I am stuck on the following problem:
Imagine that a building has been overrun with snakes and rats. To help curb the problem, the building manager decides to offer employees brownie points for capturing and relocating them. They offer 13 points for each snake and 6 points for each rat captured and removed.
- Make a conjecture about what totals for points are possible. For example, an employee could not earn 7 points in this scenario.
- Prove that your conjecture is true.
What "conjecture" am I supposed to be making? I assume I am trying to find a fact/pattern; something like "points can only be obtained in multiples of 6", but I am lost. I only managed to find that the two provided numbers, 6 and 13, are relatively prime.
Of course, I also need to figure out how to prove such conjecture, I just can't do that until after I make it.
Edit: Okay, I totally misread the example conjecture. So my conjecture will be "an employee could not earn 14 points in this scenario". I write the following proof:
Proof: The two values of obtainable points are 6 and 13. The first 5 possible combinations of sums of 6 and 13 are, in order, as follows:
{ (0), (6), (6+6), (13), (13+6), (6+6+6) } = { 0, 6, 12, 13, 18 }
There are no possible combinations of sums of 6 and 13 that equate to 14. Therefore, it is impossible for an employee to earn 14 points in this scenario.