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396   ?    556   99   872   664

I tried to solve it but to no avail. Possible answers are 666, 491, 654, 999 and 917; since, the exam was multiple choice we were offered potential answers. The helpless hints I used while trying to solve it are: 872 -> 664, there 8x8=64; 99 -> 872, there 9-1=8 and 8x9=72; but I couldn't go further. Thanks beforehand. If you read until here then I want to provide correct answer, since the company themselves showed correct answers today(13th June 2020) but I don't know why that is correct: 917.

Source: The question appeared on MetYös 2 exam yesterday (12th June 2020 7pm Istanbul time). It is a classical university entrance preparatory exam in Turkey for foreigners. I don't want to give link to their website here, and I don't want to increase their organic search ranking. I even don't want to mention their company name and exam, but rules here force me to do that which may indirectly contributes to that company's advertising. So, dear moderators, how can I cite the source without making pr of them? Thanks.

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    $\begingroup$ I don't think there is any discernable pattern other than the partial one hexomino found below. You can always find some arbitrary connection, such as one less than the sum of the digits squares plus two times the original number $396 \to 3^2+9^2+6^2+2(396)-1 = 917$. Maybe at a push, one less than the number times its first digit, minus its digit sum $99 \to 99 \times 9 - (9+9)-1 = 872$ but none of them generalise to the whole sequence. I think the test-setters should be called on to explain this 'calculation' puzzle themselves. $\endgroup$ – Sputnik Jun 13 at 12:56
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Partial answer : Observation

One thing I notice which is probably part of the solution (but not the whole thing) is that

If we add the digits of each number, we get $$18, ?, 16, 18, 17, 16 $$ which leads us to the idea that the sum of the digits of the number represented by the ? is probably $17$.
In this regard, $917$ is the only multiple choice answer which satisfies this constraint.
However, it's not clear at this point why another number (say $359$) would not also work so I feel there may be at least one more pattern which the numbers follow.

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  • $\begingroup$ I think correct answer might be yours, because we have roughly 1.5 minutes to solve each question, so theoretically, questions should require no more than 90 seconds to solve fully. However, we sometimes get stuck and can't find solution even if we spend more than 90 minutes let alone 90 seconds. Thank you, @hexomino for your answer. $\endgroup$ – garakchy Jun 13 at 13:10

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