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When I first see the question, I haven’t had a clue how this problem works. So I decided to take 10 minutes to understand the question fully and after 10 minutes of thinking I finally realized where to start the question.
There are a few steps till I understood the question. So first of all, I translated the question into my own words that makes more sense to me. Then I tried to see how the diagrams are related to the question. I figured that the length of any part of the rods (even the places without a rod) are the information needed to find the answer. And finally I thought the whole process through figured that the it’s basically using a certain numbers to make other numbers.
For the first question, the first part of it is a very basic question after you have understood the question. It said that there are rods for 4 and 6 and we wanted to make 2. The solution is simple as this. But I still thought about the question. There are only two ways to place two numbers: to put it as a line or to stack on upon the other. And the answer is simply put the rod with the length of 6 on top and put the rod with the length of 4 at the bottom. It’s very straight forward after this. You can see that there is a gap of 2 and we have successfully made 2.
But for the second part it starts to get a bit complicated. I am sure many of you know the answer but to prove it took me a relatively long time to think it through. “Can you convince yourself you cannot make 8?” By seeing the word ‘convince’ it has already made my feel sad. We all know that proving stuff is the hardest thing to do in math (at least in my opinion). By thinking it through, I think the simplest way is just to list all the possible lengths. And I did so: 4+6=10 and 6-4=2. The number 8 isn’t in there. It was easy doing this, but this wasn’t what took the other half of my time thinking about the question. I asked myself: “What if the question is a hundred times as complicated as this question?” So I decided to have a follow-able rule. And I thought of this. Turn the rods into numbers and of course we can make the length of the rod on it self. Then list them form the smallest to the largest, count the number it makes. Take one of them out and count the number it makes, then replace while taking another one out. Repeat the process until all the rods have been taken out at least once. Then take two of them out each time and make sure they are a different pair each time, then three, then four till you finish the stage of taking out all the rods except one. Repeat the process again of instead of taking it away but it at the bottom and count the blank area. This can be a very hard process but I have tried my best to think of an easier alternative.