What are the odds?

I will assume you're talking about the UK lottery, where 6 numbers are drawn.

In any one week, there is a 1 in 18,424 chance that any particular set of 3 numbers will be among those drawn. The next week, there is exactly the same chance that they will be drawn. The numbers don't remember what happened last week.

Perhaps we're interpreting the question differently, but I disagree. The probability of exactly three of this week's numbers matching three of last week's numbers is 8815/499422 (roughly 0.01765).

I agree with markr. Since it can be any three numbers from this week matching any three numbers from last week (i.e. there are a lot of ways to get a match) this isn't that rare an occurrence.