IlluminAir is a small international airline that provides service between Toronto, Ontario; Reston,
Virginia; and Doha, Qatar. There are 17 different routes from Doha to Reston, including those
that go through Toronto. There are 11 different routes from Reston to Toronto, including those
that go through Doha. How many routes are there from Doha to Toronto?
Solution:
Before delving into the problem, consider a simpler problem of routes between three cities A, B
and C
A
B
C
If we want to travel from A to B, you can take the 3 directs routes (in purple) or you can go from
A to C using one of the two routes (in green) and travel from C to B using the one route (in red).
So total number of ways to get from A to B = # of Direct Routes from A to B + (# of routes direct
from A to C) x (# of Routes direct from C to B) = 3 + 2 x 1 = 5 ways
Similarly we can write # of ways to get from A to C = # of Direct routes from A to C + (# of direct
routes from A to B) x (# of direct Routes between B to C) = 2 + 3 x 1 = 5
Similarly we can write # of ways to get from B to C = # of Direct routes from B to C + (# of direct
routes from B to A) x (# of direct Routes between A to C) = 1 + 3 x 2 = 7
Now consider the above problem of IlliminAir which has routes between the three cities
Toronto, Reston and Doha.
Let X be the number of direct routes between Toronto and Doha, Y be the number of direct
routes between Doha and Reston and Z be the number of direct routes between Reston and
Toronto. See the diagram below for representation
Toronto
T
Z
X
D
Y
Doha
R
Reston
It is given that there are 17 total routes between Doha and Reston including the direct and
indirect routes
We can write Y + XZ = 17 ------------- 1
of routes that go through Toronto
Where Y is the # of direct routes and XZ is the number
It is also given that there are 11 total routes between Reston and Toronto including the direct
and indirect routes
We can write Z + YX = 11 -------------------- 2 where Z is the # of direct routes between Reston and
Toronto and YX is the number of routes that go through Doha
Adding the two equations we get
Y + Z + XZ + YX = 28 or (Y+Z) +X(Z+Y) = 28 or (Y+Z)(1+X) = 28 -------------------- 3
Subtract equations 1 – Equation 2
Y + XZ – Z – YX = 6 or X(Z-Y) – 1(Z-Y) = 6 or (Z-Y)(X-1) = 6 -------------------------- 4
From Equation 3: 1, 2, 4, 7, 14 are the factors of 28
From Equations 4: 1, 2, 3 are the factors of 6.
We know that the difference between X+1 and X-1 has to be two and X+1 > X-1. So the only two
numbers in the factors with a difference of two that satisfy these are X+1 = 4 and X-1 = 2
So X = 3. Substituting X = 3 in Equation 1 and 2 we get
Y+3Z = 17 ----------- 5
Z+3Y = 11 ----------- 6
Multiply equation 6 by -3
-3Z -9Y = -33 -------------- 7
Add equation 5 and 7 we get
-8Y = -16 or Y = 2
Substituting Y = 2 in equation 5 we get
2+3Z = 17 or 3Z = 15 or Z = 5
So the number of routes between Doha to Toronto = X + YZ where X is the number direct routes
between Doha and Toronto and YZ is the number of routes from Doha to Toronto that go
through Reston
X+YZ = 3 + 2 x 5 = 13 Routes between Doha and Toronto