The 40th party of the new year is being held at a local mansion. The host is very rich and his success is because of one thing — his famous recipe for Linguini! So rich indeed, that 39 parties have already occurred in a span of 13 days.
The only guests that may attend are people who correctly reply to the guard at the door. Here's where you come in. You and a friend are trying to steal this recipe. You sneak by and listen to the passwords.
For $1 \le n \le 9:$
The $n$th guest arrives, whereupon the guard, holding a mirror, says $n,$ the guest says $f(n),$ and the guest is let in. Note: the 9th guest happens to be your friend.
Your hearing allows you to pick up that $3, 6, 8, 7, 10, 10, 8, 9, 4$ are $f(1), \dots, f(9)$ respectively.
It's getting late, about 7 or 8. So you pull up to the guard and he's holding a pair of dice. If anything, you could say this mansion is rare. But you don't say anything yet, for the guard has not given you your number yet.
Now the guard says "10". How do you respond, given that the only viable option is to utter another natural number?
