I have a question regarding the CountDiv problem in Codility.
The problem given is: Write a function:
class Solution { public int solution(int A, int B, int K); }
that, given three integers A, B and K, returns the number of integers within the range [A..B] that are divisible by K, i.e.:
{ i : A ≤ i ≤ B, i mod K = 0 }
My code:
class Solution {
public int solution(int A, int B, int K) {
int start=0;
if (B<A || K==0 || K>B )
return 0;
else if (K<A)
start = K * ( A/K +1);
else if (K<=B)
start = K;
return (B-start+1)/K+ 1;
}
}
I don't get why I'm wrong, specially with this test case:
extreme_ifempty
A = 10, B = 10, K in {5,7,20}
WRONG ANSWER
got 1 expected 0
if K =5 then with i=10 A<=i<=B and i%k =0 so why should I have 0? Problem statement.
This is the O(1) solution, which passed the test
int solution(int A, int B, int K) {
int b = B/K;
int a = (A > 0 ? (A - 1)/K: 0);
if(A == 0){
b++;
}
return b - a;
}
Explanation: Number of integer in the range [1 .. X] that divisible by K is X/K. So, within the range [A .. B], the result is B/K - (A - 1)/K
In case A is 0, as 0 is divisible by any positive number, we need to count it in.
Java solution with O(1) and 100% in codility, adding some test cases with solutions for those who want to try and not see others solutions:
// Test cases
// [1,1,1] = 1
// [0,99,2] = 50
// [0, 100, 3] = 34
// [11,345,17] = 20
// [10,10,5] = 1
// [3, 6, 2] = 2
// [6,11,2] = 3
// [16,29,7] = 2
// [1,2,1] = 2
public int solution(int A, int B, int K) {
int offsetForLeftRange = 0;
if ( A % K == 0) { ++offsetForLeftRange; }
return (B/K) - (A /K) + offsetForLeftRange;
}
The way to solve this problem is by Prefix Sums as this is part of that section in Codility.
https://codility.com/programmers/lessons/3/
https://codility.com/media/train/3-PrefixSums.pdf
Using this technique one can subtract the count of integers between 0 and A that are divisible by K (A/K+1) from the the count of integers between 0 and B that are divisible by K (B/K+1).
Remember that A is inclusive so if it is divisible then include that as part of the result.
Below is my solution:
class Solution {
public int solution(int A, int B, int K) {
int b = (B/K) + 1; // From 0 to B the integers divisible by K
int a = (A/K) + 1; // From 0 to A the integers divisible by K
if (A%K == 0) { // "A" is inclusive; if divisible by K then
--a; // remove 1 from "a"
}
return b-a; // return integers in range
}
}
return A==B ? (A%K==0 ? 1:0) : 1+((B-A)/K)*K /K;
Well it is a completely illegible oneliner but i posted it just because i can ;-)
complete java code here:
package countDiv;
public class Solution {
/**
* First observe that
* <li> the amount of numbers n in [A..B] that are divisible by K is the same as the amount of numbers n between [0..B-A]
* they are not the same numbes of course, but the question is a range question.
* Now because we have as a starting point the zero, it saves a lot of code.
* <li> For that matter, also A=-1000 and B=-100 would work
*
* <li> Next, consider the corner cases.
* The case where A==B is a special one:
* there is just one number inside and it either is divisible by K or not, so return a 1 or a 0.
* <li> if K==1 then the result is all the numbers between and including the borders.
* <p/>
* So the algorithm simplifies to
* <pre>
* int D = B-A; //11-5=6
* if(D==0) return B%K==0 ? 1:0;
* int last = (D/K)*K; //6
* int parts = last/K; //3
* return 1+parts;//+1 because the left part (the 0) is always divisible by any K>=1.
* </pre>
*
* #param A : A>=1
* #param B : 1<=A<=B<=2000000000
* #param K : K>=1
*/
private static int countDiv(int A, int B, int K) {
return A==B ? A%K==0 ? 1:0 : 1+((B-A)/K)*K /K;
}
public static void main(String[] args) {
{
int a=10; int b=10; int k=5; int result=1;
System.out.println( a + "..." + b + "/" + k + " = " + countDiv(a,b,k) + (result!=countDiv(a,b,k) ? " WRONG" :" (OK)" ));
}
{
int a=10; int b=10; int k=7; int result=0;
System.out.println( a + "..." + b + "/" + k + " = " + countDiv(a,b,k) + (result!=countDiv(a,b,k) ? " WRONG" :" (OK)" ));
}
{
int a=6; int b=11; int k=2; int result=3;
System.out.println( a + "..." + b + "/" + k + " = " + countDiv(a,b,k) + (result!=countDiv(a,b,k) ? " WRONG" :" (OK)" ));
}
{
int a=6; int b=2000000000; int k=1; int result=b-a+1;
System.out.println( a + "..." + b + "/" + k + " = " + countDiv(a,b,k) + (result!=countDiv(a,b,k) ? " WRONG" :" (OK)" ));
}
}
}//~countDiv
I think the answers above don't provide enough logical explanation to why each solution works (the math behind the solution) so I am posting my solution here.
The idea is to use the arithmetic sequence here. If we have first divisible number (>= A) and last divisible number (<= B) we have an arithmetic sequence with distance K. Now all we have to do is find the total number of terms in the range [newA, newB] which are total divisible numbers in range [newA, newB]
first term (a1) = newA
last/n-th term (an) = newB
distance (d) = K
Sn = a1 + (a1+K) + (a1 + 2k) + (a1 + 3k) + ... + (a1 + (n-1)K)
`n` in the above equation is what we are interested in finding. We know that
n-th term = an = a1 + (n-1)K
as an = newB, a1 = newA so
newB = newA + (n-1)K
newB = newA + nK - K
nK = newB - newA + K
n = (newB - newA + K) / K
Now that we have above formula so just apply it in code.
fun countDiv(A: Int, B: Int, K: Int): Int {
//NOTE: each divisible number has to be in range [A, B] and we can not exceed this range
//find the first divisible (by k) number after A (greater than A but less than B to stay in range)
var newA = A
while (newA % K != 0 && newA < B)
newA++
//find the first divisible (by k) number before B (less than B but greater than A to stay in range)
var newB = B
while (newB % K != 0 && newB > newA)
newB--
//now that we have final new range ([newA, newB]), verify that both newA and newB are not equal
//because in that case there can be only number (newA or newB as both are equal) and we can just check
//if that number is divisible or not
if (newA == newB) {
return (newA % K == 0).toInt()
}
//Now that both newA and newB are divisible by K (a complete arithmetic sequence)
//we can calculate total divisions by using arithmetic sequence with following params
//a1 = newA, an = newB, d = K
// we know that n-th term (an) can also be calculated using following formula
//an = a1 + (n - 1)d
//n (total terms in sequence with distance d=K) is what we are interested in finding, put all values
//newB = newA + (n - 1)K
//re-arrange -> n = (newB - newA + K) / K
//Note: convert calculation to Long to avoid integer overflow otherwise result will be incorrect
val result = ((newB - newA + K.toLong()) / K.toDouble()).toInt()
return result
}
I hope this helps someone. FYI, codility solution with 100% score
Simple solution in Python:
def solution(A, B, K):
count = 0
if A % K == 0:
count += 1
count += int((B / K) - int(A / K))
return count
explanation:
B/K is the total numbers divisible by K [1..B]
A/K is the total numbers divisible by K [1..A]
The subtracts gives the total numbers divisible by K [A..B]
if A%K == 0, then we need to add it as well.
This is my 100/100 solution:
https://codility.com/demo/results/trainingRQDSFJ-CMR/
class Solution {
public int solution(int A, int B, int K) {
return (B==0) ? 1 : B/K + ( (A==0) ? 1 : (-1)*(A-1)/K);
}
}
Key aspects of this solution:
If A=1, then the number of divisors are found in B/K.
If A=0, then the number of divisors are found in B/K plus 1.
If B=0, then there is just one i%K=0, i.e. zero itself.
Here is my simple solution, with 100%
https://app.codility.com/demo/results/trainingQ5XMG7-8UY/
public int solution(int A, int B, int K) {
while (A % K != 0) {
++A;
}
while (B % K != 0) {
--B;
}
return (B - A) / K + 1;
}
Python 3 one line solution with score 100%
from math import ceil, floor
def solution(A, B, K):
return floor(B / K) - ceil(A / K) + 1
This works with O(1) Test link
using System;
class Solution
{
public int solution(int A, int B, int K)
{
int value = (B/K)-(A/K);
if(A%K == 0)
{
value=value+1;
}
return value;
}
}
I'm not sure what are you trying to do in your code, but simpler way would be to use modulo operator (%).
public int solution(int A, int B, int K)
{
int noOfDivisors = 0;
if(B < A || K == 0 || K > B )
return 0;
for(int i = A; i <= B; i++)
{
if((i % K) == 0)
{
noOfDivisors++;
}
}
return noOfDivisors;
}
If I understood the question correctly I believe this is the solution:
public static int solution(int A, int B, int K) {
int count = 0;
if(K == 0) {
return (-1);
}
if(K > B) {
return 0;
}
for(int i = A; i <= B; ++i) {
if((i % K) == 0) {
++count;
}
}
return count;
}
returning -1 is due to an illegal operation (division by zero)
int solution(int A, int B, int K) {
int tmp=(A%K==0?1:0);
int x1=A/K-tmp ;
int x2=B/K;
return x2-x1;
}
100/100 - another variation of the solution, based on Pham Trung's idea
class Solution {
public int solution(int A, int B, int K) {
int numOfDivs = A > 0 ? (B / K - ((A - 1) / K)) : ((B / K) + 1);
return numOfDivs;
}
}
class Solution {
public int solution(int A, int B, int K) {
int a = A/K, b = B/K;
if (A/K == 0)
b++;
return b - a;
}
}
This passes the test.
It's similar to "how many numbers from 2 to 5". We all know it's (5 - 2 + 1). The reason we add 1 at the end is that the first number 2 counts.
After A/K, B/K, this problem becomes the same one above. Here we need to decide if A counts in this problem. Only if A%K == 0, it counts then we need to add 1 to the result b - a (the same with b+1).
Here's my solution, two lines of Java code.
public int solution(int A, int B, int K) {
int a = (A == 0) ? -1 : (A - 1) / K;
return B / K - a;
}
The thought is simple.
a refers to how many numbers are divisible in [1..A-1]
B / K refers to how many numbers are divisible in [1..B]
0 is divisible by any integer so if A is 0, you should add one to the answer.
Here is my solution and got 100%
public int solution(int A, int B, int K) {
int count = B/K - A/K;
if(A%K == 0) {
count++;
}
return count;
}
B/K will give you the total numbers divisible by K [1..B]
A/K will give you the total numbers divisible by K [1..A]
then subtract, this will give you the total numbers divisible by K [A..B]
check A%K == 0, if true, then + 1 to the count
Another O(1) solution which got 100% in the test.
int solution(int A, int B, int K) {
if (A%K)
A = A+ (K-A%K);
if (A>B)
return 0;
return (B-A)/K+1;
}
This is my 100/100 solution:
public int solution1(int A, int B, int K) {
return A == 0 ? B / K - A / K + 1 : (B) / K - (A - 1) / K;
}
0 is divisible by any integer so if A is 0, you should add one to the answer.
This is the O(1) solution, ( There is no check required for the divisility of a)
public static int countDiv(int a, int b, int k) {
double l1 = (double)a / k;
double l = -1 * Math.floor(-1 * l1);
double h1 = (double) b / k;
double h = Math.floor(h1);
Double diff = h-l+1;
return diff.intValue();
}
There is a lot of great answers, but I think this one has some elegance in it, also gives 100% on codility.
public int solution(int a, int b, int k) {
return Math.floorDiv(b, k) - Math.floorDiv(a-1, k);
}
Explanation: Number of integers in the range [1 .. B] that divisible by K is B/K. Range [A .. B] can be transformed to [1 .. B] - [1 .. A) (notice that round bracket after A means that A does not belong to that range). That gives as a result B/K - (A-1)/K. Math.floorDiv is used to divide numbers and skip remaining decimal parts.
I will show my code in go :)
func CountDiv(a int, b int, k int) int {
count := int(math.Floor(float64(b/k)) - math.Floor(float64(a/k)));
if (math.Mod(float64(a), float64(k)) == 0) {
count++
}
return count
}
The total score is 100%
If someone is still interested in this exercise, I share my Python solution (100% in Codility)
def solution(A, B, K):
if not (B-A)%K:
res = int((B-A)/K)
else:
res = int(B/K) - int(A/K)
return res + (not A%K)
int divB = B / K;
int divA = A / K;
if(A % K != 0) {
divA++;
}
return (divB - divA) + 1;
passed 100% in codelity
My 100% score solution with one line code in python:
def solution(A, B, K):
# write your code in Python 3.6
return int(B/K) - int(A/K) + (A%K==0)
pass
int solution(int A, int B, int K)
{
// write your code in C++14 (g++ 6.2.0)
int counter = 0;
if (A == B)
A % K == 0 ? counter++ : 0;
else
{
counter = (B - A) / K;
if (A % K == 0) counter++;
else if (B % K == 0) counter++;
else if ((counter*K + K) > A && (counter*K + K) < B) counter++;
}
return counter;
}
Assumptions:
A and B are integers within the range [0..2,000,000,000];
K is an integer within the range [1..2,000,000,000];
A ≤ B.
int from = A+(K-A%K)%K;
if (from > B) {
return 0;
}
return (B-from)/K + 1;
I need a function which can calculate the mathematical combination of (n, k) for a card game.
My current attempt is to use a function based on usual Factorial method :
static long Factorial(long n)
{
return n < 2 ? 1 : n * Factorial(n - 1);
}
static long Combinatory(long n , long k )
{
return Factorial(n) / (Factorial(k) * Factorial(n - k));
}
It's working very well but the matter is when I use some range of number (n value max is 52 and k value max is 4), it keeps me returning a wrong value. E.g :
long comb = Combinatory(52, 2) ; // return 1 which should be actually 1326
I know that it's because I overflow the long when I make Factorial(52) but the range result I need is not as big as it seems.
Is there any way to get over this issue ?
Instead of using the default combinatory formula n! / (k! x (n - k)!), use the recursive property of the combinatory function.
(n, k) = (n - 1, k) + (n - 1, k - 1)
Knowing that : (n, 0) = 1 and (n, n) = 1.
-> It will make you avoid using factorial and overflowing your long.
Here is sample of implementation you can do :
static long Combinatory(long n, long k)
{
if (k == 0 || n == k )
return 1;
return Combinatory(n - 1, k) + Combinatory(n - 1, k - 1);
}
EDIT : With a faster iterative algorithm
static long Combinatory(long n, long k)
{
if (n - k < k)
k = n - k;
long res = 1;
for (int i = 1; i <= k; ++i)
{
res = (res * (n - i + 1)) / i;
}
return res;
}
In C# you can use BigInteger (I think there's a Java equivalent).
e.g.:
static long Combinatory(long n, long k)
{
return (long)(Factorial(new BigInteger(n)) / (Factorial(new BigInteger(k)) * Factorial(new BigInteger(n - k))));
}
static BigInteger Factorial(BigInteger n)
{
return n < 2 ? 1 : n * Factorial(n - 1);
}
You need to add a reference to System.Numerics to use BigInteger.
If this is not for a homework assignment, there is an efficient implementation in Apache's commons-math package
http://commons.apache.org/proper/commons-math/apidocs/org/apache/commons/math3/util/ArithmeticUtils.html#binomialCoefficientDouble%28int,%20int%29
If it is for a homework assignment, start avoiding factorial in your implementation.
Use the property that (n, k) = (n, n-k) to rewrite your choose using the highest value for k.
Then note that you can reduce n!/k!(n-k)! to n * n-1 * n-2 .... * k / (n-k) * (n-k-1) ... * 1 means that you are multiplying every number from [k, n] inclusive, then dividing by every number [1,n-k] inclusive.
// From memory, please verify correctness independently before trusting its use.
//
public long choose(n, k) {
long kPrime = Math.max(k, n-k);
long returnValue = 1;
for(i = kPrime; i <= n; i++) {
returnValue *= i;
}
for(i = 2; i <= n - kPrime; i++) {
returnValue /= i;
}
return returnValue;
}
Please double check the maths, but this is a basic idea you could go down to get a reasonably efficient implementation that will work for numbers up to a poker deck.
The recursive formula is also known as Pascal's triangle, and IMO it's the easiest way to calculate combinatorials. If you're only going to need C(52,k) (for 0<=k<=52) I think it would be best to fill a table with them at program start. The following C code fills a table using this method:
static int64_t* pascals_triangle( int N)
{
int n,k;
int64_t* C = calloc( N+1, sizeof *C);
for( n=0; n<=N; ++n)
{ C[n] = 1;
for( k=n-1; k>0; --k)
{ C[k] += C[k-1];
}
}
return C;
}
After calling this with N=52, for example returns, C[k] will hold C(52,k) for k=0..52
package testing.project;
public class PalindromeThreeDigits {
public static void main(String[] args) {
int value = 0;
for(int i = 100;i <=999;i++)
{
for(int j = i;j <=999;j++)
{
int value1 = i * j;
StringBuilder sb1 = new StringBuilder(""+value1);
String sb2 = ""+value1;
sb1.reverse();
if(sb2.equals(sb1.toString()) && value<value1) {
value = value1;
}
}
}
System.out.println(value);
}
}
This is the code that I wrote in Java... Is there any efficient way other than this.. And can we optimize this code more??
We suppose the largest such palindrome will have six digits rather than five, because 143*777 = 111111 is a palindrome.
As noted elsewhere, a 6-digit base-10 palindrome abccba is a multiple of 11. This is true because a*100001 + b*010010 + c*001100 is equal to 11*a*9091 + 11*b*910 + 11*c*100. So, in our inner loop we can decrease n by steps of 11 if m is not a multiple of 11.
We are trying to find the largest palindrome under a million that is a product of two 3-digit numbers. To find a large result, we try large divisors first:
We step m downwards from 999, by 1's;
Run n down from 999 by 1's (if 11 divides m, or 9% of the time) or from 990 by 11's (if 11 doesn't divide m, or 91% of the time).
We keep track of the largest palindrome found so far in variable q. Suppose q = r·s with r <= s. We usually have m < r <= s. We require m·n > q or n >= q/m. As larger palindromes are found, the range of n gets more restricted, for two reasons: q gets larger, m gets smaller.
The inner loop of attached program executes only 506 times, vs the ~ 810000 times the naive program used.
#include <stdlib.h>
#include <stdio.h>
int main(void) {
enum { A=100000, B=10000, C=1000, c=100, b=10, a=1, T=10 };
int m, n, p, q=111111, r=143, s=777;
int nDel, nLo, nHi, inner=0, n11=(999/11)*11;
for (m=999; m>99; --m) {
nHi = n11; nDel = 11;
if (m%11==0) {
nHi = 999; nDel = 1;
}
nLo = q/m-1;
if (nLo < m) nLo = m-1;
for (n=nHi; n>nLo; n -= nDel) {
++inner;
// Check if p = product is a palindrome
p = m * n;
if (p%T==p/A && (p/B)%T==(p/b)%T && (p/C)%T==(p/c)%T) {
q=p; r=m; s=n;
printf ("%d at %d * %d\n", q, r, s);
break; // We're done with this value of m
}
}
}
printf ("Final result: %d at %d * %d inner=%d\n", q, r, s, inner);
return 0;
}
Note, the program is in C but same techniques will work in Java.
What I would do:
Start at 999, working my way backwards to 998, 997, etc
Create the palindrome for my current number.
Determine the prime factorization of this number (not all that expensive if you have a pre-generated list of primes.
Work through this prime factorization list to determine if I can use a combination of the factors to make 2 3 digit numbers.
Some code:
int[] primes = new int[] {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,
73,79,83,89,97,101,103,107,109,113,,127,131,137,139,149,151,157,163,167,173,
179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,
283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,
419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,
547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,
661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,
811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,
947,953,967,971,977,983,991,997};
for(int i = 999; i >= 100; i--) {
String palstr = String.valueOf(i) + (new StringBuilder().append(i).reverse());
int pal = Integer.parseInt(pal);
int[] factors = new int[20]; // cannot have more than 20 factors
int remainder = pal;
int facpos = 0;
primeloop:
for(int p = 0; p < primes.length; i++) {
while(remainder % p == 0) {
factors[facpos++] = p;
remainder /= p;
if(remainder < p) break primeloop;
}
}
// now to do the combinations here
}
We can translate the task into the language of mathematics.
For a short start, we use characters as digits:
abc * xyz = n
abc is a 3-digit number, and we deconstruct it as 100*a+10*b+c
xyz is a 3-digit number, and we deconstruct it as 100*x+10*y+z
Now we have two mathematical expressions, and can define a,b,c,x,y,z as € of {0..9}.
It is more precise to define a and x as of element from {1..9}, not {0..9}, because 097 isn't really a 3-digit number, is it?
Ok.
If we want to produce a big number, we should try to reach a 9......-Number, and since it shall be palindromic, it has to be of the pattern 9....9. If the last digit is a 9, then from
(100*a + 10*b + c) * (100*x + 10*y + z)
follows that z*c has to lead to a number, ending in digit 9 - all other calculations don't infect the last digit.
So c and z have to be from (1,3,7,9) because (1*9=9, 9*1=9, 3*3=9, 7*7=49).
Now some code (Scala):
val n = (0 to 9)
val m = n.tail // 1 to 9
val niners = Seq (1, 3, 7, 9)
val highs = for (a <- m;
b <- n;
c <- niners;
x <- m;
y <- n;
z <- niners) yield ((100*a + 10*b + c) * (100*x + 10*y + z))
Then I would sort them by size, and starting with the biggest one, test them for being palindromic. So I would omit to test small numbers for being palindromic, because that might not be so cheap.
For aesthetic reasons, I wouldn't take a (toString.reverse == toString) approach, but a recursive divide and modulo solution, but on todays machines, it doesn't make much difference, does it?
// Make a list of digits from a number:
def digitize (z: Int, nums : List[Int] = Nil) : List[Int] =
if (z == 0) nums else digitize (z/10, z%10 :: nums)
/* for 342243, test 3...==...3 and then 4224.
Fails early for 123329 */
def palindromic (nums : List[Int]) : Boolean = nums match {
case Nil => true
case x :: Nil => true
case x :: y :: Nil => x == y
case x :: xs => x == xs.last && palindromic (xs.init) }
def palindrom (z: Int) = palindromic (digitize (z))
For serious performance considerations, I would test it against a toString/reverse/equals approach. Maybe it is worse. It shall fail early, but division and modulo aren't known to be the fastest operations, and I use them to make a List from the Int. It would work for BigInt or Long with few redeclarations, and works nice with Java; could be implemented in Java but look different there.
Okay, putting the things together:
highs.filter (_ > 900000) .sortWith (_ > _) find (palindrom)
res45: Option[Int] = Some(906609)
There where 835 numbers left > 900000, and it returns pretty fast, but I guess even more brute forcing isn't much slower.
Maybe there is a much more clever way to construct the highest palindrom, instead of searching for it.
One problem is: I didn't knew before, that there is a solution > 900000.
A very different approach would be, to produce big palindromes, and deconstruct their factors.
public class Pin
{
public static boolean isPalin(int num)
{
char[] val = (""+num).toCharArray();
for(int i=0;i<val.length;i++)
{
if(val[i] != val[val.length - i - 1])
{
return false;
}
}
return true;
}
public static void main(String[] args)
{
for(int i=999;i>100;i--)
for(int j=999;j>100;j--)
{
int mul = j*i;
if(isPalin(mul))
{
System.out.printf("%d * %d = %d",i,j,mul);
return;
}
}
}
}
package ex;
public class Main {
public static void main(String[] args) {
int i = 0, j = 0, k = 0, l = 0, m = 0, n = 0, flag = 0;
for (i = 999; i >= 100; i--) {
for (j = i; j >= 100; j--) {
k = i * j;
// System.out.println(k);
m = 0;
n = k;
while (n > 0) {
l = n % 10;
m = m * 10 + l;
n = n / 10;
}
if (m == k) {
System.out.println("pal " + k + " of " + i + " and" + j);
flag = 1;
break;
}
}
if (flag == 1) {
// System.out.println(k);
break;
}
}
}
}
A slightly different approach that can easily calculate the largest palindromic number made from the product of up to two 6-digit numbers.
The first part is to create a generator of palindrome numbers. So there is no need to check if a number is palindromic, the second part is a simple loop.
#include <memory>
#include <iostream>
#include <cmath>
using namespace std;
template <int N>
class PalindromeGenerator {
unique_ptr <int []> m_data;
bool m_hasnext;
public :
PalindromeGenerator():m_data(new int[N])
{
for(auto i=0;i<N;i++)
m_data[i]=9;
m_hasnext=true;
}
bool hasNext() const {return m_hasnext;}
long long int getnext()
{
long long int v=0;
long long int b=1;
for(int i=0;i<N;i++){
v+=m_data[i]*b;
b*=10;
}
for(int i=N-1;i>=0;i--){
v+=m_data[i]*b;
b*=10;
}
auto i=N-1;
while (i>=0)
{
if(m_data[i]>=1) {
m_data[i]--;
return v;
}
else
{
m_data[i]=9;
i--;
}
}
m_hasnext=false;
return v;
}
};
template<int N>
void findmaxPalindrome()
{
PalindromeGenerator<N> gen;
decltype(gen.getnext()) minv=static_cast<decltype(gen.getnext())> (pow(10,N-1));
decltype(gen.getnext()) maxv=static_cast<decltype(gen.getnext())> (pow(10,N)-1);
decltype(gen.getnext()) start=11*(maxv/11);
while(gen.hasNext())
{
auto v=gen.getnext();
for (decltype(gen.getnext()) i=start;i>minv;i-=11)
{
if (v%i==0)
{
auto r=v/i;
if (r>minv && r<maxv ){
cout<<"done:"<<v<<" "<<i<< "," <<r <<endl;
return ;
}
}
}
}
return ;
}
int main(int argc, char* argv[])
{
findmaxPalindrome<6>();
return 0;
}
You can use the fact that 11 is a multiple of the palindrome to cut down on the search space. We can get this since we can assume the palindrome will be 6 digits and >= 111111.
e.g. ( from projecteuler ;) )
P= xyzzyx = 100000x + 10000y + 1000z + 100z + 10y +x
P=100001x+10010y+1100z
P=11(9091x+910y+100z)
Check if i mod 11 != 0, then the j loop can be subtracted by 11 (starting at 990) since at least one of the two must be divisible by 11.
You can try the following which prints
999 * 979 * 989 = 967262769
largest palindrome= 967262769 took 0.015
public static void main(String... args) throws IOException, ParseException {
long start = System.nanoTime();
int largestPalindrome = 0;
for (int i = 999; i > 100; i--) {
LOOP:
for (int j = i; j > 100; j--) {
for (int k = j; k > 100; k++) {
int n = i * j * k;
if (n < largestPalindrome) continue LOOP;
if (isPalindrome(n)) {
System.out.println(i + " * " + j + " * " + k + " = " + n);
largestPalindrome = n;
}
}
}
}
long time = System.nanoTime() - start;
System.out.printf("largest palindrome= %d took %.3f seconds%n", largestPalindrome, time / 1e9);
}
private static boolean isPalindrome(int n) {
if (n >= 100 * 1000 * 1000) {
// 9 digits
return n % 10 == n / (100 * 1000 * 1000)
&& (n / 10 % 10) == (n / (10 * 1000 * 1000) % 10)
&& (n / 100 % 10) == (n / (1000 * 1000) % 10)
&& (n / 1000 % 10) == (n / (100 * 1000) % 10);
} else if (n >= 10 * 1000 * 1000) {
// 8 digits
return n % 10 == n / (10 * 1000 * 1000)
&& (n / 10 % 10) == (n / (1000 * 1000) % 10)
&& (n / 100 % 10) == (n / (100 * 1000) % 10)
&& (n / 1000 % 10) == (n / (10 * 1000) % 10);
} else if (n >= 1000 * 1000) {
// 7 digits
return n % 10 == n / (1000 * 1000)
&& (n / 10 % 10) == (n / (100 * 1000) % 10)
&& (n / 100 % 10) == (n / (10 * 1000) % 10);
} else throw new AssertionError();
}
i did this my way , but m not sure if this is the most efficient way of doing this .
package problems;
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
public class P_4 {
/**
* #param args
* #throws IOException
*/
static int[] arry = new int[6];
static int[] arry2 = new int[6];
public static boolean chk()
{
for(int a=0;a<arry.length;a++)
if(arry[a]!=arry2[a])
return false;
return true;
}
public static void main(String[] args) throws IOException {
// TODO Auto-generated method stub
InputStreamReader ir = new InputStreamReader(System.in);
BufferedReader br = new BufferedReader(ir);
int temp,z,i;
for(int x=999;x>100;x--)
for(int y=999;y>100;y--)
{
i=0;
z=x*y;
while(z>0)
{
temp=z%10;
z=z/10;
arry[i]=temp;
i++;
}
for(int k = arry.length;k>0;k--)
arry2[arry.length- k]=arry[k-1];
if(chk())
{
System.out.print("pelindrome = ");
for(int l=0;l<arry2.length;l++)
System.out.print(arry2[l]);
System.out.println(x);
System.out.println(y);
}
}
}
}
This is code in C, a little bit long, but gets the job done.:)
#include <stdio.h>
#include <stdlib.h>
/*
A palindromic number reads the same both ways. The largest palindrome made from the product of two
2-digit numbers is 9009 = 91 99.
Find the largest palindrome made from the product of two 3-digit numbers.*/
int palndr(int b)
{
int *x,*y,i=0,j=0,br=0;
int n;
n=b;
while(b!=0)
{
br++;
b/=10;
}
x=(int *)malloc(br*sizeof(int));
y=(int *)malloc(br*sizeof(int));
int br1=br;
while(n!=0)
{
x[i++]=y[--br]=n%10;
n/=10;
}
int ind = 1;
for(i=0;i<br1;i++)
if(x[i]!=y[i])
ind=0;
free(x);
free(y);
return ind;
}
int main()
{
int i,cek,cekmax=1;
int j;
for(i=100;i<=999;i++)
{
for(j=i;j<=999;j++)
{
cek=i*j;
if(palndr(cek))
{
if(pp>cekmax)
cekmax=cek;
}
}
}
printf("The largest palindrome is: %d\n\a",cekmax);
}
You can actually do it with Python, it's easy just take a look:
actualProduct = 0
highestPalindrome = 0
# Setting the numbers. In case it's two digit 10 and 99, in case is three digit 100 and 999, etc.
num1 = 100
num2 = 999
def isPalindrome(number):
number = str(number)
reversed = number[::-1]
if number==reversed:
return True
else:
return False
a = 0
b = 0
for i in range(num1,num2+1):
for j in range(num1,num2+1):
actualProduct = i * j
if (isPalindrome(actualProduct) and (highestPalindrome < actualProduct)):
highestPalindrome = actualProduct
a = i
b = j
print "Largest palindrome made from the product of two %d-digit numbers is [ %d ] made of %d * %d" % (len(str(num1)), highestPalindrome, a, b)
Since we are not cycling down both iterators (num1 and num2) at the same time, the first palindrome number we find will be the largest. We don’t need to test to see if the palindrome we found is the largest. This significantly reduces the time it takes to calculate.
package testing.project;
public class PalindromeThreeDigits {
public static void main(String[] args) {
int limit = 99;
int max = 999;
int num1 = max, num2, prod;
while(num1 > limit)
{
num2 = num1;
while(num2 > limit)
{
total = num1 * num2;
StringBuilder sb1 = new StringBuilder(""+prod);
String sb2 = ""+prod;
sb1.reverse();
if( sb2.equals(sb1.toString()) ) { //optimized here
//print and exit
}
num2--;
}
num1--;
}
}//end of main
}//end of class PalindromeThreeDigits
I tried the solution by Tobin joy and vickyhacks and both of them produce the result 580085 which is wrong here is my solution, though very clumsy:
import java.util.*;
class ProjEu4
{
public static void main(String [] args) throws Exception
{
int n=997;
ArrayList<Integer> al=new ArrayList<Integer>();
outerloop:
while(n>100){
int k=reverse(n);
int fin=n*1000+k;
al=findfactors(fin);
if(al.size()>=2)
{
for(int i=0;i<al.size();i++)
{
if(al.contains(fin/al.get(i))){
System.out.println(fin+" factors are:"+al.get(i)+","+fin/al.get(i));
break outerloop;}
}
}
n--;
}
}
private static ArrayList<Integer> findfactors(int fin)
{
ArrayList<Integer> al=new ArrayList<Integer>();
for(int i=100;i<=999;i++)
{
if(fin%i==0)
al.add(i);
}
return al;
}
private static int reverse(int number)
{
int reverse = 0;
while(number != 0){
reverse = (reverse*10)+(number%10);
number = number/10;
}
return reverse;
}
}
Most probably it is replication of one of the other solution but it looks simple owing to pythonified code ,even it is a bit brute-force.
def largest_palindrome():
largest_palindrome = 0;
for i in reversed(range(1,1000,1)):
for j in reversed(range(1, i+1, 1)):
num = i*j
if check_palindrome(str(num)) and num > largest_palindrome :
largest_palindrome = num
print "largest palindrome ", largest_palindrome
def check_palindrome(term):
rev_term = term[::-1]
return rev_term == term
What about : in python
>>> for i in range((999*999),(100*100), -1):
... if str(i) == str(i)[::-1]:
... print i
... break
...
997799
>>>
I believe there is a simpler approach: Examine palindromes descending from the largest product of two three digit numbers, selecting the first palindrome with two three digit factors.
Here is the Ruby code:
require './palindrome_range'
require './prime'
def get_3_digit_factors(n)
prime_factors = Prime.factors(n)
rf = [prime_factors.pop]
rf << prime_factors.shift while rf.inject(:*) < 100 || prime_factors.inject(:*) > 999
lf = prime_factors.inject(:*)
rf = rf.inject(:*)
lf < 100 || lf > 999 || rf < 100 || rf > 999 ? [] : [lf, rf]
end
def has_3_digit_factors(n)
return !get_3_digit_factors(n).empty?
end
pr = PalindromeRange.new(0, 999 * 999)
n = pr.downto.find {|n| has_3_digit_factors(n)}
puts "Found #{n} - Factors #{get_3_digit_factors(n).inspect}, #{Prime.factors(n).inspect}"
prime.rb:
class Prime
class<<self
# Collect all prime factors
# -- Primes greater than 3 follow the form of (6n +/- 1)
# Being of the form 6n +/- 1 does not mean it is prime, but all primes have that form
# See http://primes.utm.edu/notes/faq/six.html
# -- The algorithm works because, while it will attempt non-prime values (e.g., (6 *4) + 1 == 25),
# they will fail since the earlier repeated division (e.g., by 5) means the non-prime will fail.
# Put another way, after repeatedly dividing by a known prime, the remainder is itself a prime
# factor or a multiple of a prime factor not yet tried (e.g., greater than 5).
def factors(n)
square_root = Math.sqrt(n).ceil
factors = []
while n % 2 == 0
factors << 2
n /= 2
end
while n % 3 == 0
factors << 3
n /= 3
end
i = 6
while i < square_root
[(i - 1), (i + 1)].each do |f|
while n % f == 0
factors << f
n /= f
end
end
i += 6
end
factors << n unless n == 1
factors
end
end
end
palindrome_range.rb:
class PalindromeRange
FIXNUM_MAX = (2**(0.size * 8 -2) -1)
def initialize(min = 0, max = FIXNUM_MAX)
#min = min
#max = max
end
def downto
return enum_for(:downto) unless block_given?
n = #max
while n >= #min
yield n if is_palindrome(n)
n -= 1
end
nil
end
def each
return upto
end
def upto
return enum_for(:downto) unless block_given?
n = #min
while n <= #max
yield n if is_palindrome(n)
n += 1
end
nil
end
private
def is_palindrome(n)
s = n.to_s
i = 0
j = s.length - 1
while i <= j
break if s[i] != s[j]
i += 1
j -= 1
end
i > j
end
end
public class ProjectEuler4 {
public static void main(String[] args) {
int x = 999; // largest 3-digit number
int largestProduct = 0;
for(int y=x; y>99; y--){
int product = x*y;
if(isPalindormic(x*y)){
if(product>largestProduct){
largestProduct = product;
System.out.println("3-digit numbers product palindormic number : " + x + " * " + y + " : " + product);
}
}
if(y==100 || product < largestProduct){y=x;x--;}
}
}
public static boolean isPalindormic(int n){
int palindormic = n;
int reverse = 0;
while(n>9){
reverse = (reverse*10) + n%10;
n=n/10;
}
reverse = (reverse*10) + n;
return (reverse == palindormic);
}
}