A party is being held at a local mansion. The host is very rich and his success is because of one thing - his famous recipe for Spaghetti! By the way, this guy is so rich that he gives a party everyday.
The only guests that may attend are people who correctly reply to the guard at the door. They always use a scheme where they concoct a function as the password, so when the guard gives a number, the guest computes the result and is allowed in if, and only if, the answer is correct.
However, uninvited people with some math degrees always were able to sneak in to every party that was being held because they were smart enough to guess the function used. They are not stopped even if some of them happen to guess it wrong and die in the process; they simply never stop.
Today the party is special - the rich man will premiere his recipe for Fettuccine to a set of very distinctive guests! Looks like that he is expanding his Spaghetti businesss into other markets. To ensure that no uninvited math nerd slips in again, he looked for some function that the math nerds would not be able to guess and would not find in their PhD books or scientific articles.
Here's where you come in. You and a friend are trying to steal this new recipe. You selected him because he won a gold medal in a competition about function regression and holds the current world record for it. Further he has a PhD thesis and numerous acclaimed Post-doc works about breaking numeric codes. Also, he works very hard, 18 hours per day and 7 day-a-week in the NSA to ensure that the US government can illegally read the e-mails of everybody in the world. You two sneak by and listen to the passwords.
The first guest arrives. The guard says "10", and the guest replies "8".
The second guest arrives. The guard says "100"; the guest replies "7".
The third guest arrives. The guard says "41"; the guest replies "5".
The forth guest arrives. The guard says "113"; the guest replies "7".
The fifth guest arrives. The guard says "79"; the guest replies "8".
Your friend then suddenly has an idea. He starts to quickly scratch some formulas on a paper and then says: "Eureka, got it! They are using a composite of the 3rd-order Friedman-Jumlach partial derivative vertex inverse function!".
So he goes to the guard and is given "39". He very confidently answers "6". Then he is poisoned by a Venomoth, eletrocuted by a Pikachu, burned by a Charizard, drowned by a Squirtle, and finally has his bones crushed to dust by a Snorlax.
Another guest arrives. The guard says "131", and the guest replies "6".
And still other guest arrives. The guard says "63"; the guest replies "4".
Another guest. The guard says "25"; the guest replies "7".
You, who never had good math scores in school because you were playing video games all day long, and don't even remember what the Bhaskara's formula or the Pythagoras theorem is, think a bit and then guess that you might have got it. So you walk up to the door and the guard says "177".
What should you respond?
