Saturday, April 11, 2009

Project Euler problems 11 - 15

Here are problems 11 - 16 for project Euler.

Problem 11: What is the greatest product of four adjacent numbers in any direction (up, down, left, right, or diagonally) in the 20 20 grid? I used an array of arrays to represent the grid.

def euler11(grid)
maxVal = 0
(0..19).each{ |i|
(0..19).each{ |j|
# calculate horizontal
if j < 15 then
val = grid[i][j] * grid[i][j+1]
* grid[i][j+2] *grid[i][j+3]
maxVal = val if maxVal < val

# calculate diag down right
if i < 15 then
val = grid[i][j] * grid[i+1][j+1]
* grid[i+2][j+2] * grid[i+3][j+3]
maxVal = val if maxVal < val
end

# calculate diag up right
if i > 3 then
val = grid[i][j] * grid[i-1][j+1]
* grid[i-2][j+2] * grid[i-3][j+3]
maxVal = val if maxVal < val
end
end

# calculate vertical
if i < 15 then
val = grid[i][j] * grid[i+1][j]
* grid[i+2][j] * grid[i+3][j]
maxVal = val if maxVal < val
end
}
}
maxVal
end
Problem 12: What is the value of the first triangle number to have over five hundred divisors?

A triangle number is calculated as Tn = (n**2 + n) / 2.

def euler12()
maxInt = 1 << 64
(500..maxInt).each{|val|
numT = (val**2 + val)/2
limit = Math.sqrt(numT)
stepVal = (0 == numT % 2) ? 1 : 2
divs = 0
(1..limit).step(stepVal){ |x|
if 0 == numT % x then
divs += 2
end
}
return numT if divs > 500
}
end
Problem 13: Find the first 10 digits (most significant) of the sum of 100 50-digit numbers.

def euler13
numArr = [...]
sum = numArr.inject(0){|s,n| n+s}
val=sum.to_s.slice(0..9)
end
Problem 14: Find the starting number, under 1,000,000, that produces the longest chain for the Collatz problem.

The Collatz conjecture, or problem, generates a chain of numbers that terminates at 1. The chain is generated as Ni+1 = { Ni / 2 if Ni is even, 3 Ni + 1 if Ni is odd. At first read this was an "egads!" problem. But after looking at it was depressing easy to solve. The immediate solution that came to mind is a recursive algorithm using memoization. As follows, this runs in just under 59 seconds on my computer.

def findLen(start, cache)
len = cache[start]
if len
return start, len
else
nextVal = (0 == start%2) ? start / 2 : 3 * start + 1
nextVal, len = findLen(nextVal, cache)
cache[start] = len + 1
return start, len +1
end
end

def euler14()
cache = { 1 => 1 }
maxVal = 1
maxLen = 1

(2..1000000).each{ |start|
val, len = findLen(start, cache)

if maxLen < len
maxLen = len
maxVal = val
end
}

return maxVal, maxLen
end
I came up with an iterative approach as well. This is the same algorithm as the recursive solution, but it pushes values with unknown chain length onto a stack, instead of pushing them onto the call stack. This runs in just over 10 seconds which give a good indication on how expensive function calls can be.

def euler14_b()
cache = {1 => 1}
maxVal = 1
maxLen = 1
stack = []

(2..1000000).each{ |start|

val = start
len = cache[val]
while !len
stack.push(val)
val = (0 == val%2) ? val / 2 : 3 * val + 1
len = cache[val]
end

val = stack.pop
while val
len += 1
cache[val] = len
val = stack.pop
end

if maxLen < len
maxLen = len
maxVal = start
end
}

return maxVal, maxLen

end
Problem 15: How many routes are there through a 20 20 grid?

This was the most interesting problem out of this set. My first idea was to start at the end and generate valid paths back to the start. My second idea was, hey this is a tree, I should be able to calculate the number of paths directly. I thought that the number of paths should be proportional to the height of the tree. A good initial thought, but a little time at the white board showed that it was incorrect. A Google search turned up a blog entries by James Horsley and Real Ultimate Programming, after reading them and cracking open my dusty old discrete math books the answer is 40! / (20! * 20!) or C(40, 20).

To describe the answer in a bit more detail. One must always make 40 moves to get from the start to the finish. 20 moves down and 20 moves right. This gives 40! possibilities (20 down + 20 right). But since moves are indistinguishable, unordered, and without replacement, the total is reduced by 20! ways to select down moves, and 20! ways to select right moves. Or 40! / (20! * 20!).