Problem 11 · No equal neighbouring digits
Free first digit and $9$ options for each neighbour: $9^3$.
Integer answer, at most 4 digitsHow many three-digit numbers have no two consecutive equal digits? (For example, $121$ works; $112$ doesn't.)
Copa Cangur · SCM
Easy
Closed answer
Reasoned solution
Key idea: count position by position: each digit only has to avoid the previous one.
The first digit can be $1$–$9$: $9$ options. The second can be any of the $10$ digits except the first: $9$ options. The third, anything except the second: $9$ options.
$$9 \cdot 9 \cdot 9 = 729.$$
Answer: 729