Triangle (void in the inside) pattern print - go

Good morning guys, I wanted to ask you help for an exercise that is literally driving me crazy... I should print a triangle of n lines with (n - r) asterisks, but with an empty center. So an exercise with nested loops.
if we had n == 5, it would be something like this:
*****
* *
* *
**
*
I tried with this one:
package main
import "fmt"
func main() {
var n int
fmt.Print("Inserire n: ")
fmt.Scan(&n)
for r := 0; r < n; r++ { // r == line
for a := 0; a < n - r; a++ { // a == asterisk
fmt.Print("*")
if r > 0 || r < n - 1 {
for s := 0; s < (n - (3 + r) +1); s++ { // s == space
fmt.Print(" ")
break
}
}
}
fmt.Println("")
}
}
My output for n == 5 looks like this:
* * * * *
* * * *
* * *
**
*
There are 2 problems that I cannot really understand how to solve:
Why does the first line of asterisks have spaces if I put as a condition that r must be greater than 0 and less than n-1?
How do you prevent unnecessary asterisks, but only spaces, from being printed on the other lines (r)?


Why does the first line of asterisks have spaces if I put as a condition that r must be greater than 0 and less than n-1?
You are correct in that "greater than 0 and less than n-1" would give you what you need but that is not what your statement currently is (if r > 0 || r < n - 1 - || = OR not AND).
How do you prevent unnecessary asterisks, but only spaces, from being printed on the other lines (r)?
Currently you have three levels of loops which makes the algorithm more complex than it needs to be:
Lines
Asterisks
Spaces
You only really need two levels (lines and columns). With this approach you just need to make a decision for each position - do I output * or . Pseudo code for this follows (have left a few bits for you to fill in):
for a := 0; a < n-r; a++ { // a == column
if [Firstline] || [first column] || [last column] {
fmt.Print("*")
} else {
fmt.Print(" ")
}
}
From comments:
why is the last condition a == n - r -1?
Consider the loop a := 0; a < n-r; a++; if n-r == 5 then a will range from 0 to 4 (inclusive - playground). This means that the last value (column) will be 4 (n-r-1). You could change the loop to a := 1; a <= n-r; a++ in which case n - r would be the last column.

Related

using while loop the page keep loading over and over (codewars problem)

heyy guys i m trying to solve Factorial decomposition (codewars task )
well some numbers work with me untill i reach number 23 the page keep looping over and over someone help me please
function decomp(n) {
let c =[]
let sum =1
for(let i=n;i>=1;i--){
sum*=i
}
let k= 2
while(k<=sum){
if(sum%k!==0){
k++}
while(sum%k==0){
c.push(k)
sum = sum/k
}
}
return c.join('*')
}
the function works good until i reach the number 23 the keep loading over and over , the tasks is about the function is decomp(n) and should return the decomposition of n! into its prime factors in increasing order of the primes, as a string.
factorial can be a very big number (4000! has 12674 digits, n can go from 300 to 4000).
In Fortran - as in any other language - the returned string is not permitted to contain any redundant trailing whitespace: you can use dynamically allocated character strings.
example
n = 12; decomp(12) -> "2^10 * 3^5 * 5^2 * 7 * 11"
since 12! is divisible by 2 ten times, by 3 five times, by 5 two times and by 7 and 11 only once.
n = 22; decomp(22) -> "2^19 * 3^9 * 5^4 * 7^3 * 11^2 * 13 * 17 * 19"
n = 25; decomp(25) -> 2^22 * 3^10 * 5^6 * 7^3 * 11^2 * 13 * 17 * 19 * 23

23! cannot be exactly expressed in double-precision floating-point format, which JavaScript uses for its numbers.
However, you don't need to compute n!. You just need to factorize each number and concat their factorization.
Actually, you don't even need to factorize each number. Note that given n and p, there are (n/p) numbers no greater than n that are multiples of p, (n/(p*p)) that are multiples of p*p, etc.
function *primes(n) {
// Sieve of Eratosthenes
const isPrime = Array(n + 1).fill(true);
isPrime[0] = isPrime[1] = false;
for (let i = 2; i <= n; i++) {
if (isPrime[i]) {
yield i;
for (let j = i * i; j <= n; j += i)
isPrime[j] = false;
}
}
}
function decomp(n) {
let s = n + '! =';
for (const p of primes(n)) {
let m = n, c = 0;
// There are (n/p) numbers no greater than n that are multiples of p
// There are (n/(p*p)) numbers no greater than n that are multiples of p*p
// ...
while (m = ((m / p) | 0)) {
c += m;
}
s += (p == 2 ? ' ' : ' * ') + p + (c == 1 ? '' : '^' + c);
}
return s;
}
console.log(decomp(12))
console.log(decomp(22))
console.log(decomp(23))
console.log(decomp(24))
console.log(decomp(25))

Check if a number from 1 to N has a number 3 in it

How do I check how many numbers from 1 to N (N < 100) have number 3 in it without converting it to a string to check it?

You can use th % mod operator to take the digits of the number one by one, and check them with 3.
Like,
int x;
while(num != 0) // num here goes from 1 to 100
{
x = num % 10;
if(x == 3)
{
//eureka
}
num /= 10;
}
EDIT
Lets check the algorithm for 35.
First iteration
//num = 35
x = num % 10; // x = 5 (35 % 10)
if(x == 3) // is x equal to 3 (NO)
{
//eureka
}
num /= 10; // num = 3 (35 / 10)
While loop check
num != 0 // num = 5
Second Iteration
//num = 35
x = num % 10; // x = 3 (5 % 10)
if(x == 3) // is x equal to 3 (YES)
{
//eureka
}
num /= 10; // num = 0 (5 / 10)
While loop check
num != 0 // num = 0
// While loop exits

I think the simplest way is by remainder and checking if number is between 30 and 39
if((x%10)==3||(x<40&&x>=30))
{
//Thats it
}

You can make use of the modulas operator % such that if(n % 3 == 1){success operation}

Caculating total combinations

I don't know how to go about this programming problem.
Given two integers n and m, how many numbers exist such that all numbers have all digits from 0 to n-1 and the difference between two adjacent digits is exactly 1 and the number of digits in the number is atmost 'm'.
What is the best way to solve this problem? Is there a direct mathematical formula?
Edit: The number cannot start with 0.
Example:
for n = 3 and m = 6 there are 18 such numbers (210, 2101, 21012, 210121 ... etc)
Update (some people have encountered an ambiguity):
All digits from 0 to n-1 must be present.

This Python code computes the answer in O(nm) by keeping track of the numbers ending with a particular digit.
Different arrays (A,B,C,D) are used to track numbers that have hit the maximum or minimum of the range.
n=3
m=6
A=[1]*n # Number of ways of being at digit i and never being to min or max
B=[0]*n # number of ways with minimum being observed
C=[0]*n # number of ways with maximum being observed
D=[0]*n # number of ways with both being observed
A[0]=0 # Cannot start with 0
A[n-1]=0 # Have seen max so this 1 moves from A to C
C[n-1]=1 # Have seen max if start with highest digit
t=0
for k in range(m-1):
A2=[0]*n
B2=[0]*n
C2=[0]*n
D2=[0]*n
for i in range(1,n-1):
A2[i]=A[i+1]+A[i-1]
B2[i]=B[i+1]+B[i-1]
C2[i]=C[i+1]+C[i-1]
D2[i]=D[i+1]+D[i-1]
B2[0]=A[1]+B[1]
C2[n-1]=A[n-2]+C[n-2]
D2[0]=C[1]+D[1]
D2[n-1]=B[n-2]+D[n-2]
A=A2
B=B2
C=C2
D=D2
x=sum(d for d in D2)
t+=x
print t

After doing some more research, I think there may actually be a mathematical approach after all, although the math is advanced for me. Douglas S. Stones pointed me in the direction of Joseph Myers' (2008) article, BMO 2008–2009 Round 1 Problem 1—Generalisation, which derives formulas for calculating the number of zig-zag paths across a rectangular board.
As I understand it, in Anirudh's example, our board would have 6 rows of length 3 (I believe this would mean n=3 and r=6 in the article's terms). We can visualize our board so:
0 1 2 example zig-zag path: 0
0 1 2 1
0 1 2 0
0 1 2 1
0 1 2 2
0 1 2 1
Since Myers' formula m(n,r) would generate the number for all the zig-zag paths, that is, the number of all 6-digit numbers where all adjacent digits are consecutive and digits are chosen from (0,1,2), we would still need to determine and subtract those that begin with zero and those that do not include all digits.
If I understand correctly, we may do this in the following way for our example, although generalizing the concept to arbitrary m and n may prove more complicated:
Let m(3,6) equal the number of 6-digit numbers where all adjacent digits
are consecutive and digits are chosen from (0,1,2). According to Myers,
m(3,r) is given by formula and also equals OEIS sequence A029744 at
index r+2, so we have
m(3,6) = 16
How many of these numbers start with zero? Myers describes c(n,r) as the
number of zig-zag paths whose colour is that of the square in the top
right corner of the board. In our case, c(3,6) would include the total
for starting-digit 0 as well as starting-digit 2. He gives c(3,2r) as 2^r,
so we have
c(3,6) = 8. For starting-digit 0 only, we divide by two to get 4.
Now we need to obtain only those numbers that include all the digits in
the range, but how? We can do this be subtracting m(n-1,r) from m(n,r).
In our case, we have all the m(2,6) that would include only 0's and 1's,
and all the m(2,6) that would include 1's and 2's. Myers gives
m(2,anything) as 2, so we have
2*m(2,6) = 2*2 = 4
But we must remember that one of the zero-starting numbers is included
in our total for 2*m(2,6), namely 010101. So all together we have
m(3,6) - c(3,6)/2 - 4 + 1
= 16 - 4 - 4 + 1
= 9
To complete our example, we must follow a similar process for m(3,5),
m(3,4) and m(3,3). Since it's late here, I might follow up tomorrow...

One approach could be to program it recursively, calling the function to add as well as subtract from the last digit.
Haskell code:
import Data.List (sort,nub)
f n m = concatMap (combs n) [n..m]
combs n m = concatMap (\x -> combs' 1 [x]) [1..n - 1] where
combs' count result
| count == m = if test then [concatMap show result] else []
| otherwise = combs' (count + 1) (result ++ [r + 1])
++ combs' (count + 1) (result ++ [r - 1])
where r = last result
test = (nub . sort $ result) == [0..n - 1]
Output:
*Main> f 3 6
["210","1210","1012","2101","12101","10121","21210","21012"
,"21010","121210","121012","121010","101212","101210","101012"
,"212101","210121","210101"]
In response to Anirudh Rayabharam's comment, I hope the following code will be more 'pseudocode' like. When the total number of digits reaches m, the function g outputs 1 if the solution has hashed all [0..n-1], and 0 if not. The function f accumulates the results for g for starting digits [1..n-1] and total number of digits [n..m].
Haskell code:
import qualified Data.Set as S
g :: Int -> Int -> Int -> Int -> (S.Set Int, Int) -> Int
g n m digitCount lastDigit (hash,hashCount)
| digitCount == m = if test then 1 else 0
| otherwise =
if lastDigit == 0
then g n m d' (lastDigit + 1) (hash'',hashCount')
else if lastDigit == n - 1
then g n m d' (lastDigit - 1) (hash'',hashCount')
else g n m d' (lastDigit + 1) (hash'',hashCount')
+ g n m d' (lastDigit - 1) (hash'',hashCount')
where test = hashCount' == n
d' = digitCount + 1
hash'' = if test then S.empty else hash'
(hash',hashCount')
| hashCount == n = (S.empty,hashCount)
| S.member lastDigit hash = (hash,hashCount)
| otherwise = (S.insert lastDigit hash,hashCount + 1)
f n m = foldr forEachNumDigits 0 [n..m] where
forEachNumDigits numDigits accumulator =
accumulator + foldr forEachStartingDigit 0 [1..n - 1] where
forEachStartingDigit startingDigit accumulator' =
accumulator' + g n numDigits 1 startingDigit (S.empty,0)
Output:
*Main> f 3 6
18
(0.01 secs, 571980 bytes)
*Main> f 4 20
62784
(1.23 secs, 97795656 bytes)
*Main> f 4 25
762465
(11.73 secs, 1068373268 bytes)

model your problem as 2 superimposed lattices in 2 dimensions, specifically as pairs (i,j) interconnected with oriented edges ((i0,j0),(i1,j1)) where i1 = i0 + 1, |j1 - j0| = 1, modified as follows:
dropping all pairs (i,j) with j > 9 and its incident edges
dropping all pairs (i,j) with i > m-1 and its incident edges
dropping edge ((0,0), (1,1))
this construction results in a structure like in this diagram:
:
the requested numbers map to paths in the lattice starting at one of the green elements ((0,j), j=1..min(n-1,9)) that contain at least one pink and one red element ((i,0), i=1..m-1, (i,n-1), i=0..m-1 ). to see this, identify the i-th digit j of a given number with point (i,j). including pink and red elements ('extremal digits') guarantee that all available diguts are represented in the number.
Analysis
for convenience, let q1, q2 denote the position-1.
let q1 be the position of a number's first digit being either 0 or min(n-1,9).
let q2 be the position of a number's first 0 if the digit at position q1 is min(n-1,9) and vv.
case 1: first extremal digit is 0
the number of valid prefixes containing no 0 can be expressed as sum_{k=1..min(n-1,9)} (paths_to_0(k,1,q1), the function paths_to_0 being recursively defined as
paths_to_0(0,q1-1,q1) = 0;
paths_to_0(1,q1-1,q1) = 1;
paths_to_0(digit,i,q1) = 0; if q1-i < digit;
paths_to_0(x,_,_) = 0; if x >= min(n-1,9)
// x=min(n-1,9) mustn't occur before position q2,
// x > min(n-1,9) not at all
paths_to_0(x,_,_) = 0; if x <= 0;
// x=0 mustn't occur before position q1,
// x < 0 not at all
and else paths_to_0(digit,i,q1) =
paths_to_0(digit+1,i+1,q1) + paths_to_0(digit-1,i+1,q1);
similarly we have
paths_to_max(min(n-1,9),q2-1,q2) = 0;
paths_to_max(min(n-2,8),q2-1,q2) = 1;
paths_to_max(digit,i,q2) = 0 if q2-i < n-1;
paths_to_max(x,_,_) = 0; if x >= min(n-1,9)
// x=min(n-1,9) mustn't occur before
// position q2,
// x > min(n-1,9) not at all
paths_to_max(x,_,_) = 0; if x < 0;
and else paths_to_max(digit,q1,q2) =
paths_max(digit+1,q1+1,q2) + paths_to_max(digit-1,q1+1,q2);
and finally
paths_suffix(digit,length-1,length) = 2; if digit > 0 and digit < min(n-1,9)
paths_suffix(digit,length-1,length) = 1; if digit = 0 or digit = min(n-1,9)
paths_suffix(digit,k,length) = 0; if length > m-1
or length < q2
or k > length
paths_suffix(digit,k,0) = 1; // the empty path
and else paths_suffix(digit,k,length) =
paths_suffix(digit+1,k+1,length) + paths_suffix(digit-1,k+1,length);
... for a grand total of
number_count_case_1(n, m) =
sum_{first=1..min(n-1,9), q1=1..m-1-(n-1), q2=q1..m-1, l_suffix=0..m-1-q2} (
paths_to_0(first,1,q1)
+ paths_to_max(0,q1,q2)
+ paths_suffix(min(n-1,9),q2,l_suffix+q2)
)
case 2: first extremal digit is min(n-1,9)
case 2.1: initial digit is not min(n-1,9)
this is symmetrical to case 1 with all digits d replaced by min(n,10) - d. as the lattice structure is symmetrical, this means number_count_case_2_1 = number_count_case_1.
case 2.2: initial digit is min(n-1,9)
note that q1 is 1 and the second digit must be min(n-2,8).
thus
number_count_case_2_2 (n, m) =
sum_{q2=1..m-2, l_suffix=0..m-2-q2} (
paths_to_max(1,1,q2)
+ paths_suffix(min(n-1,9),q2,l_suffix+q2)
)
so the grand grand total will be
number_count ( n, m ) = 2 * number_count_case_1 (n, m) + number_count_case_2_2 (n, m);
Code
i don't know whether a closed expression for number_count exists, but the following perl code will compute it (the code is but a proof of concept as it does not use memoization techniques to avoid recomputing results already obtained):
use strict;
use warnings;
my ($n, $m) = ( 5, 7 ); # for example
$n = ($n > 10) ? 10 : $n; # cutoff
sub min
sub paths_to_0 ($$$) {
my (
$d
, $at
, $until
) = #_;
#
if (($d == 0) && ($at == $until - 1)) { return 0; }
if (($d == 1) && ($at == $until - 1)) { return 1; }
if ($until - $at < $d) { return 0; }
if (($d <= 0) || ($d >= $n))) { return 0; }
return paths_to_0($d+1, $at+1, $until) + paths_to_0($d-1, $at+1, $until);
} # paths_to_0
sub paths_to_max ($$$) {
my (
$d
, $at
, $until
) = #_;
#
if (($d == $n-1) && ($at == $until - 1)) { return 0; }
if (($d == $n-2) && ($at == $until - 1)) { return 1; }
if ($until - $at < $n-1) { return 0; }
if (($d < 0) || ($d >= $n-1)) { return 0; }
return paths_to_max($d+1, $at+1, $until) + paths_to_max($d-1, $at+1, $until);
} # paths_to_max
sub paths_suffix ($$$) {
my (
$d
, $at
, $until
) = #_;
#
if (($d < $n-1) && ($d > 0) && ($at == $until - 1)) { return 2; }
if ((($d == $n-1) && ($d == 0)) && ($at == $until - 1)) { return 1; }
if (($until > $m-1) || ($at > $until)) { return 0; }
if ($until == 0) { return 1; }
return paths_suffix($d+1, $at+1, $until) + paths_suffix($d-1, $at+1, $until);
} # paths_suffix
#
# main
#
number_count =
sum_{first=1..min(n-1,9), q1=1..m-1-(n-1), q2=q1..m-1, l_suffix=0..m-1-q2} (
paths_to_0(first,1,q1)
+ paths_to_max(0,q1,q2)
+ paths_suffix(min(n-1,9),q2,l_suffix+q2)
)
my ($number_count, $number_count_2_2) = (0, 0);
my ($first, $q1, i, $l_suffix);
for ($first = 1; $first <= $n-1; $first++) {
for ($q1 = 1; $q1 <= $m-1 - ($n-1); $q1++) {
for ($q2 = $q1; $q2 <= $m-1; $q2++) {
for ($l_suffix = 0; $l_suffix <= $m-1 - $q2; $l_suffix++) {
$number_count =
$number_count
+ paths_to_0($first,1,$q1)
+ paths_to_max(0,$q1,$q2)
+ paths_suffix($n-1,$q2,$l_suffix+$q2)
;
}
}
}
}
#
# case 2.2
#
for ($q2 = 1; $q2 <= $m-2; $q2++) {
for ($l_suffix = 0; $l_suffix <= $m-2 - $q2; $l_suffix++) {
$number_count_2_2 =
$number_count_2_2
+ paths_to_max(1,1,$q2)
+ paths_suffix($n-1,$q2,$l_suffix+$q2)
;
}
}
$number_count = 2 * $number_count + number_count_2_2;

Number distribution

Problem: We have x checkboxes and we want to check y of them evenly.
Example 1: select 50 checkboxes of 100 total.
[-]
[x]
[-]
[x]
...
Example 2: select 33 checkboxes of 100 total.
[-]
[-]
[x]
[-]
[-]
[x]
...
Example 3: select 66 checkboxes of 100 total:
[-]
[x]
[x]
[-]
[x]
[x]
...
But we're having trouble to come up with a formula to check them in code, especially once you go 11/111 or something similar. Anyone has an idea?

Let's first assume y is divisible by x. Then we denote p = y/x and the solution is simple. Go through the list, every p elements, mark 1 of them.
Now, let's say r = y%x is non zero. Still p = y/x where / is integer devision. So, you need to:
In the first p-r elements, mark 1 elements
In the last r elements, mark 2 elements
Note: This depends on how you define evenly distributed. You might want to spread the r sections withx+1 elements in between p-r sections with x elements, which indeed is again the same problem and could be solved recursively.
Alright so it wasn't actually correct. I think this would do though:
Regardless of divisibility:
if y > 2*x, then mark 1 element every p = y/x elements, x times.
if y < 2*x, then mark all, and do the previous step unmarking y-x out of y checkboxes (so like in the previous case, but x is replaced by y-x)
Note: This depends on how you define evenly distributed. You might want to change between p and p+1 elements for example to distribute them better.

Here's a straightforward solution using integer arithmetic:
void check(char boxes[], int total_count, int check_count)
{
int i;
for (i = 0; i < total_count; i++)
boxes[i] = '-';
for (i = 0; i < check_count; i++)
boxes[i * total_count / check_count] = 'x';
}
total_count is the total number of boxes, and check_count is the number of boxes to check.
First, it sets every box to unchecked. Then, it checks check_count boxes, scaling the counter to the number of boxes.
Caveat: this is left-biased rather than right-biased like in your examples. That is, it prints x--x-- rather than --x--x. You can turn it around by replacing
boxes[i * total_count / check_count] = 'x';
with:
boxes[total_count - (i * total_count / check_count) - 1] = 'x';
Correctness
Assuming 0 <= check_count <= total_count, and that boxes has space for at least total_count items, we can prove that:
No check marks will overlap. i * total_count / check_count increments by at least one on every iteration, because total_count >= check_count.
This will not overflow the buffer. The subscript i * total_count / check_count
Will be >= 0. i, total_count, and check_count will all be >= 0.
Will be < total_count. When n > 0 and d > 0:
(n * d - 1) / d < n
In other words, if we take n * d / d, and nudge the numerator down, the quotient will go down, too.
Therefore, (check_count - 1) * total_count / check_count will be less than total_count, with the assumptions made above. A division by zero won't happen because if check_count is 0, the loop in question will have zero iterations.

Say number of checkboxes is C and the number of Xes is N.
You example states that having C=111 and N=11 is your most troublesome case.
Try this: divide C/N. Call it D. Have index in the array as double number I. Have another variable as counter, M.
double D = (double)C / (double)N;
double I = 0.0;
int M = N;
while (M > 0) {
if (checkboxes[Round(I)].Checked) { // if we selected it, skip to next
I += 1.0;
continue;
}
checkboxes[Round(I)].Checked = true;
M --;
I += D;
if (Round(I) >= C) { // wrap around the end
I -= C;
}
}
Please note that Round(x) should return nearest integer value for x.
This one could work for you.

I think the key is to keep count of how many boxes you expect to have per check.
Say you want 33 checks in 100 boxes. 100 / 33 = 3.030303..., so you expect to have one check every 3.030303... boxes. That means every 3.030303... boxes, you need to add a check. 66 checks in 100 boxes would mean one check every 1.51515... boxes, 11 checks in 111 boxes would mean one check every 10.090909... boxes, and so on.
double count = 0;
for (int i = 0; i < boxes; i++) {
count += 1;
if (count >= boxes/checks) {
checkboxes[i] = true;
count -= count.truncate(); // so 1.6 becomes 0.6 - resetting the count but keeping the decimal part to keep track of "partial boxes" so far
}
}
You might rather use decimal as opposed to double for count, or there's a slight chance the last box will get skipped due to rounding errors.

Bresenham-like algorithm is suitable to distribute checkboxes evenly. Output of 'x' corresponds to Y-coordinate change. It is possible to choose initial err as random value in range [0..places) to avoid biasing.
def Distribute(places, stars):
err = places // 2
res = ''
for i in range(0, places):
err = err - stars
if err < 0 :
res = res + 'x'
err = err + places
else:
res = res + '-'
print(res)
Distribute(24,17)
Distribute(24,12)
Distribute(24,5)
output:
x-xxx-xx-xx-xxx-xx-xxx-x
-x-x-x-x-x-x-x-x-x-x-x-x
--x----x----x---x----x--

Quick html/javascript solution:
<html>
<body>
<div id='container'></div>
<script>
var cbCount = 111;
var cbCheckCount = 11;
var cbRatio = cbCount / cbCheckCount;
var buildCheckCount = 0;
var c = document.getElementById('container');
for (var i=1; i <= cbCount; i++) {
// make a checkbox
var cb = document.createElement('input');
cb.type = 'checkbox';
test = i / cbRatio - buildCheckCount;
if (test >= 1) {
// check the checkbox we just made
cb.checked = 'checked';
buildCheckCount++;
}
c.appendChild(cb);
c.appendChild(document.createElement('br'));
}
</script>
</body></html>

Adapt code from one question's answer or another answer from earlier this month. Set N = x = number of checkboxes and M = y = number to be checked and apply formula (N*i+N)/M - (N*i)/M for section sizes. (Also see Joey Adams' answer.)
In python, the adapted code is:
N=100; M=33; p=0;
for i in range(M):
k = (N+N*i)/M
for j in range(p,k-1): print "-",
print "x",
p=k
which produces
- - x - - x - - x - - x - - [...] x - - x - - - x where [...] represents 25 --x repetitions.
With M=66 the code gives
x - x x - x x - x x - x x - [...] x x - x x - x - x where [...] represents mostly xx- repetitions, with one x- in the middle.
Note, in C or java: Substitute for (i=0; i<M; ++i) in place of for i in range(M):. Substitute for (j=p; j<k-1; ++j) in place of for j in range(p,k-1):.
Correctness: Note that M = x boxes get checked because print "x", is executed M times.

What about using Fisher–Yates shuffle ?
Make array, shuffle and pick first n elements. You do not need to shuffle all of them, just first n of array. Shuffling can be find in most language libraries.

Determining valid adjacent cells of a square stored as an array

I have an array (of 9 elements, say) which I must treat as a (3 by 3) square.
For the sake of simplifying the question, this is a one-based array (ie, indexing starts at 1 instead of 0).
My goal is to determine valid adjacent squares relative to a starting point.
In other words, how it's stored in memory: 1 2 3 4 5 6 7 8 9
How I'm treating it:
7 8 9
4 5 6
1 2 3
I already know how to move up and down and test for going out of bounds (1 >= current_index <= 9)
edit: I know the above test is overly general but it's simple and works.
//row_size = 3, row_step is -1, 0 or 1 depending on if we're going left,
//staying put or going right respectively.
current_index += (row_size * row_step);
How do I test for an out of bounds condition when going left or right? Conceptually I know it involves determining if 3 (for example) is on the same row as 4 (or if 10 is even within the same square as 9, as an alternate example, given that multiple squares are in the same array back to back), but I can't figure out how to determine that. I imagine there's a modulo in there somewhere, but where?
Thanks very much,
Geoff
Addendum:
Here's the resulting code, altered for use with a zero-based array (I cleaned up the offset code present in the project) which walks adjacent squares.
bool IsSameSquare(int index0, int index1, int square_size) {
//Assert for square_size != 0 here
return (!((index0 < 0) || (index1 < 0))
&& ((index0 < square_size) && (index1 < square_size)))
&& (index0 / square_size == index1 / square_size);
}
bool IsSameRow(int index0, int index1, int row_size) {
//Assert for row_size != 0 here
return IsSameSquare(index0, index1, row_size * row_size)
&& (index0 / row_size == index1 / row_size);
}
bool IsSameColumn(int index0, int index1, int row_size) {
//Assert for row_size != 0 here
return IsSameSquare(index0, index1, row_size * row_size)
&& (index0 % row_size == index1 % row_size);
}
//for all possible adjacent positions
for (int row_step = -1; row_step < 2; ++row_step) {
//move up, down or stay put.
int row_adjusted_position = original_position + (row_size * row_step);
if (!IsSameSquare(original_position, row_adjusted_position, square_size)) {
continue;
}
for (int column_step = -1; column_step < 2; ++column_step) {
if ((row_step == 0) & (column_step == 0)) { continue; }
//hold on to the position that has had its' row position adjusted.
int new_position = row_adjusted_position;
if (column_step != 0) {
//move left or right
int column_adjusted_position = new_position + column_step;
//if we've gone out of bounds again for the column.
if (IsSameRow(column_adjusted_position, new_position, row_size)) {
new_position = column_adjusted_position;
} else {
continue;
}
} //if (column_step != 0)
//if we get here we know it's safe, do something with new_position
//...
} //for each column_step
} //for each row_step

This is easier if you used 0-based indexing. These rules work if you subtract 1 from all your indexes:
Two indexes are in the same square if (a/9) == (b/9) and a >= 0 and b >= 0.
Two indexes are in the same row if they are in the same square and (a/3) == (b/3).
Two indexes are in the same column if they are in the same square and (a%3) == (b%3).

There are several way to do this, I'm choosing a weird one just for fun. Use modulus.
Ase your rows are size 3 just use modulus of 3 and two simple rules.
If currPos mod 3 = 0 and (currPos+move) mod 3 = 1 then invalid
If currPos mod 3 = 1 and (currPos+move) mod 3 = 0 then invalid
this check for you jumping two a new row, you could also do one rule like this
if (currPos mod 3)-((currPos+move) mod 3)> 1 then invalid
Cheers

You should be using a multidimensional array for this.
If your array class doesn't support multidimensional stuff, you should write up a quick wrapper that does.

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