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One is a sum (Posted on 2013-07-05) Difficulty: 3 of 5
a. How many 4 digit numbers are there such that at least one digit is a sum of 2 or 3 of the others?
b. How many 5 digit numbers are there such that at least one digit is a sum of 2,3 or 4 of the others?

The following conditions apply both for a and for b:
*No leading zeros.
*The numbers consist of distinct digits.

No Solution Yet Submitted by Ady TZIDON    
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Solution computer solution Comment 1 of 1

DEFDBL A-Z
FOR n = 1000 TO 9999
  ns$ = LTRIM$(STR$(n))
  good = 1
  FOR i = 1 TO LEN(ns$) - 1
    IF INSTR(i + 1, ns$, MID$(ns$, i, 1)) THEN good = 0: EXIT FOR
  NEXT
  IF good THEN
    good = 0
    FOR one = 0 TO 1
      t1 = one * VAL(MID$(ns$, 1, 1))
    FOR two = 0 TO 1
      t2 = t1 + two * VAL(MID$(ns$, 2, 1))
    FOR three = 0 TO 1
      t3 = t2 + three * VAL(MID$(ns$, 3, 1))
    FOR four = 0 TO 1
      t4 = t3 + four * VAL(MID$(ns$, 4, 1))
      t = one + two + three + four


      IF t > 1 THEN
        IF VAL(MID$(ns$, 1, 1)) = t4 AND one = 0 THEN good = 1
        IF VAL(MID$(ns$, 2, 1)) = t4 AND two = 0 THEN good = 1
        IF VAL(MID$(ns$, 3, 1)) = t4 AND three = 0 THEN good = 1
        IF VAL(MID$(ns$, 4, 1)) = t4 AND four = 0 THEN good = 1
      END IF
     
      
    NEXT
    NEXT
    NEXT
    NEXT
    IF good THEN
     ct = ct + 1
    END IF
  END IF
NEXT
PRINT : PRINT ct

ct = 0

FOR n = 10000 TO 99999
  ns$ = LTRIM$(STR$(n))
  good = 1
  FOR i = 1 TO LEN(ns$) - 1
    IF INSTR(i + 1, ns$, MID$(ns$, i, 1)) THEN good = 0: EXIT FOR
  NEXT
  IF good THEN
    good = 0
    FOR one = 0 TO 1
      t1 = one * VAL(MID$(ns$, 1, 1))
    FOR two = 0 TO 1
      t2 = t1 + two * VAL(MID$(ns$, 2, 1))
    FOR three = 0 TO 1
      t3 = t2 + three * VAL(MID$(ns$, 3, 1))
    FOR four = 0 TO 1
      t4 = t3 + four * VAL(MID$(ns$, 4, 1))
    FOR five = 0 TO 1
      t5 = t4 + five * VAL(MID$(ns$, 5, 1))
      t = one + two + three + four + five

      IF t > 1 THEN
        IF VAL(MID$(ns$, 1, 1)) = t4 AND one = 0 THEN good = 1
        IF VAL(MID$(ns$, 2, 1)) = t4 AND two = 0 THEN good = 1
        IF VAL(MID$(ns$, 3, 1)) = t4 AND three = 0 THEN good = 1
        IF VAL(MID$(ns$, 4, 1)) = t4 AND four = 0 THEN good = 1
        IF VAL(MID$(ns$, 5, 1)) = t5 AND five = 0 THEN good = 1
      END IF
    
     
    NEXT
    NEXT
    NEXT
    NEXT
    NEXT
    IF good THEN
     ct = ct + 1
    END IF
  END IF
NEXT n
PRINT : PRINT ct

finds the respective answers as:

2400

and

17496


  Posted by Charlie on 2013-07-07 12:25:57
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