The cation shown undergoes a 1,2-methyl shift to give a more stable cation. How many hydrogens are on sp3 carbons directly attached to the positively charged carbon of the new cation?

The cation shown is a secondary cation next to a quaternary carbon. A methyl group shifts from the quaternary carbon to the cationic carbon, giving the tertiary 2,3-dimethylbutan-2-yl cation. Its charged carbon bears two CH3 groups (6 H) and one CH carbon with 1 H, so the total is 7 (checked with RDKit).