勉強会当日は時間内に解けなかったので、家で解き直しました。
もっと精進します。
object Hena {
  def makeBox(src:String, w:Int, h:Int) = { 
    src.map(arg => {
      val i = arg.toString.toInt
      List.fill((i - h).abs)('0') ++ List.fill(i)('1')
    }).toList.transpose
  }

  def overflowFromLeft(i:Int, l:List[Char]) = { 
    List.fill(i)(' ') ++ l.mkString.drop(i).toList
  }

  def overflowFromRight(i:Int, l:List[Char]) = { 
    l.mkString.dropRight(i).toList ++ List.fill(i)(' ')
  }
  
  def overflow(arg:List[Char]) = { 
    val l = overflowFromLeft(arg.indexOf('1'), arg)
    overflowFromRight(l.size - arg.lastIndexOf('1') - 1, l)
  }

  def solve(src:String) = { 
    val items = src.split("0").filter(!_.isEmpty)
    items.map(item => {
      val (w, h) = (item.length, item.max.toInt)
      makeBox(item, w, h).map(arg => arg match {
        case _ if(arg.head == '0' || arg.last == '0') => overflow(arg)
        case _ => arg 
      }).flatten
    }).mkString.count(_ == '0').toString
  }

  def test(src:String, expected:String) = { 
    val actual = solve(src)
    println(if(expected == actual) "ok" else "***NG***")
  }

  def main(args:Array[String]) = { 
    /*0*/ test( "83141310145169154671122", "24" ) 
    // 以下略
  }
}