Find nine single digit numbers (other than 1, 2, 3, ..., and 9) with a sum of 45 and a product of 9! (362,880).

For instance, 2, 2, 3, 4, 5, 6, 7, 8, 8 add up to 45, but their product is 645,120.

Try finding the answer without the use of a program.

I tried, using the thoughts in my previous post, but heck, that was beating my head against the wall for a D2 problem.

CLS
FOR a = 1 TO 5
FOR b = a TO (45 - a) / 8
tb = a + b: pb = a * b
FOR c = b TO (45 - tb) / 7
tc = tb + c: pc = pb * c
FOR d = c TO (45 - tc) / 6
td = tc + d: pd = pc * d
FOR e = d TO (45 - td) / 5
te = td + e: pe = pd * e
FOR f = 9 - te TO (45 - te) / 4
 IF f >= e THEN
  tf = te + f: pf = pe * f
  FOR g = 27 - tf TO (45 - tf) / 3
   IF g >= f THEN
    tg = tf + g: pg = pf * g
    FOR h = 36 - tg TO (45 - tg) / 2
     IF h >= g THEN
      th = tg + h: ph = pg * h
      i = 45 - th
      IF i < 10 AND i >= h THEN
        pi = ph * i
        IF pi = 362880 THEN
          PRINT a; b; c; d; e; f; g; h; i
        END IF
      END IF
     END IF
    NEXT h
   END IF
  NEXT g
 END IF
NEXT f
NEXT e
NEXT d
NEXT c
NEXT b
NEXT a

finds

 1  2  3  4  5  6  7  8  9
 1  2  4  4  4  5  7  9  9

the first being 9! form and the second the desired set.

In terms of my previous thoughts, that involves, relative to the 9 factorial form: 

1. splitting apart the 8 into 4 and 2; losing 2 from the total and adding one number

2. splitting 6 into 3 and 2; losing 1 more from the total and adding another number

3. combining the two new 2's into 4; reducing the number of numbers by 1

4. combining the two new 3's into 9; losing all the gained 3 from the total and reducing the number of numbers by 1