With the U.S.'s large $1.5 Billion lottery this week, I wrote a function in Ruby to make Powerball picks. In Powerball, you choose 5 numbers from the range 1..69 (with no duplicates) and 1 number from the range 1..26.
This is what I came up with:
def pball
Array(1..69).shuffle[0..4].sort + [rand(1..26)]
end
It works by creating an array of integers from 1 to 69, shuffling that array, choosing the first 5 numbers, sorting those, and finally adding on a number from 1 to 26.
To do this in Swift takes a bit more work since Swift doesn't have the built-in shuffle method on Array.
This was my attempt:
func pball() -> [Int] {
let arr = Array(1...69).map{($0, drand48())}.sort{$0.1 < $1.1}.map{$0.0}[0...4].sort()
return arr + [Int(arc4random_uniform(26) + 1)]
}
Since there is no shuffle method, it works by creating an [Int] with values in the range 1...69. It then uses map to create [(Int, Double)], an array of tuple pairs that contain the numbers and a random Double in the range 0.0 ..< 1.0. It then sorts this array using the Double values and uses a second map to return to [Int] and then uses the slice [0...4] to extract the first 5 numbers and sort() to sort them.
In the second line, it appends a number in the range 1...26. I tried adding this to the first line, but Swift gave the error:
Expression was too complex to be solved in reasonable time; consider
breaking up the expression into distinct sub-expressions.
Can anyone suggest how to turn this into a 1-line function? Perhaps there is a better way to choose the 5 numbers from 1...69.
Xcode 8.3 • Swift 3.1
import GameKit
var powerballNumbers: [Int] {
return (GKRandomSource.sharedRandom().arrayByShufflingObjects(in: Array(1...69)) as! [Int])[0..<5].sorted() + [Int(arc4random_uniform(26) + 1)]
}
powerballNumbers // [5, 9, 62, 65, 69, 2]
Swift 2.x
import GameKit
var powerballNumbers: [Int] {
return (GKRandomSource.sharedRandom().arrayByShufflingObjectsInArray(Array(1...69)) as! [Int])[0...4].sort() + [Int(arc4random_uniform(26).successor())]
}
powerballNumbers // [21, 37, 39, 42, 65, 23]
I don't find the "one-liner" concept very compelling. Some languages lend themselves to it; others don't. I would suggest giving Swift a shuffle method to start with:
extension Array {
mutating func shuffle () {
for var i = self.count - 1; i != 0; i-- {
let ix1 = i
let ix2 = Int(arc4random_uniform(UInt32(i+1)))
(self[ix1], self[ix2]) = (self[ix2], self[ix1])
}
}
}
But since I made this mutating, we still need more than one line to express the entire operation because we have to have a var reference to our starting array:
var arr = Array(1...69)
(1...4).forEach {_ in arr.shuffle()}
let result = Array(arr[0..<5]) + [Int(arc4random_uniform(26)) + 1]
If you really insist on the one-liner, and you don't count the code needed to implement shuffle, then you can do it, though less efficiently, by defining shuffle more like this:
extension Array {
func shuffle () -> [Element] {
var arr = self
for var i = arr.count - 1; i != 0; i-- {
let ix1 = i
let ix2 = Int(arc4random_uniform(UInt32(i+1)))
(arr[ix1], arr[ix2]) = (arr[ix2], arr[ix1])
}
return arr
}
}
And here's your one-liner:
let result = Array(1...69).shuffle().shuffle().shuffle().shuffle()[0..<5] + [Int(arc4random_uniform(26)) + 1]
But oops, I omitted your sort. I don't see how to do that without getting the "too complex" error; to work around that, I had to split it into two lines:
var result = Array(1...69).shuffle().shuffle().shuffle().shuffle()[0..<5].sort(<)
result.append(Int(arc4random_uniform(26)) + 1)
How about this:
let winningDraw = (1...69).sort{ _ in arc4random_uniform(2) > 0}[0...4].sort() + [Int(arc4random_uniform(26)+1)]
[edit] above formula wasn't random. but this one will be
(1...69).map({Int(rand()%1000*70+$0)}).sort().map({$0%70})[0...4].sort() + [Int(rand()%26+1)]
For the fun of it, a non-GameplayKit (long) one-liner for Swift 3, using the global sequence(state:next:) function to generate random elements from the mutable state array rather than shuffling the array (although mutating the value array 5 times, so some extra copy operations here...)
let powerballNumbers = Array(sequence(state: Array(1...69), next: {
(s: inout [Int]) -> Int? in s.remove(at: Int(arc4random_uniform(UInt32(s.count))))})
.prefix(5).sorted()) + [Int(arc4random_uniform(26) + 1)]
... broken down for readability.
(Possible in future Swift version)
If the type inference weren't broken inout closure parameters (as arguments to closures), we could reduce the above to:
let powerballNumbers = Array(sequence(state: Array(1...69), next: {
$0.remove(at: Int(arc4random_uniform(UInt32($0.count)))) })
.prefix(5).sorted()) + [Int(arc4random_uniform(26) + 1)]
If we'd also allow the following extension
extension Int {
var rand: Int { return Int(arc4random_uniform(UInt32(exactly: self) ?? 0)) }
}
Then, we could go on to reduce the one-line to:
let powerballNumbers = Array(sequence(state: Array(1...69), next: { $0.remove(at: $0.count.rand) }).prefix(5).sorted()) + [26.rand + 1]
Xcode 10 • Swift 4.2
Swift now has added shuffled() to ClosedRange and random(in:) to Int which now makes this easily accomplished in one line:
func pball() -> [Int] {
return (1...69).shuffled().prefix(5).sorted() + [Int.random(in: 1...26)]
}
Further trimmings:
Because of the return type of pball(), the Int can be inferred in the random method call. Also, .prefix(5) can be replaced with [...4]. Finally, return can be omitted from the one-line function:
func pball() -> [Int] {
(1...69).shuffled()[...4].sorted() + [.random(in: 1...26)]
}
I'm working on a Texas Holdem game and i need to generate all possible k subsets from an Array of cards (represented as numbers in this example). This is how it looks so far:
public function getKSubsetsFromArray(arr:Array, k:int):Array {
var data:Array = new Array();
var result:Array = new Array();
combinations(arr, data, 0, arr.length - 1, 0, k, result, 0);
return result;
}
public function combinations(arr:Array, data:Array, start:int, end:int, index:int, r:int, resultArray:Array, resultIndex:int):int {
if (index == r) {
trace(resultIndex, data);
resultArray[resultIndex] = data;
return ++resultIndex;
}
for (var i:int = start; i<=end && end-i+1 >= r-index; i++) {
data[index] = arr[i];
resultIndex = combinations(arr, data, i + 1, end, index + 1, r, resultArray, resultIndex);
}
return resultIndex;
}
I am new to Actionscript, my idea is to have a function that takes an array of number and a parameter k, and returns an Array of arrays each of size k. However once i test the functions I get an array containing only the last combination nCk times. For example:
var testArray:Array = new Array(1, 2, 3, 4, 5);
trace(getKSubsetsFromArray(testArray, 3));
Returns:
0 1,2,3
1 1,2,4
2 1,2,5
3 1,3,4
4 1,3,5
5 1,4,5
6 2,3,4
7 2,3,5
8 2,4,5
9 3,4,5
The function output is
3,4,5,3,4,5,3,4,5,3,4,5,3,4,5,3,4,5,3,4,5,3,4,5,3,4,5,3,4,5
Of course it should print an array containing all the combinations listed before but it only prints the last one the right amount of times.
Thank your for your help.
The reason for the error is that when you are making array of arrays you are actually using the reference of the same array (data) so when the last combination is executed the contains of data array become 3,4,5 and each of index of resultArray points to data array so it prints out same values.
Solution :-
if (index == r) {
trace(resultIndex, data);
var result = new Array();
copy(result,data)
resultArray[resultIndex] = result;
return ++resultIndex;
}
Note :-
The above is pseudo code as i am not familiar with actionscript but you can implement copy function that copies values of data into result in actionscript syntax.
Does anyone know a simple algorithm to check if a Sudoku-Configuration is valid? The simplest algorithm I came up with is (for a board of size n) in Pseudocode
for each row
for each number k in 1..n
if k is not in the row (using another for-loop)
return not-a-solution
..do the same for each column
But I'm quite sure there must be a better (in the sense of more elegant) solution. Efficiency is quite unimportant.
You need to check for all the constraints of Sudoku :
check the sum on each row
check the sum on each column
check for sum on each box
check for duplicate numbers on each row
check for duplicate numbers on each column
check for duplicate numbers on each box
that's 6 checks altogether.. using a brute force approach.
Some sort of mathematical optimization can be used if you know the size of the board (ie 3x3 or 9x9)
Edit: explanation for the sum constraint: Checking for the sum first (and stoping if the sum is not 45) is much faster (and simpler) than checking for duplicates. It provides an easy way of discarding a wrong solution.
Peter Norvig has a great article on solving sudoku puzzles (with python),
https://norvig.com/sudoku.html
Maybe it's too much for what you want to do, but it's a great read anyway
Check each row, column and box such that it contains the numbers 1-9 each, with no duplicates. Most answers here already discuss this.
But how to do that efficiently? Answer: Use a loop like
result=0;
for each entry:
result |= 1<<(value-1)
return (result==511);
Each number will set one bit of the result. If all 9 numbers are unique, the lowest 9
bits will be set.
So the "check for duplicates" test is just a check that all 9 bits are set, which is the same as testing result==511.
You need to do 27 of these checks.. one for each row, column, and box.
Just a thought: don't you need to also check the numbers in each 3x3 square?
I'm trying to figure out if it is possible to have the rows and columns conditions satisfied without having a correct sudoku
This is my solution in Python, I'm glad to see it's the shortest one yet :)
The code:
def check(sud):
zippedsud = zip(*sud)
boxedsud=[]
for li,line in enumerate(sud):
for box in range(3):
if not li % 3: boxedsud.append([]) # build a new box every 3 lines
boxedsud[box + li/3*3].extend(line[box*3:box*3+3])
for li in range(9):
if [x for x in [set(sud[li]), set(zippedsud[li]), set(boxedsud[li])] if x != set(range(1,10))]:
return False
return True
And the execution:
sudoku=[
[7, 5, 1, 8, 4, 3, 9, 2, 6],
[8, 9, 3, 6, 2, 5, 1, 7, 4],
[6, 4, 2, 1, 7, 9, 5, 8, 3],
[4, 2, 5, 3, 1, 6, 7, 9, 8],
[1, 7, 6, 9, 8, 2, 3, 4, 5],
[9, 3, 8, 7, 5, 4, 6, 1, 2],
[3, 6, 4, 2, 9, 7, 8, 5, 1],
[2, 8, 9, 5, 3, 1, 4, 6, 7],
[5, 1, 7, 4, 6, 8, 2, 3, 9]]
print check(sudoku)
Create an array of booleans for every row, column, and square. The array's index represents the value that got placed into that row, column, or square. In other words, if you add a 5 to the second row, first column, you would set rows[2][5] to true, along with columns[1][5] and squares[4][5], to indicate that the row, column, and square now have a 5 value.
Regardless of how your original board is being represented, this can be a simple and very fast way to check it for completeness and correctness. Simply take the numbers in the order that they appear on the board, and begin building this data structure. As you place numbers in the board, it becomes a O(1) operation to determine whether any values are being duplicated in a given row, column, or square. (You'll also want to check that each value is a legitimate number: if they give you a blank or a too-high number, you know that the board is not complete.) When you get to the end of the board, you'll know that all the values are correct, and there is no more checking required.
Someone also pointed out that you can use any form of Set to do this. Arrays arranged in this manner are just a particularly lightweight and performant form of a Set that works well for a small, consecutive, fixed set of numbers. If you know the size of your board, you could also choose to do bit-masking, but that's probably a little overly tedious considering that efficiency isn't that big a deal to you.
Create cell sets, where each set contains 9 cells, and create sets for vertical columns, horizontal rows, and 3x3 squares.
Then for each cell, simply identify the sets it's part of and analyze those.
You could extract all values in a set (row, column, box) into a list, sort it, then compare to '(1, 2, 3, 4, 5, 6, 7, 8, 9)
I did this once for a class project. I used a total of 27 sets to represent each row, column and box. I'd check the numbers as I added them to each set (each placement of a number causes the number to be added to 3 sets, a row, a column, and a box) to make sure the user only entered the digits 1-9. The only way a set could get filled is if it was properly filled with unique digits. If all 27 sets got filled, the puzzle was solved. Setting up the mappings from the user interface to the 27 sets was a bit tedious, but made the rest of the logic a breeze to implement.
It would be very interesting to check if:
when the sum of each row/column/box equals n*(n+1)/2
and the product equals n!
with n = number of rows or columns
this suffices the rules of a sudoku. Because that would allow for an algorithm of O(n^2), summing and multiplying the correct cells.
Looking at n = 9, the sums should be 45, the products 362880.
You would do something like:
for i = 0 to n-1 do
boxsum[i] := 0;
colsum[i] := 0;
rowsum[i] := 0;
boxprod[i] := 1;
colprod[i] := 1;
rowprod[i] := 1;
end;
for i = 0 to n-1 do
for j = 0 to n-1 do
box := (i div n^1/2) + (j div n^1/2)*n^1/2;
boxsum[box] := boxsum[box] + cell[i,j];
boxprod[box] := boxprod[box] * cell[i,j];
colsum[i] := colsum[i] + cell[i,j];
colprod[i] := colprod[i] * cell[i,j];
rowsum[j] := colsum[j] + cell[i,j];
rowprod[j] := colprod[j] * cell[i,j];
end;
end;
for i = 0 to n-1 do
if boxsum[i] <> 45
or colsum[i] <> 45
or rowsum[i] <> 45
or boxprod[i] <> 362880
or colprod[i] <> 362880
or rowprod[i] <> 362880
return false;
Some time ago, I wrote a sudoku checker that checks for duplicate number on each row, duplicate number on each column & duplicate number on each box. I would love it if someone could come up one with like a few lines of Linq code though.
char VerifySudoku(char grid[81])
{
for (char r = 0; r < 9; ++r)
{
unsigned int bigFlags = 0;
for (char c = 0; c < 9; ++c)
{
unsigned short buffer = r/3*3+c/3;
// check horizontally
bitFlags |= 1 << (27-grid[(r<<3)+r+c])
// check vertically
| 1 << (18-grid[(c<<3)+c+r])
// check subgrids
| 1 << (9-grid[(buffer<<3)+buffer+r%3*3+c%3]);
}
if (bitFlags != 0x7ffffff)
return 0; // invalid
}
return 1; // valid
}
if the sum and the multiplication of a row/col equals to the right number 45/362880
First, you would need to make a boolean, "correct". Then, make a for loop, as previously stated. The code for the loop and everything afterwards (in java) is as stated, where field is a 2D array with equal sides, col is another one with the same dimensions, and l is a 1D one:
for(int i=0; i<field.length(); i++){
for(int j=0; j<field[i].length; j++){
if(field[i][j]>9||field[i][j]<1){
checking=false;
break;
}
else{
col[field[i].length()-j][i]=field[i][j];
}
}
}
I don't know the exact algorithim to check the 3x3 boxes, but you should check all the rows in field and col with "/*array name goes here*/[i].contains(1)&&/*array name goes here*/[i].contains(2)" (continues until you reach the length of a row) inside another for loop.
def solution(board):
for i in board:
if sum(i) != 45:
return "Incorrect"
for i in range(9):
temp2 = []
for x in range(9):
temp2.append(board[i][x])
if sum(temp2) != 45:
return "Incorrect"
return "Correct"
board = []
for i in range(9):
inp = raw_input()
temp = [int(i) for i in inp]
board.append(temp)
print solution(board)
Here's a nice readable approach in Python:
from itertools import chain
def valid(puzzle):
def get_block(x,y):
return chain(*[puzzle[i][3*x:3*x+3] for i in range(3*y, 3*y+3)])
rows = [set(row) for row in puzzle]
columns = [set(column) for column in zip(*puzzle)]
blocks = [set(get_block(x,y)) for x in range(0,3) for y in range(0,3)]
return all(map(lambda s: s == set([1,2,3,4,5,6,7,8,9]), rows + columns + blocks))
Each 3x3 square is referred to as a block, and there are 9 of them in a 3x3 grid. It is assumed as the puzzle is input as a list of list, with each inner list being a row.
Let's say int sudoku[0..8,0..8] is the sudoku field.
bool CheckSudoku(int[,] sudoku)
{
int flag = 0;
// Check rows
for(int row = 0; row < 9; row++)
{
flag = 0;
for (int col = 0; col < 9; col++)
{
// edited : check range step (see comments)
if ((sudoku[row, col] < 1)||(sudoku[row, col] > 9))
{
return false;
}
// if n-th bit is set.. but you can use a bool array for readability
if ((flag & (1 << sudoku[row, col])) != 0)
{
return false;
}
// set the n-th bit
flag |= (1 << sudoku[row, col]);
}
}
// Check columns
for(int col= 0; col < 9; col++)
{
flag = 0;
for (int row = 0; row < 9; row++)
{
if ((flag & (1 << sudoku[row, col])) != 0)
{
return false;
}
flag |= (1 << sudoku[row, col]);
}
}
// Check 3x3 boxes
for(int box= 0; box < 9; box++)
{
flag = 0;
for (int ofs = 0; ofs < 9; ofs++)
{
int col = (box % 3) * 3;
int row = ((int)(box / 3)) * 3;
if ((flag & (1 << sudoku[row, col])) != 0)
{
return false;
}
flag |= (1 << sudoku[row, col]);
}
}
return true;
}
Let's assume that your board goes from 1 - n.
We'll create a verification array, fill it and then verify it.
grid [0-(n-1)][0-(n-1)]; //this is the input grid
//each verification takes n^2 bits, so three verifications gives us 3n^2
boolean VArray (3*n*n) //make sure this is initialized to false
for i = 0 to n
for j = 0 to n
/*
each coordinate consists of three parts
row/col/box start pos, index offset, val offset
*/
//to validate rows
VArray( (0) + (j*n) + (grid[i][j]-1) ) = 1
//to validate cols
VArray( (n*n) + (i*n) + (grid[i][j]-1) ) = 1
//to validate boxes
VArray( (2*n*n) + (3*(floor (i/3)*n)+ floor(j/3)*n) + (grid[i][j]-1) ) = 1
next
next
if every array value is true then the solution is correct.
I think that will do the trick, although i'm sure i made a couple of stupid mistakes in there. I might even have missed the boat entirely.
array = [1,2,3,4,5,6,7,8,9]
sudoku = int [][]
puzzle = 9 #9x9
columns = map []
units = map [] # box
unit_l = 3 # box width/height
check_puzzle()
def strike_numbers(line, line_num, columns, units, unit_l):
count = 0
for n in line:
# check which unit we're in
unit = ceil(n / unit_l) + ceil(line_num / unit_l) # this line is wrong - rushed
if units[unit].contains(n): #is n in unit already?
return columns, units, 1
units[unit].add(n)
if columns[count].contains(n): #is n in column already?
return columns, units, 1
columns[count].add(n)
line.remove(n) #remove num from temp row
return columns, units, line.length # was a number not eliminated?
def check_puzzle(columns, sudoku, puzzle, array, units):
for (i=0;i< puzzle;i++):
columns, units, left_over = strike_numbers(sudoku[i], i, columns, units) # iterate through rows
if (left_over > 0): return false
Without thoroughly checking, off the top of my head, this should work (with a bit of debugging) while only looping twice. O(n^2) instead of O(3(n^2))
Here is paper by math professor J.F. Crook: A Pencil-and-Paper Algorithm for Solving Sudoku Puzzles
This paper was published in April 2009 and it got lots of publicity as definite Sudoku solution (check google for "J.F.Crook Sudoku" ).
Besides algorithm, there is also a mathematical proof that algorithm works (professor admitted that he does not find Sudoku very interesting, so he threw some math in paper to make it more fun).
I'd write an interface that has functions that receive the sudoku field and returns true/false if it's a solution.
Then implement the constraints as single validation classes per constraint.
To verify just iterate through all constraint classes and when all pass the sudoku is correct. To speedup put the ones that most likely fail to the front and stop in the first result that points to invalid field.
Pretty generic pattern. ;-)
You can of course enhance this to provide hints which field is presumably wrong and so on.
First constraint, just check if all fields are filled out. (Simple loop)
Second check if all numbers are in each block (nested loops)
Third check for complete rows and columns (almost same procedure as above but different access scheme)
Here is mine in C. Only pass each square once.
int checkSudoku(int board[]) {
int i;
int check[13] = { 0 };
for (i = 0; i < 81; i++) {
if (i % 9 == 0) {
check[9] = 0;
if (i % 27 == 0) {
check[10] = 0;
check[11] = 0;
check[12] = 0;
}
}
if (check[i % 9] & (1 << board[i])) {
return 0;
}
check[i % 9] |= (1 << board[i]);
if (check[9] & (1 << board[i])) {
return 0;
}
check[9] |= (1 << board[i]);
if (i % 9 < 3) {
if (check[10] & (1 << board[i])) {
return 0;
}
check[10] |= (1 << board[i]);
} else if (i % 9 < 6) {
if (check[11] & (1 << board[i])) {
return 0;
}
check[11] |= (1 << board[i]);
} else {
if (check[12] & (1 << board[i])) {
return 0;
}
check[12] |= (1 << board[i]);
}
}
}
Here is what I just did for this:
boolean checkers=true;
String checking="";
if(a.length/3==1){}
else{
for(int l=1; l<a.length/3; l++){
for(int n=0;n<3*l;n++){
for(int lm=1; lm<a[n].length/3; lm++){
for(int m=0;m<3*l;m++){
System.out.print(" "+a[n][m]);
if(a[n][m]<=0){
System.out.print(" (Values must be positive!) ");
}
if(n==0){
if(m!=0){
checking+=", "+a[n][m];
}
else{
checking+=a[n][m];
}
}
else{
checking+=", "+a[n][m];
}
}
}
System.out.print(" "+checking);
System.out.println();
}
}
for (int i=1;i<=a.length*a[1].length;i++){
if(checking.contains(Integer.toString(i))){
}
else{
checkers=false;
}
}
}
checkers=checkCol(a);
if(checking.contains("-")&&!checking.contains("--")){
checkers=false;
}
System.out.println();
if(checkers==true){
System.out.println("This is correct! YAY!");
}
else{
System.out.println("Sorry, it's not right. :-(");
}
}
private static boolean checkCol(int[][]a){
boolean checkers=true;
int[][]col=new int[][]{{0,0,0},{0,0,0},{0,0,0}};
for(int i=0; i<a.length; i++){
for(int j=0; j<a[i].length; j++){
if(a[i][j]>9||a[i][j]<1){
checkers=false;
break;
}
else{
col[a[i].length-j][i]=a[i][j];
}
}
}
String alia="";
for(int i=0; i<col.length; i++){
for(int j=1; j<=col[i].length; j++){
alia=a[i].toString();
if(alia.contains(""+j)){
alia=col[i].toString();
if(alia.contains(""+j)){}
else{
checkers=false;
}
}
else{
checkers=false;
}
}
}
return checkers;
}
You can check if sudoku is solved, in these two similar ways:
Check if the number is unique in each row, column and block.
A naive solution would be to iterate trough every square and check if the number is unique in the row, column block that number occupies.
But there is a better way.
Sudoku is solved if every row, column and block contains a permutation of the numbers (1 trough 9)
This only requires to check every row, column and block, instead of doing that for every number. A simple implementation would be to have a bitfield of numbers 1 trough 9 and remove them when you iterate the columns, rows and blocks. If you try to remove a missing number or if the field isn't empty when you finish then sudoku isn't correctly solved.
Here's a very concise version in Swift, that only uses an array of Ints to track the groups of 9 numbers, and only iterates over the sudoku once.
import UIKit
func check(_ sudoku:[[Int]]) -> Bool {
var groups = Array(repeating: 0, count: 27)
for x in 0...8 {
for y in 0...8 {
groups[x] += 1 << sudoku[x][y] // Column (group 0 - 8)
groups[y + 9] += 1 << sudoku[x][y] // Row (group 9 - 17)
groups[(x + y * 9) / 9 + 18] += 1 << sudoku[x][y] // Box (group 18 - 27)
}
}
return groups.filter{ $0 != 1022 }.count == 0
}
let sudoku = [
[7, 5, 1, 8, 4, 3, 9, 2, 6],
[8, 9, 3, 6, 2, 5, 1, 7, 4],
[6, 4, 2, 1, 7, 9, 5, 8, 3],
[4, 2, 5, 3, 1, 6, 7, 9, 8],
[1, 7, 6, 9, 8, 2, 3, 4, 5],
[9, 3, 8, 7, 5, 4, 6, 1, 2],
[3, 6, 4, 2, 9, 7, 8, 5, 1],
[2, 8, 9, 5, 3, 1, 4, 6, 7],
[5, 1, 7, 4, 6, 8, 2, 3, 9]
]
if check(sudoku) {
print("Pass")
} else {
print("Fail")
}
One minor optimization you can make is that you can check for duplicates in a row, column, or box in O(n) time rather than O(n^2): as you iterate through the set of numbers, you add each one to a hashset. Depending on the language, you may actually be able to use a true hashset, which is constant time lookup and insertion; then checking for duplicates can be done in the same step by seeing if the insertion was successful or not. It's a minor improvement in the code, but going from O(n^2) to O(n) is a significant optimization.