+---+   +---+   +---+   +---+   +---+   
| 4 |-->| ● |-->| 9 |-->| ● |-->| 7 |
+---+   +---+   +---+   +---+   +---+
  ^       ^       ^       ^       ^
  |       |       |       |       |      
+---+   +---+   +---+   +---+   +---+   
| ● |-->| 7 |-->| ● |-->| 5 |-->| ● |
+---+   +---+   +---+   +---+   +---+
  ^       ^       ^       ^       ^
  |       |       |       |       |      
+---+   +---+   +---+   +---+   +---+   
| 5 |-->| ● |-->| 9 |-->| ● |-->| 4 |
+---+   +---+   +---+   +---+   +---+  
  ^       ^       ^       ^       ^
  |       |       |       |       |      
+---+   +---+   +---+   +---+   +---+   
| ● |-->| 8 |-->| ● |-->| 6 |-->| ● |
+---+   +---+   +---+   +---+   +---+
  ^       ^       ^       ^       ^ 
  |       |       |       |       |      
+---+   +---+   +---+   +---+   +---+   
| 6 |-->| ● |-->| 8 |-->| ● |-->| 5 |
+---+   +---+   +---+   +---+   +---+

Move from the bottom left-hand 6 to the top right-hand 7 in the direction of the arrows, adding together all the five numbers .

Each black circle is worth minus 5 and this will be taken from the total each time you meet one.

How many different routes are there- each giving a total of 10 ?

DECLARE SUB moveIt ()
CLEAR , , 25000
CLS
DIM SHARED gr(5, 5), totWays, row, col, totval, lv, hist(10)

OPEN "diffrtes.txt" FOR OUTPUT AS #2

DATA 4,-5,9,-5,7
DATA -5,7,-5,5,-5
DATA 5,-5,9,-5,4
DATA -5,8,-5,6,-5
DATA 6,-5,8,-5,5

FOR row = 1 TO 5
FOR col = 1 TO 5
  READ gr(row, col)
NEXT
NEXT

row = 5: col = 1
totval = gr(row, col)
lv = 1: hist(1) = 6

moveIt

PRINT totWays

SUB moveIt
  lv = lv + 1
  IF row = 1 AND col = 5 THEN
     IF totval = 10 THEN
       totWays = totWays + 1
       FOR i = 1 TO lv - 1: PRINT hist(i); : NEXT: PRINT
       FOR i = 1 TO lv - 1
         IF hist(i) <> -5 THEN PRINT #2, hist(i);
       NEXT: PRINT #2,
     END IF
  ELSE
    FOR delcol = 0 TO 1
      delrow = delcol - 1
      IF col + delcol < 6 AND delcol > 0 THEN
        col = col + delcol
        totval = totval + gr(row, col)
        hist(lv) = gr(row, col)
        moveIt
        totval = totval - gr(row, col)
        col = col - delcol
      END IF
      IF row + delrow > 0 AND delrow < 0 THEN
        row = row + delrow
        totval = totval + gr(row, col)
        hist(lv) = gr(row, col)
        moveIt
        totval = totval - gr(row, col)
        row = row - delrow
      END IF
    NEXT
  END IF
  lv = lv - 1
END SUB

finds five routes:

 6  5  7  5  7
 6  5  7  5  7
 6  5  7  5  7
 6  5  7  5  7
 6  8  5  4  7
 
The numbers add up to 30, so the four dots each incurs along the way subtract a total of 20, leaving 10.

The sequence 6, 5, 7, 5, 7 occurs four times as there are two ways of getting from each 5 to each 7. The sole way of getting 6, 8, 5, 4, 7 is along the bottom and then up the right side.

Posted by Charlie on 2013-09-09 12:11:38