Fill in the missing number style question

Question

This nice question from a math competition doesn't seem to make any sense.

What number should replace the question mark? $$\begin{matrix} 22 & (4) & 82 \\ 73 & (3) & 97 \\ 93 & (?) & 99 \end{matrix}$$

Answer 1

$22\equiv 82\mod 4$

$73\equiv 97\mod 3$

$93\equiv 99\mod 6$

So you could fill in ?=6. But without any context you could probably justify anything.

Answer 2

I can't really verify if this is the correct pattern, but it's definitely a pattern here. In the first number in each row, swap the two digits. Then multiply the swapped first digit by the number in the parentheses to get the first digit in the last number in each row.

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$$ 22 \quad (4) \quad 82 $$

Swap the $2$ and $2$ (so nothing changes, really..) and multiply the first $2$ by $(4)$ to get $8$. Replace first digit with $8$ and you have $82$.

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$$ 73 \quad (3) \quad 97 $$

Swap the $7$ and $3$ and multiply the $3$ by $(3)$ to get $9$. Replace first digit with $9$ and you have $97$.

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$$ 93 \quad (?) \quad 99 $$

Swap the $9$ and $3$ and multiply the $3$ by $(?)$ to get $9$. Replace first digit with $9$ and you have $99$. Thus the $(?)$ must be $(3)$.

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But as the (currently only) other answer says, without context you can probably justify anything.

Answer 3

The following procedure is probably the right one:

1. Take the first number, split it to digits $73 \rightarrow (7,3)$
2. swap the order of the digits $(7,3)\rightarrow(3,7)$
3. multiply the (now) first digit by the number in the ()'s: $(3,7) \rightarrow (3\times(3),7)$
4. evaluate the numbers $(3\times 3,7)\rightarrow (9,7)$
5. Combine the digits again to single number: $(9,7) \rightarrow 97$

Now the trick for the last one is to figure out what goes to the ()'s, so we get $$ 93 \rightarrow (9,3)$$ $$ (9,3)\rightarrow (3,9)$$ $$ (3,9)\rightarrow (3 \times(x),9)$$ Now at this point, we might start from the end, going backwards(from step5): $$ 99 \rightarrow (9,9) $$ $$ (9,9) \rightarrow (3 \times(x),9) $$ $$ 9 = 3 \times(x)$$ $$ x=3 $$

Result: question mark is replaced by "$3$".