That Old Troll Problem

Algebra Level 3

You are on your way to pay a visit to your Grandma, who lives at the end of the valley. It's her birthday, and you want to give her two of the cakes you've made. Between your house and her house, you have to cross 7 bridges, and as it goes in the land of make believe, there is troll under every bridge! Each troll, quite rightly, insists that you pay a troll toll. Before you can cross their bridge, you have to give them half of the cakes you are carrying, but as they are kind trolls, they each give you back a single cake. What is the least number of cakes you have to leave home with?

Assumptions

  • Your Grandma does not have the necessary items to make a cake at her place and she has no neighbours

  • There is only one possible route from your house to your Grandma's

  • The only troll toll accepted is in the form of cakes

  • The toll must be paid each time a bridge is crossed

P.S. Remember you are paying a visit. You will surely need to return


The answer is 258.

Nehemiah Osei by Nehemiah Osei , 23, Ghana

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1 solution

Nehemiah Osei
Aug 27, 2015

I started working this from the back.

Since we are paying our Grandma a visit, we will surely be returning home by the same route. So we came prepared for the journey forward and backward

Assuming we went to stay with Grandma, we will have to arrive with two cakes

Meaning the 7th troll took half the previous amount and gave one back to arrive at that 2

Let the previous amount be x x . We have,

( 1 2 ) x + 1 = 2 (\frac{1}{2})x+1=2

that gives our x x to be 2

Continuously doing this, we will realise she started with 2

Remember, this was just for the case where she would stay at her Grandma's place.

This will surely help us on her return home. Now we know that on her return, she will need 2 cakes

This obviously means that she should arrive at her Grandma's place with two extra cakes

Making 4 in total

Taking this from the back,

Let the amount (before the troll takes its share) at each stage be x x

7th troll - ( 1 2 ) x + 1 = 4 (\frac{1}{2})x+1=4

x = 6 x=6

6th troll - ( 1 2 ) x + 1 = 6 (\frac{1}{2})x+1=6

x = 10 x=10

5th troll - ( 1 2 ) x + 1 = 10 (\frac{1}{2})x+1=10

x = 18 x=18

4th troll - ( 1 2 ) x + 1 = 18 (\frac{1}{2})x+1=18

x = 34 x=34

3rd troll - ( 1 2 ) x + 1 = 34 (\frac{1}{2})x+1=34

x = 66 x=66

2nd troll - ( 1 2 ) x + 1 = 66 (\frac{1}{2})x+1=66

x = 130 x=130

1st troll - ( 1 2 ) x + 1 = 130 (\frac{1}{2})x+1=130

x = 258 x=258

This implies that the least number of cakes she could take to fully complete her journey back and forth is 258 258

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