Penny Piles
As the problem goes, we essentially have to organize the distribution of pennies in correspondence to the two rules we are given. In other words we can only transfer pennies from one of the drawers to the other if the number of pennies in the drawers is even. My plan at first was to try and find some sort of way to prove that not all the numbers blow our threshold were obtainable. In addition, we know that we only have to prove half of the values below the threshold as when you prove one half you prove the other. For instance if you prove that you can get the number 63, you know that you can get the number one in the other drawer and so on. From here I denoted the amount of pennies in the left drawer as x and the amount of pennies in the right drawer as y with their sum having to add to 64. Then I essentially created a factor tree.
As we must follow those two rules given to us, we can say that division is closed for these integers otherwise it would break the whole problem. If it weren't closed then we'd be able to take half of odd amounts of pennies and move them to the other side. After this point I began thinking about how we could prove that we could obtain any combination. My thought process is similar to proofs we did in class in regards to denoting some new variable as an instance of the x + y = 64 relationship. As such since integer division in this case is closed this would be correct. However the problem I'm having is proving that this applies to every value less than the threshold. Aside from showing that we can obtain every combination by actually doing it out, I haven't found a way to prove this. The route I'm thinking of taking is first showing that we can obtain all the prime numbers below the threshold and from here get every other result.
As we must follow those two rules given to us, we can say that division is closed for these integers otherwise it would break the whole problem. If it weren't closed then we'd be able to take half of odd amounts of pennies and move them to the other side. After this point I began thinking about how we could prove that we could obtain any combination. My thought process is similar to proofs we did in class in regards to denoting some new variable as an instance of the x + y = 64 relationship. As such since integer division in this case is closed this would be correct. However the problem I'm having is proving that this applies to every value less than the threshold. Aside from showing that we can obtain every combination by actually doing it out, I haven't found a way to prove this. The route I'm thinking of taking is first showing that we can obtain all the prime numbers below the threshold and from here get every other result.
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