Hard Question from UMD Math Competition 2014

Question

There are three underground gopher nests in a field; call them A, B, C. There are tunnels between A and B, between A and C, and between B and C. A gopher counts that there are 89 ways to get from A to B and there are 82 ways to get from A to C. The gopher tries to count how many ways to get from B to C, but the number is greater than 99. Since the gopher cannot work with 3-digit numbers, it cannot count them. Help out the gopher and determine how many ways there are to get from B to C. (Note: A way to get from A to C is either a tunnel directly from A to C or a tunnel from A to B and then a tunnel from B to C. Similar definitions apply for ways to get from A to B and ways to get from B to C.)

 

a. 120

b. 171

c. 178

d. 302

e. 802

Answer 1

To solve this I drew a triangle to represent the situation. I labeled the vertices A, B, and C. The length of (number of routes between) BC I labeled x, AB y, and AC z.

 

You can calculate the total number of routes by adding the direct routes plus the product of the two indirect routes. The number of routes between A and B is y + xz, between A and C is z + xy, and between B and C is x + yz.

 

You know that

y+xz = 89

z+xy = 82

 

x+yz > 99

 

If you subtract the first two equations you get

(y-z) + x(z-y) = 7

-1 + x = 7/(z-y)     (divide by (z-y). remember that y-z = -(z-y) )

x-1 = 7/(z-y)

Since x has to be a whole number, (z-y) must be a factor of 7. The only choices are 1 and 7.

 

IF z-y = 1

z = y+1

x-1 = 7

x = 8

 

y+xz = 89

y+8(y+1) = 89

9y + 8 = 81

y=9

z=9+1=10

 

The number of routes between B&C is

x+yz = 8 + 9*10 = 98.

This is too small! This answer is incorrect.

 

IF z-y = 7

z=y+7

x-1=1

x=2

y+xz = 89

y+2(y+7) = 89

3y + 14 = 89

y = 25

z = 25+7 = 32

The number of routes between B&C is
x+yz = 2 + 25*32 = 802

 

The answer is (e)