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


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 $1 \rightarrow 1 \rightarrow 1 \rightarrow \ldots$

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

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