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You are given the first 20 digits of π: 31415926535897932384. In each move, you can select a contiguous group of 5 digits and increase/decrease them all by the same integer, provided that each resulting digit stays between 0 and 9 inclusive. For example you can increase the first 5 digits by 2, giving you 53637926535897932384. However you cannot decrease the same 5 digits by 2 as that would result in negative digits. Can you bring every digit to 0?


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This is a variant of the question Reducing $\pi$ to zero by Dmitry Kamenetsky.

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    $\begingroup$ Is there any practical usefulness in solving such a problem? $\endgroup$ Sep 26, 2023 at 5:27
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    $\begingroup$ @webadventurer I enjoy puzzles. I don’t care if they have any real world application. I would characterize myself as a pure mathematician not an applied mathematician. I do math because I enjoy its intrinsic beauty. $\endgroup$ Sep 26, 2023 at 8:01

1 Answer 1

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At the time of writing, I am currently at 196 reputation, so I am just under the limit.

I believe that you cannot reduce the first 20 digits of $\pi$ to 0.

We start by adding all the digits in the question together, to get the number $97$. This is our starting sum $S$. We also notice that each time we increase a contiguous group of 5 digits by a number $k$, we increase $S$ by $5k$, where $k$ is some integer. (Reducing a group of 5 digits is the same as having a negative $k$).

Therefore, if we are only able to increase or reduce $S$ by multiples of 5 at each step, the last digit of $S$ can only ever be $7$ or $2$. However, for $\pi$ to be reduced to 0, $S$ must equal $0$. Therefore, it is impossible to reduce $\pi$ to 0 by only increasing/decreasing groups of 5 digits.

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    $\begingroup$ Your answer is correct even if the 5 digits are not contiguous. $\endgroup$ Sep 23, 2023 at 21:43
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    $\begingroup$ Yes, I was just following the wording of the question $\endgroup$ Sep 23, 2023 at 23:43
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    $\begingroup$ It is common to use spoilers to give others a chance at completing it themselves. I believe you can add them with >! in front of your text. $\endgroup$ Sep 25, 2023 at 6:15
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    $\begingroup$ @infinitezero I have now spoilered the explanation, do you think I should spoiler the answer as well? $\endgroup$ Sep 25, 2023 at 6:28
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    $\begingroup$ I think it's good this way :) $\endgroup$ Sep 25, 2023 at 7:36

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