On a particular street in Waterloo, there are exactly 14 houses, each numbered with an integer between 500 and 599, inclusive. The 14 house numbers form an arithmetic sequence in which 7 terms are even and 7 terms are odd. One of the houses is numbered 555 and none of the remaining 13 numbers has two equal digits. What is the smallest of the 14 house numbers?

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 is an arithmetic sequence with four terms.)
Since this arithmetic sequence of integers includes both even and odd integers, then the common difference in the sequence must be odd. (If it were even, then all of the terms in the sequence would be even or all would be odd.)

Suppose that this common difference is $d$.

Note that 555 is one of the terms in the sequence.

If $d=1$, then either or both of 554 and 556 would be in the sequence. Since no term other than 555 can contain repeated digits, then $d \neq 1$.

If $d=3$, then either or both of 552 and 558 would be in the sequence. Since no term other than 555 can contain repeated digits, then $d \neq 3$.

If $d=5$, then 550 cannot be in the sequence, so 555 would have to be the smallest term in the sequence. This would mean that 560 and 565 were both in the sequence. Since no term other than 555 can contain repeated digits, then $d \neq 5$.

We can also rule out the possibility that $d \geq 9$ :

Since there are 14 terms in the sequence and the common difference is $d$, then the largest term is $13 d$ larger than the smallest term.

Since all terms are between 500 and 599 , inclusive, then the maximum possible difference between two terms is 99 .

This means that $13 d \leq 99$. Since $d$ is an odd integer, then cannot be 9 or greater.

Since $d$ is odd, $d$ cannot be 1,3 or 5 , and $d \leq 7$, then it must be the case that $d=7$.

Can we actually construct a sequence with the desired properties?

Starting with 555 and adding and subtracting $7 \mathrm{~s}$, we get the sequence

$$
506,513,520,527,534,541,548,555,562,569,576,583,590,597
$$

This sequence has 14 terms in the desired range, including 7 even and 7 odd terms, and no term other than 555 has repeated digits.

(Note that this sequence cannot be extended in either direction and stay in the desired range.) Therefore, the smallest of the 14 house numbers must be 506 .