understand the steps in the algorithm based on the definition of the algorithm. Going by the Wiki of the algorithm: http://en.wikipedia.org/wiki/Pohlig%E2%80%93Hellman_algorithm I know the first part 1) is to calculate the prime factor of p-1 - which is fine. Howeever, I am not sure what I need to do in steps 2) and 3). Can someone help with explaining this in plain english (i) - or pseudocode. I want to code the solution myself obviously but I cannot make any more progress unless i understand the algorithm. Note: I have done a lot of searching for this and I read S. Pohlig and M. Hellman (1978). "An Improved Algorithm for Computing Logarithms over GF(p) and its Cryptographic Significance but its still not really making sense to me. Thanks in advance

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• Static analysis of multiple if statements (conditions)

  - by koppernickus

  I have code similar to: if conditionA(x, y, z) then doA() else if conditionB(x, y, z) then doB() ... else if conditionZ(x, y, z) then doZ() else throw ShouldNeverHappenException I would like to validate two things (using static analysis): If all conditions conditionA, conditionB, ..., conditionZ are mutually exclusive (i.e. it is not possible that two or more conditions are true in the same time). All possible cases are covered, i.e. "else throw" statement will never be called. Could you recommend me a tool and/or a way I could (easily) do this? I would appreciate more detailed informations than "use Prolog" or "use Mathematica"... ;-)

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• Fractional to Decimal Form.

  - by ThePower

  Hi, there probably isn't an answer to this apart from "Create it yourself", but you never know, there might be some string representation for this. Basically, I would like to display number values as fractional instead of decimal when displaying the values as a string. Instead of a value displaying as: 1.1428571428571428571428571428571 I would prefer it to display as 8/7 Is there any way of doing this without writing the functionality myself? Regards Lloyd

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• Encode complex number as RGB pixel

  - by Vi

  How is it better to encode a complex number into RGB pixel and vice versa? Probably (logarithm of) an absolute value goes to brightness and an argument goes to hue. Desaturated pixes should receive randomized argument in reverse transformation. Something like: 0 - (0,0,0) 1 - (255,0,0) -1 - (0,255,255) 0.5 - (128,0,0) i - (255,255,0) -i - (255,0,255) (0,0,0) - 0 (255,255,255) - e^(i * random) (128,128,128) - 0.5 * e^(i *random) (0,128,128) - -0.5 Are there ready-made formulas for that?

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• Scared of Calculus - Required to pass Differential Calculus as part of my Computer science major

  - by ke3pup

  Hi guys I'm finishing my Computer science degree in university but my fear of maths (lack of background knowledge) made me to leave all my maths units til' the very end which is now. i either take them on and pass or have to give up. I've passed all my programming units easily but knowing my poor maths skills won't do i've been staying clear of the maths units. I have to pass Differential Calculus and Linear Algebra first. With a help of book named "Linear Algebra: A Modern Introduction" i'm finding myself on track and i think i can pass the Linear Algebra unit. But with differential calculus i can't find a book to help me. They're either too advanced or just too simple for what i have to learn. The things i'm required to know for this units are: Set notation, the real number line, Complex numbers in cartesian form. Complex plane, modulus. Complex numbers in polar form. De Moivre’s Theorem. Complex powers and nth roots. Definition of ei? and ez for z complex. Applications to trigonometry. Revision of domain and range of a function Working in R3. Curves and surfaces. Functions of 2 variables. Level curves.Partial derivatives and tangent planes. The derivative as a difference quotient. Geometric significance of the derivative. Discussion of limit. Higher order partial derivatives. Limits of f(x,y). Continuity. Maxima and minima of f(x,y). The chain rule. Implicit differentiation. Directional derivatives and the gradient. Limit laws, l’Hoˆpital’s rule, composition law. Definition of sinh and cosh and their inverses. Taylor polynomials. The remainder term. Taylor series. Is there a book to help me get on track with the above? Being a student i can't buy too many books hence why i'm looking for a book that covers topics I need to know. The University library has a fairly limited collection which i took as loan but didn't find useful as it was too complex.

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• Python, dictionaries, and chi-square contingency table

  - by rohanbk

  I have a file which contains several lines in the following format (word, time that the word occurred in, and frequency of documents containing the given word within the given instance in time): #inputfile <word, time, frequency> apple, 1, 3 banana, 1, 2 apple, 2, 1 banana, 2, 4 orange, 3, 1 I have Python class below that I used to create 2-D dictionaries to store the above file using as the key, and frequency as the value: class Ddict(dict): ''' 2D dictionary class ''' def __init__(self, default=None): self.default = default def __getitem__(self, key): if not self.has_key(key): self[key] = self.default() return dict.__getitem__(self, key) wordtime=Ddict(dict) # Store each inputfile entry with a <word,time> key timeword=Ddict(dict) # Store each inputfile entry with a <time,word> key # Loop over every line of the inputfile for line in open('inputfile'): word,time,count=line.split(',') # If <word,time> already a key, increment count try: wordtime[word][time]+=count # Otherwise, create the key except KeyError: wordtime[word][time]=count # If <time,word> already a key, increment count try: timeword[time][word]+=count # Otherwise, create the key except KeyError: timeword[time][word]=count The question that I have pertains to calculating certain things while iterating over the entries in this 2D dictionary. For each word 'w' at each time 't', calculate: The number of documents with word 'w' within time 't'. (a) The number of documents without word 'w' within time 't'. (b) The number of documents with word 'w' outside time 't'. (c) The number of documents without word 'w' outside time 't'. (d) Each of the items above represents one of the cells of a chi-square contingency table for each word and time. Can all of these be calculated within a single loop or do they need to be done one at a time? Ideally, I would like the output to be what's below, where a,b,c,d are all the items calculated above: print "%s, %s, %s, %s" %(a,b,c,d)

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• can i generate multiplication table of 1-9 from a single for loop??

  - by Abid

  i just want to know that whether i can generate a multiplication table from 1 - 9 with a single for loop??

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• Calculating negative fractions in Objective C

  - by Mark Reid

  I've been coding my way through Steve Kochan's Programming in Objective-C 2.0 book. I'm up to an exercise in chapter 7, ex 4, in case anyone has the book. The question posed by the exercise it will the Fraction class written work with negative fractions such as -1/2 + -2/3? Here's the implementation code in question - @implementation Fraction @synthesize numerator, denominator; -(void) print { NSLog(@"%i/%i", numerator, denominator); } -(void) setTo: (int) n over: (int) d { numerator = n; denominator = d; } -(double) convertToNum { if (denominator != 0) return (double) numerator / denominator; else return 1.0; } -(Fraction *) add: (Fraction *) f { // To add two fractions: // a/b + c/d = ((a * d) + (b * c)) / (b * d) // result will store the result of the addition Fraction *result = [[Fraction alloc] init]; int resultNum, resultDenom; resultNum = (numerator * f.denominator) + (denominator * f.numerator); resultDenom = denominator * f.denominator; [result setTo: resultNum over: resultDenom]; [result reduce]; return result; } -(Fraction *) subtract: (Fraction *) f { // To subtract two fractions: // a/b - c/d = ((a * d) - (b * c)) / (b * d) // result will store the result of the addition Fraction *result = [[Fraction alloc] init]; int resultNum, resultDenom; resultNum = numerator * f.denominator - denominator * f.numerator; resultDenom = denominator * f.denominator; [result setTo: resultNum over: resultDenom]; [result reduce]; return result; } -(Fraction *) multiply: (Fraction *) f { // To multiply two fractions // a/b * c/d = (a*c) / (b*d) // result will store the result of the addition Fraction *result = [[Fraction alloc] init]; int resultNum, resultDenom; resultNum = numerator * f.numerator; resultDenom = denominator * f.denominator; [result setTo: resultNum over: resultDenom]; [result reduce]; return result; } -(Fraction *) divide: (Fraction *) f { // To divide two fractions // a/b / c/d = (a*d) / (b*c) // result will store the result of the addition Fraction *result = [[Fraction alloc] init]; int resultNum, resultDenom; resultNum = numerator * f.denominator; resultDenom = denominator * f.numerator; [result setTo: resultNum over: resultDenom]; [result reduce]; return result; } -(void) reduce { int u = numerator; int v = denominator; int temp; while (v != 0) { temp = u % v; u = v; v = temp; } numerator /= u; denominator /= u; } @end My question to you is will it work with negative fractions and can you explain how you know? Part of the issue is I don't know how to calculate negative fractions myself so I'm not too sure how to know. Many thanks.

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• Rotation Interpolation

  - by Rob

  Hello, NB: I'll present this question in degrees purely for simplicity, radians, degrees, different zero-bearing, the problem is essentially the same. Does anyone have any ideas on the code behind rotational interpolation? Given a linear interpolation function: Lerp(from, to, amount), where amount is 0...1 which returns a value between from and to, by amount. How could I apply this same function to a rotational interpolation between 0 and 360 degrees? Given that degrees should not be returned outside 0 and 360. Given this unit circle for degrees: where from = 45 and to = 315, the algorithm should take the shortest path to the angle, i.e. it should go through zero, to 360 and then to 315 - and not all the way round 90, 180, 270 to 315. Is there a nice way to achieve this? Or is it going to just be a horrid mess of if() blocks? Am I missing some well understood standard way of doing this? Any help would be appreciated.

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• Is it possible to implement bitwise operators using integer arithmetic?

  - by Statement

  Hello World! I am facing a rather peculiar problem. I am working on a compiler for an architecture that doesn't support bitwise operations. However, it handles signed 16 bit integer arithmetics and I was wondering if it would be possible to implement bitwise operations using only: Addition (c = a + b) Subtraction (c = a - b) Division (c = a / b) Multiplication (c = a * b) Modulus (c = a % b) Minimum (c = min(a, b)) Maximum (c = max(a, b)) Comparisons (c = (a < b), c = (a == b), c = (a <= b), et.c.) Jumps (goto, for, et.c.) The bitwise operations I want to be able to support are: Or (c = a | b) And (c = a & b) Xor (c = a ^ b) Left Shift (c = a << b) Right Shift (c = a b) (All integers are signed so this is a problem) Signed Shift (c = a b) One's Complement (a = ~b) (Already found a solution, see below) Normally the problem is the other way around; how to achieve arithmetic optimizations using bitwise hacks. However not in this case. Writable memory is very scarce on this architecture, hence the need for bitwise operations. The bitwise functions themselves should not use a lot of temporary variables. However, constant read-only data & instruction memory is abundant. A side note here also is that jumps and branches are not expensive and all data is readily cached. Jumps cost half the cycles as arithmetic (including load/store) instructions do. On other words, all of the above supported functions cost twice the cycles of a single jump. Some thoughts that might help: I figured out that you can do one's complement (negate bits) with the following code: // Bitwise one's complement b = ~a; // Arithmetic one's complement b = -1 - a; I also remember the old shift hack when dividing with a power of two so the bitwise shift can be expressed as: // Bitwise left shift b = a << 4; // Arithmetic left shift b = a * 16; // 2^4 = 16 // Signed right shift b = a >>> 4; // Arithmetic right shift b = a / 16; For the rest of the bitwise operations I am slightly clueless. I wish the architects of this architecture would have supplied bit-operations. I would also like to know if there is a fast/easy way of computing the power of two (for shift operations) without using a memory data table. A naive solution would be to jump into a field of multiplications: b = 1; switch (a) { case 15: b = b * 2; case 14: b = b * 2; // ... exploting fallthrough (instruction memory is magnitudes larger) case 2: b = b * 2; case 1: b = b * 2; } Or a Set & Jump approach: switch (a) { case 15: b = 32768; break; case 14: b = 16384; break; // ... exploiting the fact that a jump is faster than one additional mul // at the cost of doubling the instruction memory footprint. case 2: b = 4; break; case 1: b = 2; break; }

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• How to convert closed bezier curves to Bitmaps?

  - by Sorush Rabiee

  I need an algorithm to convert a closed bezier curve (perhaps self-crossing) to a binary bitmap: 0 for inside pixels and 1 for outside. I'm writing a code that needs to implement some operations on bezier curves, could anybody give me some resources or tutorials about beziere? Wikipedia and others didn't say anything about optimization, subtracting, union, knot insertion and deletion and other operations :-)

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• Encode complex number as RGB pixel and back

  - by Vi

  How is it better to encode a complex number into RGB pixel and vice versa? Probably (logarithm of) an absolute value goes to brightness and an argument goes to hue. Desaturated pixes should receive randomized argument in reverse transformation. Something like: 0 - (0,0,0) 1 - (255,0,0) -1 - (0,255,255) 0.5 - (128,0,0) i - (255,255,0) -i - (255,0,255) (0,0,0) - 0 (255,255,255) - e^(i * random) (128,128,128) - 0.5 * e^(i *random) (0,128,128) - -0.5 Are there ready-made formulas for that? Edit: Looks like I just need to convert RGB to HSB and back. Edit 2: Existing RGB - HSV converter fragment: if (hsv.sat == 0) { hsv.hue = 0; // ! return hsv; } I don't want 0. I want random. And not just if hsv.sat==0, but if it is lower that it should be ("should be" means maximum saturation, saturation that is after transformation from complex number).

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• Is there any Mathematical Model or Theory behind Programming Languages?

  - by Padmarag

  RDBMS are based on Relational Algebra as well as Codd's Model. Do we have something similar to that for Programming languages or OOP? One concept may be Turing Machine.

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• Mapping a BigInteger to a circle

  - by Martin

  I have a C# system using 160 bit numbers, stored in a BigInteger. I want to display these things on a circle, which means mapping the 0-2^160 range into the 0-2Pi range. How would I do this? The approach that jumps instantly to mind is angle = (number / pow(2, 160)) * TwoPi; However, that has complexities because the division will truncate the result into an integer.

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• Min and Max value of integral type in C#

  - by Jonathan

  Whats the mathematical formulae to calculate the MIN and MAX value of an integral type using your calculator. I know you can use Integer.Max or Integer.Min etc or look it up on msdn however I want to know how to calculate it.

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• finding ALL cycles in a huge sparse matrix

  - by Andy

  Hi there, First of all I'm quite a Java beginner, so I'm not sure if this is even possible! Basically I have a huge (3+million) data source of relational data (i.e. A is friends with B+C+D, B is friends with D+G+Z (but not A - i.e. unmutual) etc.) and I want to find every cycle within this (not necessarily connected) directed graph. I've found this thread (http://stackoverflow.com/questions/546655/finding-all-cycles-in-graph/549402#549402) which has pointed me to Donald Johnson's (elementary) cycle-finding algorithm which, superficially at least, looks like it'll do what I'm after (I'm going to try when I'm back at work on Tuesday - thought it wouldn't hurt to ask in the meanwhile!). I had a quick scan through the code of the Java implementation of Johnson's algorithm (in that thread) and it looks like a matrix of relations is the first step, so I guess my questions are: a) Is Java capable of handling a 3+million*3+million matrix? (was planning on representing A-friends-with-B by a binary sparse matrix) b) Do I need to find every connected subgraph as my first problem, or will cycle-finding algorithms handle disjoint data? c) Is this actually an appropriate solution for the problem? My understanding of "elementary" cycles is that in the graph below, rather than picking out A-B-C-D-E-F it'll pick out A-B-F, B-C-D etc. but that's not the end of the world given the task. E / \ D---F / \ / \ C---B---A d) If necessary, I can simplify the problem by enforcing mutuality in relations - i.e. A-friends-with-B <== B-friends-with-A, and if really necessary I can maybe cut down the data size, but realistically it is always going to be around the 1mil mark. z) Is this a P or NP task?! Am I biting off more than I can chew? Thanks all, any help appreciated! Andy

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• What most efficient method to find a that triangle which contains the given point?

  - by Christo

  Given the triangle with vertices (a,b,c): c / \ / \ / \ a - - - b Which is then subdivided into four triangles by halving each of the edges: c / \ / \ ca / \ bc /\ - - - /\ / \ / \ / \ / \ a- - - - ab - - - -b Wich results in four triangles (a, ab, ca), (b, bc, ab), (c, ca, bc), (ab, bc, ca). Now given a point p. How do I determine in which triangle p lies, given that p is within the outer triangle (a, b, c)? Currently I intend to use ab as the origin. Check whether it is to the left of right of the line "ca - ab" using the perp of "ca - ab" and checking the sign against the dot product of "ab - a" and the perp vector and the vector "p - ab". If it is the same or the dot product is zero then it must be in (a, ab, ca)... Continue with this procedure with the other outer triangles (b, ba, ab) & (c, ca, ba). In the end if it didn't match with these it must be contained within the inner triangle (ab, bc, ca). Is there a better way to do it?

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• Perturb vector by some angle

  - by Myx

  Hello: I have a unit vector in 3D space whose direction I wish to perturb by some angle within the range 0 to theta, with the position of the vector remaining the same. What is a way I can accomplish this? Thanks.

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• How many times can you randomly generate a GUID before you risk duplicates? (.NET)

  - by SLC

  Mathematically I suppose it's possible that even two random GUIDs generated using the built in method in the .NET framework are identical, but roughly how likely are they to clash if you generate hundreds or thousands? If you generated one for every copy of Windows in the world, would they clash? The reason I ask is because I have a program that creates a lot of objects, and destroys some too, and I am wondering about the likelihood of any of those objects (including the destroyed ones) having identical GUIDs.

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• how to find distinct digit set numbers over a range of integers?

  - by evil.coder

  Suppose i have a unsigned integer, call it low and one another call it high such that highlow. The problem is to find distinct digit set numbers over this range. For example, suppose low is 1 and high is 20 then the answer is 20, because all the numbers in this range are of distinct digit sets. If suppose low is 1 and high is 21, then the answer is 20, because 12 and 21 have same digit set i.e.1, 2. I am not looking for a bruteforce algo., if anyone has a better solution then a usual bruteforce approach, please tell..

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• The most efficient way of implementing pow() function in floating point

  - by user319873

  Hi, I am trying to implement my own version of pow() and sqrt() function as my custom library doesn't have pow()/sqrt() floating point support. Can anyone help?

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• Naive Bayesian classification (spam filtering) - Doubt in one calculation? Which one is right? Plz c

  - by Microkernel

  Hi guys, I am implementing Naive Bayesian classifier for spam filtering. I have doubt on some calculation. Please clarify me what to do. Here is my question. In this method, you have to calculate P(S|W) - Probability that Message is spam given word W occurs in it. P(W|S) - Probability that word W occurs in a spam message. P(W|H) - Probability that word W occurs in a Ham message. So to calculate P(W|S), should I do (1) (Number of times W occuring in spam)/(total number of times W occurs in all the messages) OR (2) (Number of times word W occurs in Spam)/(Total number of words in the spam message) So, to calculate P(W|S), should I do (1) or (2)? (I thought it to be (2), but I am not sure, so plz clarify me) I am refering http://en.wikipedia.org/wiki/Bayesian_spam_filtering for the info by the way. I got to complete the implementation by this weekend :( Thanks and regards, MicroKernel :) @sth: Hmm... Shouldn't repeated occurrence of word 'W' increase a message's spam score? In the your approach it wouldn't, right?. Lets take a scenario and discuss... Lets say, we have 100 training messages, out of which 50 are spam and 50 are Ham. and say word_count of each message = 100. And lets say, in spam messages word W occurs 5 times in each message and word W occurs 1 time in Ham message. So total number of times W occuring in all the spam message = 5*50 = 250 times. And total number of times W occuring in all Ham messages = 1*50 = 50 times. Total occurance of W in all of the training messages = (250+50) = 300 times. So, in this scenario, how do u calculate P(W|S) and P(W|H) ? Naturally we should expect, P(W|S) P(W|H)??? right. Please share your thought...

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• Simple encryption - Sum of Hashes in C

  - by Dogbert

  I am attempting to demonstrate a simple proof of concept with respect to a vulnerability in a piece of code in a game written in C. Let's say that we want to validate a character login. The login is handled by the user choosing n items, (let's just assume n=5 for now) from a graphical menu. The items are all medieval themed: eg: _______________________________ | | | | | Bow | Sword | Staff | |-----------|-----------|-------| | Shield | Potion | Gold | |___________|___________|_______| The user must click on each item, then choose a number for each item. The validation algorithm then does the following: Determines which items were selected Drops each string to lowercase (ie: Bow becomes bow, etc) Calculates a simple string hash for each string (ie: `bow = b=2, o=15, w=23, sum = (2+15+23=40) Multiplies the hash by the value the user selected for the corresponding item; This new value is called the key Sums together the keys for each of the selected items; this is the final validation hash IMPORTANT: The validator will accept this hash, along with non-zero multiples of it (ie: if the final hash equals 1111, then 2222, 3333, 8888, etc are also valid). So, for example, let's say I select: Bow (1) Sword (2) Staff (10) Shield (1) Potion (6) The algorithm drops each of these strings to lowercase, calculates their string hashes, multiplies that hash by the number selected for each string, then sums these keys together. eg: Final_Validation_Hash = 1*HASH(Bow) + 2*HASH(Sword) + 10*HASH(Staff) + 1*HASH(Shield) + 6*HASH(Potion) By application of Euler's Method, I plan to demonstrate that these hashes are not unique, and want to devise a simple application to prove it. in my case, for 5 items, I would essentially be trying to calculate: (B)(y) = (A_1)(x_1) + (A_2)(x_2) + (A_3)(x_3) + (A_4)(x_4) + (A_5)(x_5) Where: B is arbitrary A_j are the selected coefficients/values for each string/category x_j are the hash values for each string/category y is the final validation hash (eg: 1111 above) B,y,A_j,x_j are all discrete-valued, positive, and non-zero (ie: natural numbers) Can someone either assist me in solving this problem or point me to a similar example (ie: code, worked out equations, etc)? I just need to solve the final step (ie: (B)(Y) = ...). Thank you all in advance.

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