Solved 68 and rewrote 69 in Python.

python/e068.py

@@ -1,8 +1,76 @@
+from lib_misc import permutations
+
+
+def all_equal(xs):
+    x_1 = xs[0]
+    for x in xs[1:]:
+        if x_1 != x:
+            return False
+    return True
+
+
+def is_3_gon_ring(r):
+    if not (r[2] > r[0] and r[1] > r[0]):
+        return False
+    line_034 = r[0] + r[3] + r[4]
+    line_154 = r[1] + r[5] + r[4]
+    line_253 = r[2] + r[5] + r[3]
+    return all_equal([line_034, line_154, line_253])
+
+
+def is_5_gon_ring(r):
+    if not (r[1] > r[0] and r[2] > r[0] and
+            r[3] > r[0] and r[4] > r[0]):
+        return False
+    l_056 = r[0] + r[5] + r[6]
+    l_167 = r[1] + r[6] + r[7]
+    l_278 = r[2] + r[7] + r[8]
+    l_389 = r[3] + r[8] + r[9]
+    l_495 = r[4] + r[9] + r[5]
+    return all_equal([l_056, l_167, l_278, l_389, l_495])
+
+
+def five_gon_ring_to_group_presentation(r):
+    xs = [r[0], r[5], r[6],
+          r[1], r[6], r[7],
+          r[2], r[7], r[8],
+          r[3], r[8], r[9],
+          r[4], r[9], r[5]]
+    xs = "".join(map(str, xs))
+    return xs
+
+
 def euler_068():
-    # XXX: continue here
-    return 0
+    """
+    This one was really fun. The hardest part was to get the
+    check for the representation right.  This is how my five
+    gon ring is indexed:
+
+    #    0    1   #
+    #      5      #
+    #    9   6    #
+    #  4          #
+    #     8 7 2   #
+    #             #
+    #      3      #
+
+    There is probably and automatic indexing scheme that would
+    make the check and presentation function easier to write, but not
+    necessarily easier to read. Also generating all permutations creates
+    a lot of over head. For example, the lowest number cannot be higher
+    than six so that removes about 40% of the permutations right away and
+    there are probably more heuristics.
+    """
+    rs = []
+    for p in permutations(list(range(1, 11))):
+        if is_5_gon_ring(p):
+            r = five_gon_ring_to_group_presentation(p)
+            if len(r) == 16:
+                print(r)
+                rs.append(int(r))
+    return max(rs)
 
 
 if __name__ == "__main__":
     print("e068.py: " + str(euler_068()))
-    assert(euler_068() == 0)
+    assert(euler_068() == 6531031914842725)