+---+   +---+   +---+   +---+   +---+   
| 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 ?

You know you will hit 4 black circles (-20) and have to use the 6 and 7 (+13) so to end with 10 you need 17 more from 3 numbers.

The last of these 3 number is either 9, 5 of 4.
The rest is easy by inspection:
To use the 9 you'd need the others to add to 8.  Not possible.
To use the 5 you'd need the others to add to 12.  Only 5 and 7 works.
To use the 4 you'd need the others to ass to 13.  Only 8 and 5 works.
So the number sequences are
6,5,7,5,7 and 6,8,5,4,7

The first has 4 routes (two each from 5 to 7)
The second has only 1 route.

There are 5 routes total.

Posted by Jer on 2013-09-09 14:12:50