Can anyone explain me this: assume that b and c are two integers that are greater than one. in base b, c^2 is written as 10. then b^2, when written in base c is

i know that the solution is 1_c

i also found a similar problem: suppose b and c are positive integers. when b^2 is written in base c, the result is 121_c. when c^2 is written in base b, the result is 71_b.
what is b+c?
and the solution to it (attached file)

i understand the conversion between decimal, binary, octal and hexadecimal expansions. so for example,(making a link between what i know and the 2nd problem), if b is 8, 71_8 = 7*8^1 + 1*8^0 or 71_b = 7*b^1 + 1*b^0 = 7*b + 1 equals some variable c. but how the problem should be solved if c is squared?

i couldn't find any explanation and we were not taught this at school (i am from kazakhstan btw), so can anyone explain or share the link with the explanation of this topic.