1. When completed, this crossnumber puzzle has exactly one digit in each box.
2. None of the a,b,c or d boxes (whether down or across) can contain any leading zero.
3. Both concatenations ab and cd consist of nonzero digits.

       +-----+-----+
       |  a  |  b  |
 +-----+-----+-----+
 |  c  |     |     |
 +-----+-----+-----+
 |  d  |     |      
 +-----+-----+


    ACROSS                                         DOWN                

a. Abigail's age                           a. Sum of three of the ages
                                             in c "across".
c. Sum of Abigail's age, 
Blanche's age, Cynthia's                   b. Cynthia's age.
age, and Darlene's age.                    
                                           c. Darlene's age.
d. Blanche's age                            

CLS
FOR abi = 10 TO 99
FOR bla = 10 TO 99
FOR cyn = 10 TO 99
FOR dar = 10 TO 99
  ab$ = LTRIM$(STR$(abi))
  IF RIGHT$(ab$, 1) <> "0" THEN
    bl$ = LTRIM$(STR$(bla))
    cy$ = LTRIM$(STR$(cyn))
    da$ = LTRIM$(STR$(dar))
    IF RIGHT$(da$, 1) <> "0" AND RIGHT$(da$, 1) = LEFT$(bl$, 1) AND RIGHT$(ab$, 1) = LEFT$(cy$, 1) THEN
      tot = abi + bla + cyn + dar
      IF tot > 99 AND tot < 1000 THEN
        t$ = LTRIM$(STR$(tot))
        IF LEFT$(t$, 1) = LEFT$(da$, 1) AND RIGHT$(t$, 1) = RIGHT$(cy$, 1) THEN
          aDown$ = LEFT$(ab$, 1) + MID$(t$, 2, 1) + RIGHT$(bl$, 1)
          aDn = VAL(aDown$)
          IF aDn = tot - abi OR aDn = tot - bla OR aDn = tot - cyn OR aDn = tot - dar THEN
           PRINT " "; ab$
           PRINT t$
           PRINT bl$;
           SELECT CASE tot
              CASE abi + aDn
               PRINT "   Abigail"
              CASE bla + aDn
               PRINT "   Blanche"
              CASE cyn + aDn
               PRINT "   Cynthia"
              CASE dar + aDn
               PRINT "   Darlene"
           END SELECT
           PRINT
          END IF
        END IF
      END IF
    END IF
  END IF
NEXT
NEXT
NEXT
NEXT

finds

 16
200
95   Blanche