An old member returns...

[Valentin Albillo]

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Hi, all:

Thanks for your interest and kind replies. I'm sorry for the delay in posting my own reply but this is about the faster I can manage. The answer to my explicit mini-challenge, i.e.:

" If 9x9=29, what's the square root of 69 ? "

was correctly given by Gerson W. Barbosa and it's "F". First one would need to realize that for the 9x9=29 equality to hold the numbers given can't be in base 10, then compute the actual base B, like this:

9x9 = 81 (base 10) = 29 (base B) = 2*B+9 => B=(81-9)/2 = 36

now, knowing the base B, we have that 69 (base 36) = 6*36+9 = 225 (base 10) and finally:

sqrt(69 (base 36)) = sqrt(225 (base 10)) = 15 (base 10 ) = F (base 36)

which is the answer. I think that the statement:

" If 9x9 = 29 then sqrt(69) = F "

is pretty nice and seemingly nonsensical at first sight.As for the second, implicit mini-challenge, namely:

" ... if VA is square, what's the square root of me ? "

you first need to realize that, again, VA is a number in some base which has V and A as digits and which happens to be a square, and me is also a number in the same base, digits m and e this time, which is also a perfect square.

Don't be surprised by the first number being expressed with uppercase letters for its digits while the second number uses lowercase letters, this is akin to "23AF46E5" and "23af446e5" being equally valid and used representations in base 16, the case of the letters is irrelevant.

The second mini-challenge then is to find numbers VA and ME (or va and me, if you prefer) that are simultaneously perfect squares in some base B that allows for those digits. The programming isn't complicated at all and I'll give just the answers for you to check your results:

First solution:

VA (base 65) = 2025 (base 10) => sqrt(2025 (base 10)) = 45 (base 10)
ME (base 65) = 1444 (base 10) => sqrt(1444 (base 10)) = 38 (base 10)

Second solution:

VA (base 539431265) = 16722369225 (base 10), whose sqrt is 129315 (base 10)
ME (base 539431265) = 11867487844 (base 10), whose sqrt is 108938 (base 10)

and there are no other solutions up to base 194,576,544,937 (almost 200 billion)

PS: I need to know how to properly format math expressions in a post, most obviously subscripts and tables.

Best regards.
V.
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