Determine numbers written on a chessboard with the fewest number of questions. You have a standard chessboard with 64 squares with each square having a different number written on it (say from 1 to 64), but in no particular order.
You are allowed to ask questions. A single question has the following form: "Give me the numbers written on squares $d5,b2,c4,c8,...,h6$". You can specify as many different squares as you want in a single question. 
The answer provides you with a list of numbers (like 12,43,11,62,...,53) but the numbers are returned in an arbitrary order (so you cannot say that number 12 belongs to square $d5$, for example).
What is the minimum number of questions that you have to ask in order to determine the number on each square?
My effort? I have a fairly simple proof that you can "decrypt" the board with only 6 questions. But the trick is to prove that you cannot do the same with 5 or 4 questions. That's something that I'm still struggling with.
 A: Each particular number will either be included or not included in the answers to each of your questions.  With $5$ questions, there are $2^5=32$ possible sequences of responses you can get for a each particular number.
By the pigeon-hole principle you will have at least two numbers you are curious of who have received the same sequence of responses back and thus are still ambiguous as to which is located where.
Note, $2^6=64$ is the number of squares on the chess board.  $6$ questions are then required at a minimum to fully decrypt the locations of everything.  I trust that you got a correct sequence of questions you can ask and fully expect it to be possible to construct such a sequence of questions but have not gone through the effort of doing so yet.
A: You can determine it using just 1 question:


*

*ask a1 once

*ask a2 twice

*ask a3 3 times


i.e. the list of squares to ask looks like { a1, a2,a2, a3,a3,a3, a4,a4,a4,a4, ... },
then once you get the result list, you can count how many times it's included in the result list and determine it's position on the board based on that number...
