Proposing mechanisms for carbocation rearearrangements involving hydride or alkyl shifts

Question

For each in the attached image,

https://ibb.co/6W8Hdw2

how can I draw plausible mechanisms for carbocation rearrangements involving hydride or alkyl shifts, where the carbocations can undergo rearrangements to form more stable ions by the transfer of a hydride ion or alkyl group?

Answer 1

Here's the 3rd rearrangement, where a 1,2-methyl shift converts a 1° carbocation to a more stable 3° carbocation.

The standard way to show this is to use a curved arrow starting somewhere in the middle of the C–C bond of the methyl group that is undergoing the 1,2-shift (along with the pair of electrons from its C–C bond). The head of the curved arrow points to the original carbocation center (where the C–C bond is moving to).

After the carbocation rearrangement has taken place the original carbocation center becomes a neutral carbon with 4 bonds, and the carbon that the methyl group (plus the electrons in its bond) shifted away from becomes the new carbocation center.

I cannot include additional structures for either rearrangement (a) or (b) because this would exceed some arbitrary limit imposed on the "word count" of the response.