# $1$ of $100$ Homes

On Crossfield Housing Estate, there are exactly $$100$$ homes numbered from $$1$$ to $$100$$. Rook is living in one of them. Kaito hasn't met with him for a long time and wants to pay a visit. Here is some transcripts of their call.

Rook : ".. Haha.. Yep, it's been a while.. You are welcomed to visit my home tomorrow."
Kaito : "Seriously? Alright I'll come tomorrow morning! But.. I guess I forgot which number is your house, ahaha."
Rook : "Well, in that case, let's play a puzzle!"

Kaito : "Hmm.. Is the number more than $$50$$?"
Rook answers with either yes or no.
Kaito : "Is it a square number?"
Rook answers with either yes or no.
Kaito : "Hmm.. Is it an odd number?"
Rook answers with either yes or no.

Kaito : "Is the number.."
Rook: "Hey, hey, enough, haha."
Kaito : "Huh? I'm still clueless Rook.."
Rook : "Haha, I thought you still remember that Freecell is also living in this estate and you know his number. All of these information will be perfect for you to figure out my number."
Kaito : "Oh! I remember now! I remember his home number and it must be greater than yours, right?"
Rook : "Haha, I know you will get it! See you, I'll prepare a nice breakfast for you if you want."
Kaito : "Yosh! Thanks and see you too!"

So on which number is the Rook's home?

Rook's home is number $$64$$

Reasoning

If house # were less than $$50$$ there would have been no way for Kaito to figure it out from the following 2 questions - there are 4 odd squares ($$1, 9, 25, 49$$) and 3 even squares ($$4, 16, 36$$).

So the house # is more than $$50$$. It also has to be a square. Once again it would have been impossible for Kaito to figure it out otherwise.

That leaves Kaito with three choices ($$64, 81, 100$$).

If the house # were odd, Kaito would not have asked any more questions, but he still was not sure. So the house # must be even.

Then Kaito remembers that Freecell's house has a bigger number than Rook's house. This leaves $$64$$ as the only possible number.

• I don't get it. What if, say Rook lives at 1 and Freecell lives at 2? – Ankoganit Jun 5 at 4:24
• @Ankoganit Then the answers would have been No Yes Yes and it wouldn't be possible to figure out...? – Rubio Jun 5 at 4:27
• The way I see it, there are two ways to interpret that line in the transcript: (1)Kaito remembers Freecell's house number AND the fact that it's greater than Rook's (in which case there are multiple solutions), or (2)Kaito just remembers Freecell's house number, uses that to figure out Rook's number and asks if it's indeed smaller than Freecell's for confirmation (in which case you're right, 64 is the only solution). – Ankoganit Jun 5 at 4:44
• @Ankoganit, thank you for clearly identifying two possible interpretations of the transcript. As you said, in the first interpretation there are clearly multiple possible answers. I think the second interpretation is slightly more likely because of the question mark in the Kaito's statement "Oh! I remember now! I remember his home number and it should be larger than yours, right?" – ppgdev Jun 5 at 4:58
• ppgdev indeed your answer is the intended one, good job! However, thanks and sorry @Ankoganit for pointing out another interpretation >< the second interpretation should be the intended one, but your answer (esp. of 1) is definitely an impressive alternative answer! – athin Jun 5 at 8:34

I think I found more possible solutions. All of them are compatible with the puzzle, though it wouldn't make a lot of sense for Kaito to ask for the number, yet.

no, yes, yes, Kaito 1, Freecell 2-9
no, yes, no, Kaito 4, Freecell 5-16
no, no, yes, Kaito 1, Freecell 2 or 3
no, no, no, Kaito 2, Freecell 3 or 4
yes, yes, no, Kaito 64, Freecell 65-100
yes, no, yes, Kaito 51, Freecell 52 or 53
yes, no, no, Kaito 52, Freecell 53 or 54