Impossible Puzzles

Puzzles on this Page:
The Impossible Puzzle
The Sphere minus Cylinder

The Impossible Puzzle

This is quite an old problem sometimes called the Impossible Puzzle because on first sight, it looks like there is not enough information provided to solve it. But it is solvable, by using clear logical thinking and simple arithmetic. However, the required clear and logical thinking often gets muddled! 
Here is the problem;
Sandra and Peter are mathematicians and need to find two numbers, A and B.
They are told:
Both numbers are integers (whole numbers) each greater than one, where their combined SUM is not more than 100 and B is bigger than A.


Sandra is told the SUM of the two numbers (A+B) and Peter is told the PRODUCT of the two numbers (AxB).
Neither knows what numbers the other has been told, but they both know that Sandra knows the SUM and Peter knows the PRODUCT.

No further information is shared before the pair have the following conversation;

 Sandra:   “I don’t know the two numbers.”
 Peter:   “Neither do I”
 Sandra:   “Before you spoke, I knew you didn’t know”
 Peter:   “But I do now”
 Sandra:   “I also now know the two numbers”

What are the numbers A and B?
Does not seem to be a lot to be going on with but give it a start and see how you get on. Try starting with each comment in turn, take your time and see what you can glean from each statement!

This is probably one of the most difficult problems in the A Hen and a Half book and the comprehensive and sometimes complex solution aspects are detailed step by step in the book (over many pages) with additional insight provided in the Ahaah eBook.

It is just too long to provide all of that information here, so, if you are interested, go get the A Hen and a Half book!

The Sphere minus the Cylinder

The following problem also on first sight appears to be impossible to solve as there does not appear to be enough information provided to allow a solution to be found.

A cylindrical hole 6cm long from shoulder to shoulder, has been cut precisely through the middle of a solid sphere, like this:  

What is the solid volume left in the remnants of the sphere?

As said, there does not appear to be enough information provided to allow a solution to be found, as we do not know the diameter of the cylinder! But there is. Can you work it out?

If not, the somewhat fascinating solution can be checked out here!