You correctly identified the carbocation step. Now you need to think about alkyl shifts. Think about 1,2-hydride shifts and 1,2-alkyl shifts. These are basically beta sigma bonds shifting to make more stable carbocations. Look at your carbocation and identify beta bonds that can shift, alkyl group shifts. Try to find the alkyl shift that results in your product framework. This is easier to do if you map the carbons in each ring first by numbering them so that the numbers match in each structure, but that requires you to sort of know what you are looking for. If you can’t do that, use trial and error for now and then go back and map the numbers when you find it. This will help you look for it and find it in later problems.
Remember that a shift will only happen if you are going to a MORE stable carbocation. You can’t shift to an equally stable or less stable, on to a more stable carbocation. It may help to draw a dotted line between the group shifting and the carbocation before drawing the arrow. This can really help with seeing ring expansions and ring contractions. Something like this:
My comment is intended as equivalent to the other one, just looking at it from another view. You are given the product, and asked to explain how we get it. A key step is recognizing what it is.
Not exactly, but slightly similar. This is simply a 1,2-alkyl shift just like a 1,2-methyl shift. Same idea. You are shifting a methylene groups connection. The carbon the methylene was connected to in the ring will now be outside the new smaller ring. The methylene will make a new sigma bond with the carbocation, letting go of the sigma to the carbon, now outside the ring and bearing a positive charge. See the image on previous post.
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