Note that Lisa does not know that William sometimes lies.
Lisa reasons as if William speaks the truth.
Because Lisa says after her third question, that she knows his number if he tells her whether the first digit is a 3,
we can conclude that after her first three questions, Lisa still needs to choose between two numbers, one of which starts with a 3.
A number that starts with a 3 must, in this case, be smaller than 50,
so William's (lied) answer to Lisa's first question was "No".
Now there are four possibilities:
||number is a square
||number is not a square
|number is a multiple of 4
||8, 12, 20, and more
|number is not a multiple of 4
||9, 25, 49
||10, 11, 13, and more
Only the combination "number is a multiple of 4" and "number is a square" results in two numbers, of which one starts with a 3.
William's (lied) answer to Lisa's second question therefore was "Yes",
and William's (true) answer to Lisa's third question was "Yes" too.
In reality, William's number is larger than 50, not a multiple of 4, and a square.
Of the squares larger than 50 and at most 100 (these are 64, 81, and 100), this only holds for 81.
Conclusion: William's real house-number is 81.
back to the puzzle