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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! Tonight he is trying a new variation on his signature creation which is expected to be a big hit in his restaurants all over town: Spaghetti Pie! However, after several successful attempts by a couple of party crashers to steal his recipe, the host has implemented an enhanced security protocol.

As usual, the only guests that may attend are people who correctly reply to the guard at the door.

As usual, you and a friend are trying to steal this recipe, and are hiding in some bushes by the gate listening to the challenge/responses between guests and the guards.

The first guest arrives. The guards says "33", the guest thinks for a moment, then replies "8". The guard grunts, and then says "35". The guest thinks again for a shorter amount of time, and then replies "8" again. The guard lets the guest in.

The second guest arrives. The security guard says "92". The immediately blurts out "6". The guard grunts, and then says "26". The guest immediately say "5" and is let in.

The third guest arrives. The guard thinks for a minute, and then says "50". The guest appears to be reciting something in her head, and after a minute says "Oh sure, give me the hard one. 2". The guard nods, then says "14", to which the guest immediately replies "1" and gets let in.

"Wow this is really hard", says your friend. "Not only are there two challenge/responses, but they don't seem to be following any sort of pattern."

"Yes," you reply, "it almost seems as if the numbers are just chosen from some random distribution. But that isn't at all in keeping with our host's tendency to have some kind of pattern in the challenge/response sequence!"

So you both decide to sit around a bit longer and do some more eavesdropping.

A few minutes pass and the guard changes. The next guest then arrives and the guard barks out "14" to which the guest immediately responds, "1". The guard nods and then asks "and 15?" to which the guest almost immediately responds, "9". The guard lets them in the gate.

"Hmm," your friend says. A few minutes later the next guest gets "92" to which he responds "6" in short order, after which the guard says "65" and the guest replies "3".

At this, your friend remarks, "This guard is definitely not doing his job very well, is he?" and with that he slips away and reappears at the gates a moment later. "8!" your friend announces to the guard, without waiting for him to gives his challenge. "Err, yes" the guard replies, looking puzzles, "eighty..." "7" your friend cuts him off before he can finish. "Quite," the guard agrees, scratching his head as he absentmindedly lets your friend through the gate.

Deciding that maybe your friend was onto something, you leave your hiding places and circle around to approach the gate from the main path. But when you arrive, the guard has been changed again.

"62 and 65" the guard says. You give him a strange look. As if expecting your surprise, the guard replies, "Yes, I know the e-mail said the second number would only be given after you answered the first one. Never mind that. Just give me both numbers."

What numbers do you give the guard?

This is the latest installment in the Party Security series.

Answers in spoiler tags please.

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The replies:

Reply to 62: 6
Reply to 65: 3

The reason:

You need to look at the digits of Pi: the next digit after 33 is 8, the next digit after 35 is 8, the next digit after 92 is 6 and so on. The same counts for 62 (next digit is 6) and 65 (next digit is 3).

Why the friend could reply before a number was said (reply to Mr.Mindor's comment):
After the guard changes, the other guard was just taking the next two digits from Pi: first 14, then 15, then 92, then 65 and so on, so the friend knew what the guard was going to ask and could reply before he asked something.

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  • 2
    $\begingroup$ that is the correct answer $\endgroup$ – Michael Nov 11 '14 at 18:39
  • $\begingroup$ Next question, how was your friend able to respond without the guard giving the prompts. What pattern was the guard using in selecting them? $\endgroup$ – Mr.Mindor Nov 11 '14 at 18:51
  • $\begingroup$ never mind, I see the guard did get lazy. $\endgroup$ – Mr.Mindor Nov 11 '14 at 18:55
  • $\begingroup$ @Mr.Mindor I expanded my spoiler. $\endgroup$ – ProgramFOX Nov 11 '14 at 18:56

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