baldhair

[con-amca]

Here is a logic problem I came across many years ago. I cannot now remember the solution, nor can I work it out. Anyone seen it before and can remember the answer? Or can anyone work it out? There is a logical solution.

Smith lives in 13th Avenue, which has houses numbered from 13 to 1300. Jones wants to know the number of Smith's house.
Jones asks: "Is it less than 500?"
Smith answers, but he lies.
Jones asks: "Is it a perfect square?"
Smith answers, but he lies.
Jones asks: "Is it a perfect cube?"
Smith answers, and he tells the truth.
Jones says: "If I knew whether the second figure was a 1, I could tell you the number of your house."
Smith tells him, and Jones announces the number of the house, but he is wrong.
What is the number of Smith's house?

[Salty]

Possible squares are: 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 441 484 529 576 625 676 729 784 841 900 961 1024 1089 1156 1225 1296

Possible cubes are: 27 64 125 216 343 512 729 1000

To have any chance of getting the answer, Jones must think that the last question distinguishes two numbers, one with second digit 1 and the other not. If Smith had said it was a perfect square and a perfect cube, that leaves just 64 (<500) and 729 (>500), so Jones' last question would not have been required. Smith therefore said no to either square or cube. He must have said yes to one of these, though or the number of potential answers is too high.

If Smith said it is a square but not a cube, again the number of possible answers is too high, so Smith must have said it was a cube but not a square. There are four of these cubes <500 (27 125 216 343 but not 64) and two >=500 (512 1000 but not 729). For Jones to ask the second digit = 1 question, Smith must have said >=500, and Jones would then pick 512 or 1000 dependent on the answer to the last question.

Smith must therefore have said
">=500", which is a lie so it is in fact <500
"Not a perfect square", a lie so it is in fact a perfect square
"Is a perfect cube", true so it is in fact a perfect cube

The only perfect square which is also a perfect cube which is less than 500 is 64.

The answer is therefore 64!

(I'm quite prepared to be proved wrong here.......)

[con-amca]

Bravo Salty! This is the answer as I remember it. Since I rediscovered this problem a year or so ago, I've cudgelled
my brains trying to work it out again,but without success.

I congratulate you on getting there so quickly. Makes me
feel quite inadequate!