Programming

$x = 1000;
echo 1.5*(int)(($x-1)/3)*(int)(($x+2)/3) + 2.5*(int)(($x-1)/5)*(int)(($x+4)/5) - 7.5*(int)(($x-1)/15)*(int)(($x+14)/15);

int total = 0;
int x = 1, y = 1;
do {
	if (x % 2 == 0)
		total += x;
	int z = x;
	x += y;
	y = z;
} while (x > 1000000);

int max = 0;
for(int i = 100; i <= 999; i++)
	for(int j = 100; j <= 999; j++)
		if (palin(j * i))
			if(j * i > max)
				max = j * i;
System.out.println(max); 

This does not require programming at all. 
Compute the prime factorization of each 
number from 1 to 20, and multiply the 
greatest power of each prime together:

20 = 2^2 * 5
19 = 19
18 = 2 * 3^2
17 = 17
16 = 2^4
15 = 3 * 5
14 = 2 * 7
13 = 13
11 = 11

All others are included in the previous numbers.

ANSWER: 2^4 * 3^2 * 5 * 7 * 11 * 13 * 17 * 19 
	= 232792560

require 'mathn'
n = 100
puts "Difference is #{ (n * (n**2 - 1) * (n / 4 + 1 / 6)).abs }."

$bignum = "73167...63450";
$largesum = 0;
 
for ($i = 0; $i < 994; $i++)
{
	$sum = $bignum[$i] * $bignum[$i+1] * $bignum[$i+2] * $bignum[$i+3] * $bignum[$i+4];

	if ($sum > $largesum)
	{
		echo "new = $sum, old = $largesum\n";
		$largesum = $sum;
	}
}
 
echo $largesum . "\n";

n = 1000
a = (1..n / 2).to_a.find { |a|
  (n * (n / 2 - a) % (n - a)).zero?
}
b = n * (n / 2 - a) / (n - a)
puts "Product is #{a * b * (n - a - b)}."

matrix =
"08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65
52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91
22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80
24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50
32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70
67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21
24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72
21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95
78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92
16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57
86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58
19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40
04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66
88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69
04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36
20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16
20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54
01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48"
matrix = matrix.split("\n").collect { |line|
  line.split(' ').collect { |n| n.to_i }
}
line = 4
len = matrix.length
matrix.each { |row| row.concat(Array.new(line - 1) { 0 }) }
matrix.concat(Array.new(line - 1) { Array.new(len + line - 1) { 0 } })
max = (0...len).to_a.collect { |row|
  (0...len).to_a.collect { |col|
    [
      matrix[row][col, line].inject(1) { |prod, n| prod * n },
 
      matrix[row, line].collect { |r|
        r[col]
      }.inject(1) { |prod, n| prod * n },
 
      (0...line).to_a.collect { |n|
        matrix[row + n][col + n]
      }.inject(1) { |prod, n| prod * n },
 
      (0...line).to_a.collect { |n|
        matrix[row + line - n - 1][col + n]
      }.inject(1) { |prod, n| prod * n }
    ].max
  }.max
}.max
puts "Maximum is #{max}."

part = "37107...31690
".split("\n").inject(0) { |sum, n| sum + n.to_i }.to_s[0, 10]
puts "Sum part is #{part}."