3 files changed, 107 insertions, 0 deletions
diff --git a/julia/015-lattice_paths.jl b/julia/015-lattice_paths.jl
new file mode 100644
index 0000000..c155da6
--- /dev/null
+++ b/julia/015-lattice_paths.jl
@@ -0,0 +1,30 @@
+###
+# Lattice paths
+# Problem 15
+#
+# Starting in the top left corner of a 2×2 grid, and only being able to move to the right
+# and down, there are exactly 6 routes to the bottom right corner.
+# How many such routes are there through a 20×20 grid?
+###
+
+
+# memoization is not enough for going deeper in the triangle,
+# there is a formula to get the nth row without computing the previous ones
+cache = Dict()
+
+function pascal_triangle(n, k)
+    key = (n, k)
+    if haskey(cache, key)
+        return cache[key]
+    end
+    if n == 0 || k == 0 || n == k
+        return 1
+    end
+    cache[key] = pascal_triangle(n - 1, k - 1) + pascal_triangle(n - 1, k)
+    return cache[key]
+end
+
+const LATTICE_LENGTH = 20
+result = pascal_triangle(2LATTICE_LENGTH, LATTICE_LENGTH)
+
+println(result)
diff --git a/julia/016-power_digit_sum.jl b/julia/016-power_digit_sum.jl
new file mode 100644
index 0000000..848accf
--- /dev/null
+++ b/julia/016-power_digit_sum.jl
@@ -0,0 +1,14 @@
+###
+# Power digit sum
+# Problem 16
+#
+# 2^15 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.
+# What is the sum of the digits of the number 2^1000?
+###
+
+
+const NUMBER = big(2) ^  1000
+
+result = sum(parse(Int, digit) for digit in string(NUMBER))
+
+println(result)
diff --git a/julia/018-maximum_path_sum_i.jl b/julia/018-maximum_path_sum_i.jl
new file mode 100644
index 0000000..4ba802e
--- /dev/null
+++ b/julia/018-maximum_path_sum_i.jl
@@ -0,0 +1,63 @@
+###
+# Maximum path sum I
+# Problem 18
+#
+# By starting at the top of the triangle below and moving to adjacent numbers on the row
+# below, the maximum total from top to bottom is 23.
+# 37 4
+# 2 4 6
+# 8 5 9 3
+# That is, 3 + 7 + 4 + 9 = 23.
+# Find the maximum total from top to bottom of the triangle below:
+# 75
+# 95 64
+# 17 47 82
+# 18 35 87 10
+# 20 04 82 47 65
+# 19 01 23 75 03 34
+# 88 02 77 73 07 63 67
+# 99 65 04 28 06 16 70 92
+# 41 41 26 56 83 40 80 70 33
+# 41 48 72 33 47 32 37 16 94 29
+# 53 71 44 65 25 43 91 52 97 51 14
+# 70 11 33 28 77 73 17 78 39 68 17 57
+# 91 71 52 38 17 14 91 43 58 50 27 29 48
+# 63 66 04 68 89 53 67 30 73 16 69 87 40 31
+# 04 62 98 27 23 09 70 98 73 93 38 53 60 04 23
+# NOTE: As there are only 16384 routes, it is possible to solve this problem by trying
+# every route. However, Problem 67, is the same challenge with a triangle containing one-
+# hundred rows; it cannot be solved by brute force, and requires a clever method! ;o)
+###
+
+
+const TRIANGLE = [
+                                [75],
+                              [95, 64],
+                            [17, 47, 82],
+                          [18, 35, 87, 10],
+                        [20, 04, 82, 47, 65],
+                      [19, 01, 23, 75, 03, 34],
+                    [88, 02, 77, 73, 07, 63, 67],
+                  [99, 65, 04, 28, 06, 16, 70, 92],
+                [41, 41, 26, 56, 83, 40, 80, 70, 33],
+              [41, 48, 72, 33, 47, 32, 37, 16, 94, 29],
+            [53, 71, 44, 65, 25, 43, 91, 52, 97, 51, 14],
+          [70, 11, 33, 28, 77, 73, 17, 78, 39, 68, 17, 57],
+        [91, 71, 52, 38, 17, 14, 91, 43, 58, 50, 27, 29, 48],
+      [63, 66, 04, 68, 89, 53, 67, 30, 73, 16, 69, 87, 40, 31],
+    [04, 62, 98, 27, 23, 09, 70, 98, 73, 93, 38, 53, 60, 04, 23],
+]
+
+function triangle_sum(triangle)
+    if length(triangle) == 0
+        return 0
+    end
+    low   = triangle[2:end]
+    left  = map(row -> row[1:end - 1], low)
+    right = map(row -> row[2:end], low)
+    return triangle[1][1] + max(triangle_sum(left), triangle_sum(right))
+end
+
+result = triangle_sum(TRIANGLE)
+
+println(result)