100. 3n+1 problem

>> শনিবার, ২১ নভেম্বর, ২০০৯

Background

Problems in Computer Science are often classified as belonging to a certain class of problems (e.g., NP, Unsolvable, Recursive). In this problem you will be analyzing a property of an algorithm whose classification is not known for all possible inputs.

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10055 - Hashmat the Brave Warrior


Hashmat is a brave warrior who with his group of young soldiers moves from one place to another to fight against his opponents. Before fighting he just calculates one thing, the difference between his soldier number and the opponent's soldier number. From this difference he decides whether to fight or not. Hashmat's soldier number is never greater than his opponent.

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103 Stacking Boxes


                                   Longest Increasing Subsequence
According to Arefin book
Input: Given a sequence
Output: The longest subsequence of the given sequence such that all values in this
longest  subsequence is strictly increasing/decreasing.
O(N^2) DP solution for LIS problem (this code check for increasing values):

for i = 1 to total-1
  for j = i+1 to total
    if height[j] > height[i] then
      if length[i] + 1 > length[j] then
        length[j] = length[i] + 1
        predecessor[j] = i

Example of LIS
height sequence: 1,6,2,3,5
length initially: [1,1,1,1,1] - because max length is at least 1 rite...
predecessor initially: [nil,nil,nil,nil,nil] - assume no predecessor so far


After first loop of j: 
  length: [1,2,2,2,2], because 6,2,3,5 are all > 1
  predecessor: [nil,1,1,1,1]
After second loop of j: (No change)
  length: [1,2,2,2,2], because 2,3,5 are all < 6
  predecessor: [nil,1,1,1,1]
After third loop:
  length: [1,2,2,3,3], because 3,5 are all > 2
  predecessor: [nil,1,1,3,3]
  After fourth loop:
  length: [1,2,2,3,4], because 5 > 3
  predecessor: [nil,1,1,3,4]

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440 Eeny Meeny Moo

>> বুধবার, ১৮ নভেম্বর, ২০০৯

Surely you have made the experience that when too many people use the Internet simultaneously, the net becomes very, very slow.

To put an end to this problem, the University of Ulm has developed a contingency scheme for times of peak load to cut off net access for some cities of the country in a systematic, totally fair manner. Germany's cities were enumerated randomly from 1 to n. Freiburg was number 1, Ulm was number 2, Karlsruhe was number 3, and so on in a purely random order.

Then a number m would be picked at random, and Internet access would first be cut off in city 1 (clearly the fairest starting point) and then in every mth city after that, wrapping around to 1 after n, and ignoring cities already cut off. For example, if n=17 and m=5, net access would be cut off to the cities in the order [1,6,11,16,5,12,2,9,17,10,4,15,14,3,8,13,7]. The problem is that it is clearly fairest to cut off Ulm last (after all, this is where the best programmers come from), so for a given n, the random number m needs to be carefully chosen so that city 2 is the last city selected.

Your job is to write a program that will read in a number of cities n and then determine the smallest integer m that will ensure that Ulm can surf the net while the rest of the country is cut off.

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151 power crisis

During the power crisis in New Zealand this winter (caused by a shortage of rain and hence low levels in the hydro dams), a contingency scheme was developed to turn off the power to areas of the country in a systematic, totally fair, manner. The country was divided up into N regions (Auckland was region number 1, and Wellington number 13). A number, m, would be picked `at random', and the power would first be turned off in region 1 (clearly the fairest starting point) and then in every m'th region after that, wrapping around to 1 after N, and ignoring regions already turned off. For example, if N = 17 and m = 5, power would be turned off to the regions in the order:1,6,11,16,5,12,2,9,17,10,4,15,14,3,8,13,7.

The problem is that it is clearly fairest to turn off Wellington last (after all, that is where the Electricity headquarters are), so for a given N, the `random' number m needs to be carefully chosen so that region 13 is the last region selected.

Write a program that will read in the number of regions and then determine the smallest number m that will ensure that Wellington (region 13) can function while the rest of the country is blacked out.

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136-ugly number

During the power crisis in New Zealand this winter (caused by a shortage of rain and hence low levels in the hydro dams), a contingency scheme was developed to turn off the power to areas of the country in a systematic, totally fair, manner. The country was divided up into N regions (Auckland was region number 1, and Wellington number 13). A number, m, would be picked `at random', and the power would first be turned off in region 1 (clearly the fairest starting point) and then in every m'th region after that, wrapping around to 1 after N, and ignoring regions already turned off. For example, if N = 17 and m = 5, power would be turned off to the regions in the order:1,6,11,16,5,12,2,9,17,10,4,15,14,3,8,13,7.

The problem is that it is clearly fairest to turn off Wellington last (after all, that is where the Electricity headquarters are), so for a given N, the `random' number m needs to be carefully chosen so that region 13 is the last region selected.

Write a program that will read in the number of regions and then determine the smallest number m that will ensure that Wellington (region 13) can function while the rest of the country is blacked out.

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10254.the priest mathematician


The ancient folklore behind the "Towers of Hanoi" puzzle invented by E. Lucas in 1883 is quite well known to us. One more recent legend tells us that the Brahmin monks from Benares never believed that the world could vanish at the moment they finished to transfer the 64 discs from the needle on which they were to one of the other needles, and they decided to finish the task as soon as possible.


Fig: The Four Needle (Peg) Tower of Hanoi

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10258.Contest Scoreboard


Think the contest score boards are wrong? Here's your chance to come up with the right rankings.Contestants are ranked first by the number of problems solved (the more the better), then by decreasing amounts of penalty time. If two or more contestants are tied in both problems solved and penalty time, they are displayed in order of increasing team numbers.
A problem is considered solved by a contestant if any of the submissions for that problem was judged correct. Penalty time is computed as the number of minutes it took for the first correct submission for a problem to be received plus 20 minutes for each incorrect submission received prior to the correct solution. Unsolved problems incur no time penalties.

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10252.common permutation

Given two strings of lowercase letters, a and b, print the longest string x of lowercase letters such that there is a permutation of x that is a subsequence of a and there is a permutation of x that is a subsequence of b.

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10213.How many pieces of land

You are given an elliptical shaped land and you are asked to choose n arbitrary points on its boundary. Then you connect all these points with one another with straight lines (that’s n*(n-1)/2 connections for n points). What is the maximum number of pieces of land you will get by choosing the points on the boundary carefully?



Fig: When the value of n is 6.

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10137.the trip

>> মঙ্গলবার, ১৭ নভেম্বর, ২০০৯

A number of students are members of a club that travels annually to exotic locations. Their destinations in the past have included Indianapolis, Phoenix, Nashville, Philadelphia, San Jose, and Atlanta. This spring they are planning a trip to Eindhoven.
The group agrees in advance to share expenses equally, but it is not practical to have them share every expense as it occurs. So individuals in the group pay for particular things, like meals, hotels, taxi rides, plane tickets, etc. After the trip, each student's expenses are tallied and money is exchanged so that the net cost to each is the same, to within one cent. In the past, this money exchange has been tedious and time consuming. Your job is to compute, from a list of expenses, the minimum amount of money that must change hands in order to equalize (within a cent) all the students' costs.

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10116.robot motion

A robot has been programmed to follow the instructions in its path. Instructions for the next direction the robot is to move are laid down in a grid. The possible instructions are
N north (up the page)
S south (down the page)
E east (to the right on the page)
W west (to the left on the page)
For example, suppose the robot starts on the north (top) side of Grid 1 and starts south (down). The path the robot follows is shown. The robot goes through 10 instructions in the grid before leaving the grid.
Compare what happens in Grid 2: the robot goes through 3 instructions only once, and then starts a loop through 8 instructions, and never exits.
You are to write a program that determines how long it takes a robot to get out of the grid or how the robot loops around.
There will be one or more grids for robots to navigate. The data for each is in the following form. On the first line are three integers separated by blanks: the number of rows in the grid, the number of columns in the grid, and the number of the column in which the robot enters from the north. The possible entry columns are numbered starting with one at the left. Then come the rows of the direction instructions. Each grid will have at least one and at most 10 rows and columns of instructions. The lines of instructions contain only the characters N, S, E, or W with no blanks. The end of input is indicated by a row containing 0 0 0.
For each grid in the input there is one line of output. Either the robot follows a certain number of instructions and exits the grid on any one the four sides or else the robot follows the instructions on a certain number of locations once, and then the instructions on some number of locations repeatedly. The sample input below corresponds to the two grids above and illustrates the two forms of output. The word "step" is always immediately followed by "(s)" whether or not the number before it is 1.

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10062.Tell me the frequencies!

Given a line of text you will have to find out the frequencies of the ASCII characters present in it. The given lines will contain none of the first 32 or last 128 ASCII characters. Of course lines may end with ‘\n’ and ‘\r’ but always keep those out of consideration.

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10042.Smith Numbers

Background
While skimming his phone directory in 1982, Albert Wilansky, a mathematician of Lehigh University , noticed that the telephone number of his brother-in-law H. Smith had the following peculiar property: The sum of the digits of that number was equal to the sum of the digits of the prime factors of that number. Got it? Smith's telephone number was 493-7775. This number can be written as the product of its prime factors in the following way:



The sum of all digits of the telephone number is 4+9+3+7+7+7+5=42, and the sum of the digits of its prime factors is equally 3+5+5+6+5+8+3+7=42. Wilansky was so amazed by his discovery that he named this type of numbers after his brother-in-law: Smith numbers.
As this observation is also true for every prime number, Wilansky decided later that a (simple and unsophisticated) prime number is not worth being a Smith number and he excluded them from the definition.

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10041 -Vito's Family

Background
The world-known gangster Vito Deadstone is moving to New York. He has a very big family there, all of them living in Lamafia Avenue. Since he will visit all his relatives very often, he is trying to find a house close to them.

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10035- Primary Arithmetic

Children are taught to add multi-digit numbers from right-to-left one digit at a time. Many find the "carry" operation - in which a 1 is carried from one digit position to be added to the next - to be a significant challenge. Your job is to count the number of carry operations for each of a set of addition problems so that educators may assess their difficulty.

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10013 - Super long sums

The Problem

The creators of a new programming language D++ have found out that whatever limit for SuperLongInt type they make, sometimes programmers need to operate even larger numbers. A limit of 1000 digits is so small... You have to find the sum of two numbers with maximal size of 1.000.000 digits.

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102 Ecological Bin Packing

Background
Bin packing, or the placement of objects of certain weights into different bins subject to certain constraints, is an historically interesting problem. Some bin packing problems are NP-complete but are amenable to dynamic programming solutions or to approximately optimal heuristic solutions.
In this problem you will be solving a bin packing problem that deals with recycling glass.

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583.Prime Factors

>> সোমবার, ২ নভেম্বর, ২০০৯

The most relevant definition for this problem is 2a: An integer g>1 is said to be prime if and only if its only positive divisors are itself and one (otherwise it is said to be composite). For example, the number 21 is composite; the number 23 is prime. Note that the decompositon of a positive number g into its prime factors, i.e., is unique if we assert that fi > 1 for all i and for i
One interesting class of prime numbers are the so-called Mersenne primes which are of the form 2p- 1. Euler proved that 231 - 1 is prime in 1772 -- all without the aid of a computer.

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10183: How many Fibs?[First Java Code Accepted]

>> রবিবার, ১ নভেম্বর, ২০০৯

The most relevant definition for this problem is 2a: An integer g>1 is said to be prime if and only if its only positive divisors are itself and one (otherwise it is said to be composite). For example, the number 21 is composite; the number 23 is prime. Note that the decompositon of a positive number g into its prime factors, i.e., is unique if we assert that fi > 1 for all i and for i
One interesting class of prime numbers are the so-called Mersenne primes which are of the form 2p- 1. Euler proved that 231 - 1 is prime in 1772 -- all without the aid of a computer.

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