Square Number Triangle

Solution

This can be deduced through exhaustion. Shortcuts can be taken, such as A not being a multiple of 10 (otherwise B has a leading 0) and each number only being able to end in certain digits because they are square.

A=144, B=441 and C=121. 12 is a factor of A and 21 is a factor of B.

A+B+C= 706.

Here's another problem which is more open-ended.

Solution

A and C have the same first digit and are both 3 digits long. If A is a factor of C then A=C. By the same logic, B=C as they share the same last digit. So A=B=C.

But what's special about C? 

Well, the first and last digit of B are the same, because A and C end in the same digit. Then, as A=B=C, we know that C is of the form XYX where X and Y represent digits and X is not 0.

Here's a similar problem that's a bit less open-ended. 

Solution

If A+B=C then A must end with the digit 0, because B and C share a common last digit. But B cannot start with 0 as it's a properly written 3 digit number.

For the subtraction proof, B is a 3 digit number, meaning that at least 100 is being subtracted from A. The first digit of C must be less than the first digit of A, but A and C share a common first digit so this cannot be true. Hence, A-B=C cannot be true.