diff -r d39b8733c6ea -r 293ea84d82ca progs/roman_sol.scala
--- a/progs/roman_sol.scala	Thu Jan 26 01:43:31 2017 +0000
+++ b/progs/roman_sol.scala	Fri Jan 27 14:55:56 2017 +0000
@@ -81,7 +81,7 @@
 }
 
 // some test cases
-RomanNumeral2Int(List(I,I,I,I))         // 4
+RomanNumeral2Int(List(I,I,I,I))         // 4 (invalid roman number)
 RomanNumeral2Int(List(I,V))             // 4
 RomanNumeral2Int(List(V,I))             // 6
 RomanNumeral2Int(List(I,X))             // 9
@@ -96,8 +96,8 @@
 def String2Int(s: String): Option[Int] = 
   String2RomanNumeral(s).map(RomanNumeral2Int(_))
 
-
-String2Int("IIII")      // 4 invalid 
+// some test cases
+String2Int("IIII")      // 4 (though invalid roman numeral)  
 String2Int("IV")        // 4
 String2Int("VI")        // 6
 String2Int("IX")        // 9
@@ -112,10 +112,6 @@
 String2Int("MMXI")      // 2011 
 String2Int("MIM")       // 1999
 String2Int("MCMLVI")    // 1956 
-String2Int("XXCIII")    // ?? 103 / 83
-String2Int("LXXIIX")    // ?? 80 / 78
-String2Int("IIIIX")     // ?? 12 invalid  
-String2Int("IIXX")      // ?? 20 / 18 invalid
 
 String2Int("III") 	// 3
 String2Int("XXX") 	// 30
@@ -128,17 +124,16 @@
 String2Int("IV") 	// 4
 String2Int("IX") 	// 9
 String2Int("XC") 	// 90
-String2Int("ICM") 	// ?? 901
-String2Int("CIM") 	// ?? 899
 String2Int("MDCLXVI")	// 1666
 
-String2Int("VIV")       // 9, but should be written as IX 
-String2Int("IVX")       // 14 invalid
+String2Int("VIV")       // 9 (but should be written as IX) 
+String2Int("IVX")       // 14 (also invalid)
 
 // error cases
 String2Int("MC?I")
 String2Int("abc")
 
+
 // (5) The file roman.txt contains a list of roman numerals. 
 // Read in these numerals, convert them into integers and then 
 // add them all up.
@@ -158,26 +153,111 @@
 addromanfile("roman.txt")
 
 
+// Part 2 about Validation of Roman Numerals
+//===========================================
 
-
+def Digit2Int(r: RomanDigit) = r match {
+  case I => 1
+  case V => 5
+  case X => 10
+  case L => 50
+  case C => 100
+  case D => 500
+  case M => 1000
+}
 
 def runsAllowed(r: RomanDigit) = r match {
   case I | X | C | M => true
   case V | L | D => false
 }
 
-def norunsAllowed(r: RomanDigit) = !runsAllowed(r)
-
+def subtractable(r1: RomanDigit, r2: RomanDigit) = (r1, r2)  match {
+  case (I, V) | (I, X) => true
+  case (X, L) | (X, C) => true
+  case (C, D) | (C, M) => true
+  case _ => false
+}
 
-def isValidNumeral(digitList: RomanNumeral): Bool = digitList match {
+def isValidNumeral(digitList: RomanNumeral): Boolean = digitList match {
 
-    // empty list is valid
+  // empty list is valid
   case Nil => true
 
-    // a following digit that is equal or larger is an error
-  case d1::d2::_ if (d1 <= d2)  => false 
+  // no more than three runnables in succession
+  case d1::d2::d3::d4::_ if (d1 == d2 && d1 == d3 && d1 == d4 && runsAllowed(d1)) => false
+
+  // no more than one non-runnables in succession
+  case d1::d2::_ if (d1 == d2 && !runsAllowed(d1)) => false
+
+  // subtractable
+  case d1::d2::_ if (Digit2Int(d1) < Digit2Int(d2) && !subtractable(d1, d2)) => false
+
+  // followable1
+  case d1::d2::d3::_ if (Digit2Int(d2) < Digit2Int(d3) && subtractable(d2,d3) && 
+                         Digit2Int(d1) < Digit2Int(d2) * 10) => false
+
+  // followable2 
+  case d1::d2::d3::_ if (Digit2Int(d1) < Digit2Int(d2) && subtractable(d1,d2) && 
+                         Digit2Int(d1) <= Digit2Int(d3))  => false 
 
     // A single digit is always allowed
   case _::ds => isValidNumeral(ds) 
 
 }
+
+
+val invalids = List("IXC", "XCX", "IIII", "IIIII", "DD", "VL", "MIM", "XXCIII", "LXXIIX", "IIIIX",
+                    "IIXX", "ICM", "CIM", "VIV", "IVX", "MCMC", "XIIX", "IIXX")
+invalids.map(String2RomanNumeral(_)).flatten.map(isValidNumeral(_))
+invalids.map(String2RomanNumeral(_)).flatten.map((r) => (r, isValidNumeral(r)))
+
+
+val valids = List("IV", "VI", "IX", "MCMLXXIX", "MCMXLIV", "", "MDCLXI",
+                  "MMMCMXCIX", "XLVIII", "MMVIII", "MMXI", "MCMLVI", "III",
+                  "XXX", "CCC", "MMM", "VII", "LXVI", "CL", "MCC", "XC",
+                  "MDCLXVI")
+valids.map(String2RomanNumeral(_)).flatten.map(isValidNumeral(_))
+
+
+
+
+
+def Int2Roman(n: Int): RomanNumeral = n match {
+  case 0 => Nil
+  case n if (1000 <= n) => M :: Int2Roman(n - 1000)
+  case n if (900 <= n) => C :: M :: Int2Roman(n - 900)
+  case n if (500 <= n) => D :: Int2Roman(n - 500)
+  case n if (400 <= n) => C :: D :: Int2Roman(n - 400)
+  case n if (100 <= n) => C :: Int2Roman(n - 100)
+  case n if (90 <= n) => X :: C :: Int2Roman(n - 90)
+  case n if (50 <= n) => L :: Int2Roman(n - 50)
+  case n if (40 <= n) => X :: L :: Int2Roman(n - 40)
+  case n if (10 <= n) => X :: Int2Roman(n - 10)
+  case n if (9 <= n) => I :: X :: Int2Roman(n - 9)
+  case n if (5 <= n) => V :: Int2Roman(n - 5)
+  case n if (4 <= n) => I :: V :: Int2Roman(n - 4)
+  case n if (1 <= n) => I :: Int2Roman(n - 1)
+}
+
+
+Int2Roman(4)
+Int2Roman(6)
+Int2Roman(9)
+Int2Roman(1979)
+Int2Roman(1944)
+
+RomanNumeral2Int(List(I,V))             
+RomanNumeral2Int(List(V,I))             
+RomanNumeral2Int(List(I,X))             
+RomanNumeral2Int(List(M,C,M,L,X,X,I,X)) 
+RomanNumeral2Int(List(M,C,M,X,L,I,V))
+
+
+def addvalidromanfile(filename: String) = {
+  val lines = Source.fromFile(filename)("ISO-8859-9").getLines.toList.map(_.trim)
+  val ints = lines.map(String2RomanNumeral(_)).flatten.filter(isValidNumeral(_)).map(RomanNumeral2Int(_))
+  ints.sum
+}
+
+
+addvalidromanfile("roman2.txt")