1. The problem statement, all variables and given/known data 1. A grasshopper sitting on the number line can jump p or q inches right or left. Find all points on the line the grasshopper can reach starting from the origin (i.e. at zero). 2.Say exactly eight people show up to your birthday party, where there are two types of desserts: apple pie and birthday cake. As the host of the party, you have to make sure that one or more of the guests get a piece of pie, and one or more of them get a piece of cake. In how many different ways can you distribute the desserts, given that at least one person gets both a piece of cake and a piece of pie? 3. An alien lock has 16 keys arranged in a 4 × 4 grid, each key is either pointing horizontally or vertically. In order to open the lock, you must make sure that all the keys must be vertically oriented, by switching the orientation of one key at a time. When a key is switched to another position, all the other keys in the same row and column automatically switch their orientations too (i.e. vertical to horizontal, horizontal to vertical). Is it true that no matter what the initial positions of the 16 keys are, it is always possible to open this lock? 3. The attempt at a solution 1. i think the grasshopper can jump infinite points on the line but I can't prove? 2. is it right to divide each cake and pie into 8 pieces and distribute to the 8 people so each of them can get both a piece of cake and pie? 3.someone please help me to solve this question because i think it's impossible to open the lock, but there may be another way to solve it (this reminds me of the rubik cube) Thank you very much!