A few researchers are trying to crack a code which involves discovering the values of three integers. They know they are between 1 and 100 (inclusive), and that they may be the same.
They each have a different piece of information;
Alice knows the geometric mean
Bob knows the arithmetic mean
Chris knows the arithmetic mean of the squares
They get together to share information and crack the code, but they are being very secretive in an attempt to conceal their results from anyone else. You listen in to their conversation;
Chris: “I don't know all of the numbers”
Alice: “I don't know any of the numbers”
Chris: “You didn't need to tell me that, I knew that already”
Alice: “Well now I know all the numbers!”
Chris: “I still don't know all the numbers”
Bob: “I don't know all of the numbers either”
Chris: “Argh, I still don't know all the numbers”
Bob: “Ah - now I know all of the numbers”
“A ha!” you cry. For they have accidentally revealed the numbers to you. What are they?
You should assume these researchers always tell the truth and make perfect deductions and calculations. However, they might not always be as precise as they could be; “I don't know all the numbers” does not necessarily mean that they know any of the numbers.
Note: You might want to use a computer to solve this.