Question

Harry first faces Hagrid’s Puzzle. Hagrid believes that the best way to teach Harry about magical creatures is by getting him to interact with them. Hagrid decides to make this a little interesting. Hagrid takes Harry to the Enchanted Forest. Hagrid with some help from Dumbledore divides the forest into several cells, each with it’s own coordinates as shown below. Each charm has a magical creature. Harry will start at (8,10). A spell has been laid so that movement between these cells is restricted. If your position is (a,b), you can only move to (a,a+b), (a,a-b),(a+b,b), (a-b,b), (b-a,b) and (a,b-a). Harry can contin- ue until there is no other magical creature under his reach. Harry wants to make the most of this exercise and learn as much as possible. What is the largest possible number of Magical Creatures he will be able to meet?

Answer 1

This is based on the fact that HCF or GCD of (a,a,-b) or (a,-b,b) is the same that of (a,b). This follows from the fact that every mutual factor of a and b are also the factor of a+b and `|a-b|`. And so becomes thr grid. This will happens at each stage of reach for some (a,b),
Thus lion can only reach those points whose gcd or hcf is (8,10) = =2, Therefore the answer is all the coordinates whose hcf is 2,
`{(2,2)}**1`
`{(2,4)(4,6)(6,8)(8,10)(10,12)`
`(2,6)(6,10)`
`(2,8)(4,10)`
`(2,10) (2,12)}**2` (As these could be both a,b and b,a)