# Reducing π to zero (again)

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?

To allow new users to solve this puzzle and earn reputation points, I encourage all users whose reputation is 200 or more to not post an answer until 48 hours after this question is posted. Thank you!

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

• Is there any practical usefulness in solving such a problem? Sep 26 at 5:27
• @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. Sep 26 at 8:01

## 1 Answer

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.

• Your answer is correct even if the 5 digits are not contiguous. Sep 23 at 21:43
• Yes, I was just following the wording of the question Sep 23 at 23:43
• 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. Sep 25 at 6:15
• @infinitezero I have now spoilered the explanation, do you think I should spoiler the answer as well? Sep 25 at 6:28
• I think it's good this way :) Sep 25 at 7:36