RogerBW's Blog

I’ve been doing the Weekly Challenges. The latest involved numbers excluding 1, and Minesweeper. (Note that this is open until 22 August 2021.)

TASK #1 › Count Numbers

You are given a positive integer $N.

Write a script to print count of numbers from 1 to $N that don’t contain digit 1.

I came up with four separate approaches to this.

Simplest: generate all the numbers in the range, ignore anything with a 1 in it, count what's left. This is doable with standard Unix commands:

seq 1 $N |grep -vc 1

But for example given the range 1..200 that will generate and then throw away all the numbers from 100 to 199 inclusive. Which seems wasteful.

Next thought: generate the numbers, but regex replace a 1 with a 2 whenever it shows up. That jumps over the gap from 99 to 200 in a single step. (If I'm incrementing by 1 each time, the only numbers with 1 in them that I see will take the form xxx1000, with arbitrary numbers of xs and 0s.) That still starts to bog down a bit when it's counting numbers from 1 to 100,000,000.

Using regular expressions on numbers is ugly, so version three: generate the numbers, split then into digits with integer division, then do basically the same thing as above. On Perl at least, this turns out to be rather slower.

All of these approaches seem as though they're basically O(n log n): we have to iterate through something like each number, and do something related to the number of digits in that number.

But look at it another way. What we have here is a sequence (specifically OEIS #A052383, though we're ignoring the leading zero). Given a number (that may lie in that sequence or may not), we want to find out how many terms come before it.

If it is a number in that sequence, then that's simply the input interpreted as base 9, with 2..9 in the input becoming 1..8.

If it's not in that sequence (i.e. it has a 1 in it), we need the highest number in the sequence that's less than it. The highest number in the sequence that's less than xxx1yyy (in our special input format) is xxx0888 (base 9) – assuming none of the xs is a 1.

And I think this is more like O(log n), because it just scales with the length of the number.

Without arbitrary-length arrays this wasn't amenable to PostScript. I'll use the Rust version to explain:

fn cn(n: i64) -> i64 {
    let mut k=n;
    let mut digits: Vec<i64>=vec![];

I'm going to break down the input, pushing its digits into digits, least significant first.

    while k>0 {
        let mut d=k % 10;

If the digit is a 1, then all the less-significant digits become 8. (Which is slightly wasteful, I suppose, because I've already calcuated them; if I did it left-to-right I could short-circuit the process. But this was simpler.)

        if d==1 {
            digits=vec![8;digits.len()];
        }

If the digit wasn't a 0, then subtract 1 (i.e. 1 becomes 0 with the special case dealt with above, 2 becomes 1, etc.)

        if d>0 {
            d -= 1;
        }
        digits.push(d);
        k /= 10;
    }

Now we have an array of base-9 digits, and we build those back up into a number.

    let mut tc: i64=0;
    for i in (0..=digits.len()-1).rev() {
        tc *= 9;
        tc += digits[i];
    }
    return tc;
}

TASK #2 › Minesweeper Game

You are given a rectangle with points marked with either x or *. Please consider the x as a land mine.

Write a script to print a rectangle with numbers and x as in the Minesweeper game.

A number in a square of the minesweeper game indicates the number of mines within the neighbouring squares (usually 8), also implies that there are no bombs on that square.

Well, sadly it's not really amenable to my normal test-based approach…

my @in=(
  [qw(x * * * x * x x x x)],
  [qw(* * * * * * * * * x)],
  [qw(* * * * x * x * x *)],
  [qw(* * * x x * * * * *)],
  [qw(x * * * x * * * * x)],
    );

my $ymax=$#in;
my $xmax=$#{$in[0]};

I have a second array mn ("mine neighbours") initialised to 0.

my @mn;
foreach (0..$ymax) {
  push @mn,[(0) x ($xmax+1)];
}

I scan across each cell, and for each one that has a mine in it I add 1 to the neighbouring values in mn. (Basically the same approach as the naïve Conway's Life algorithm. I constrain the edges with max and min.)

foreach my $y (0..$ymax) {
  my @sy=(max(0,$y-1)..min($ymax,$y+1));
  foreach my $x (0..$xmax) {
    my @sx=(max(0,$x-1)..min($xmax,$x+1));
    if ($in[$y][$x] eq 'x') {
      foreach my $yi (@sy) {
        foreach my $xi (@sx) {
          if ($xi==$x && $yi==$y) {
            next;
          }
          $mn[$yi][$xi]++;
        }
      }
    }
  }
}

Then replace the values in the mine cells with x for display purposes, and print each line.

foreach my $y (0..$ymax) {
  foreach my $x (0..$xmax) {
    if ($in[$y][$x] eq 'x') {
      $mn[$y][$x]='x';
    }
  }
  print join(' ',@{$mn[$y]}),"\n";
}

Full code on github.