The envelope and the number

We will find out Pi Han's answers by logical reasoning. Let us assume that both Calvin and Pi Han are perfectly logical.

First, we know Pi Han's answer to (1) must be "Yes". Otherwise, Calvin would not have needed to ask (3), knowing that it could not have been 2-digit.

Now that we know it was less than 100, Pi Han's answer to (2) must also be "Yes". Otherwise, Calvin would not have needed to ask (4), knowing that an odd number could not have a last digit of 6.

Pi Han's answer to (3) must also be "Yes". Otherwise, Calvin would not have needed to ask (4), knowing that it was 1-digit.

Now we do some casework: the even 2-digit perfect squares are 16, 36, and 64. Therefore, Pi Han's answer to (4) must be "No". Otherwise, Calvin would not be able to come up with a unique answer.

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Now we negate all. From the negations, we know that it was an odd perfect square greater than 100 and less than 300 with a second digit of 6.

There is only one number that satisfied all these conditions: 169.

Solution by diagram

• The numbers shown are the square roots of the numbers to be guessed at.

• For each question, yes = $\nearrow$ and no = $\searrow$ .

• Orange: Answer found in three questions. Calvin would not have asked any more.

• Green: Answer found in four questions. These numbers fit the answers Calvin was given.

• Yellow: There is no such number. The choices are incompatible.

• Red outline: Calvin would not have gone here, because he does not ask questions to which he already knows the answer.

• Arrows on the right side: Mirror image (swapping yes and no).

We see that Calvin must have concluded that the number was $8^2 = 64$ . Its mirror image is $13^2 = \boxed{169}$ .

The important point to first note would be that Calvin does not ask questions to which he already knows the answer, meaning if the answer given to one of the questions is obvious enough to imply the answer to a later question (in the list of four), this later question would be deemed unnecessary and Calvin would NOT ask it.

Therefore, if all four questions had to be asked, we have to avoid a situation as illustrated above. This will help us determine the individual yes/no responses to the four questions. To illustrate what I exactly mean, let's do a walkthrough. For the first question, if Pi Han had answered no, Calvin would know that the number has to be three digit which renders the third question redundant as the answer to whether the number is two-digit would definitely be no too. However, if Pi Han had answered yes to the first question, the third question would be necessary as a number below 100 can be one or two digit.

Hence Pi Han's answer to the first question must be yes . So the number, according to Pi Han, is less than 100 (one or two digit).

For the second question, if Pi Han had answered no, Calvin would know that the number is odd which renders the last question unnecessary. Why? If the number is one-digit, there is no second digit in the number to speak of. If the number is two-digit, the second digit would actually be the units digit of the number and if the number is odd, there is no way the units digit can be 6 which is an even number. Either way, the answer to the last question would obviously be no.

Hence Pi Han's answer to the second question must be yes too, which implies an even number.

For the third question, if Pi Han had answered no, Calvin would know that it has to be a one digit number. Considering that it is an even number, the only perfect square that satisfies these conditions would be four, which renders the last question unnecessary too.

Hence Pi Han's answer to the third question must also be yes , which implies a two digit number.

Should Pi Han's answer to the final question be yes or no? Well, by now, Calvin would have narrowed down the possibilities to 16, 36 and 64. There are two possible perfect squares which end with digit 6 (the second digit from the left is the units digit for a two digit number) but only one which does not, so Pi Han's answer to the final question should be no for Calvin to conclusively determine the answer to be 64.

But considering that Pi Han's four answers were all wrong, we simply take the negation and we have no , no , no and yes as the actual answers to the four questions. This means the unknown number is three digit (not any less than 100), odd (not even) and has 6 as its middle digit (the second digit from the left is actually the middle digit for a three digit number). The only perfect square not more than 300 which satisfies all these conditions is $\boxed{169}$ .