Constant puzzle

A constant is hiding behind this puzzle, can you tell which?

• 4 $$\rightarrow$$ 2 $$\rightarrow$$ 1 $$\rightarrow$$ 1 $$\rightarrow$$ 1 $$\rightarrow$$ $$\dots$$

• ? $$\rightarrow$$ ? $$\rightarrow$$ ? $$\rightarrow$$ $$\dots$$

• 4 $$\rightarrow$$ 3 $$\rightarrow$$ 6 $$\rightarrow$$ 0

• 2 $$\rightarrow$$ 1 $$\rightarrow$$ 1 $$\rightarrow$$ 1 $$\rightarrow$$ $$\dots$$

• 1 $$\rightarrow$$ 1 $$\rightarrow$$ 1 $$\rightarrow$$ $$\dots$$

• 3 $$\rightarrow$$ 6 $$\rightarrow$$ 0

• 5 $$\rightarrow$$ 7 $$\rightarrow$$ 4 $$\rightarrow$$ 0

• 6 $$\rightarrow$$ 0

• $$\dots$$

The second sequence has been deliberately partially hidden, otherwise the puzzle could have been too easy for the community :)

Only the first eight sequences have been given. It may be possible that the next sequences will be given as hints.

I think that the constant hiding behind all of this is

$$\sqrt{2}$$

Notice first that

Reading the first element of each sequence in order gives $$4,?,4,2,1,3,5,6$$ - the first digits after the decimal point in the decimal expansion of $$\sqrt{2}$$.

We generate each sequence as follows:
(i) For the $$m$$th sequence begin at the $$m$$th place after the decimal point.
(ii) Record the value $$n$$ and move $$n-1$$ places to the right.
(iii) Repeat (ii) indefinitely or until you arrive at a $$0$$.

For example, the third sequence starts at the second $$4$$. Moving $$3$$ places to the right of this we get to a $$3$$ (the sixth digit after the decimal point). Moving $$2$$ places to the right again, we get a $$6$$ (eighth digit after the decimal point). Moving $$5$$ places to the right, we get a $$0$$ (thirteenth after the decimal point) at which point the sequence must stop.

This means that the missing sequence is just $$1 \rightarrow 1 \rightarrow 1 \rightarrow \ldots$$

• Congrats! Maybe I should have hidden more sequences haha
– JKHA
Nov 5 '20 at 19:36