Problem 124

Solution 1
Let the digits be $A, B, C$ so that $B = \frac {A + C}{2}$. In order for this to be an integer, $A$ and $C$ have to have the same parity. There are $9$ possibilities for $A$, and $5$ for $C$. $B$ depends on the value of both $A$ and $C$ and is unique for each $(A,C)$. Thus our answer is $9 \cdot 5 \cdot 1 = 45 \implies E$.

Solution 2
Thus, the three digits form an arithmetic sequence.

If the numbers are all the same, then there are $9$ possible three-digit numbers.
If the numbers are different, then we count the number of strictly increasing arithmetic sequences between $0$ and $10$ and multiply by 2 for the decreasing ones:

Common difference
Sequences possible
Number of sequences

1
$012, \ldots, 789$
8

2
$024, \ldots, 579$
6

3
$036, \ldots, 369$
4

4
$048, \ldots, 159$
2

This gives us $2(8+6+4+2) = 40$. However, the question asks for three-digit numbers, so we have to ignore the four sequences starting with $0$. Thus our answer is $40 + 9 - 4 = 45 \Longrightarrow \mathrm{(E)}$.