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Paths to Equality

 Published on Sunday, 29th November 2020, 01:00 am; Solved by 145;
Difficulty rating: 50%

Problem 736

Define two functions on lattice points:

$r(x,y) = (x+1,2y)$
$s(x,y) = (2x,y+1)$

A path to equality of length $n$ for a pair $(a,b)$ is a sequence $\Big((a_1,b_1),(a_2,b_2),\ldots,(a_n,b_n)\Big)$, where:

  • $(a_1,b_1) = (a,b)$
  • $(a_k,b_k) = r(a_{k-1},b_{k-1})$ or $(a_k,b_k) = s(a_{k-1},b_{k-1})$ for $k > 1$
  • $a_k \ne b_k$ for $k < n$
  • $a_n = b_n$

$a_n = b_n$ is called the final value.

For example,

$(45,90)\xrightarrow{r} (46,180)\xrightarrow{s}(92,181)\xrightarrow{s}(184,182)\xrightarrow{s}(368,183)\xrightarrow{s}(736,184)\xrightarrow{r}$
$(737,368)\xrightarrow{s}(1474,369)\xrightarrow{r}(1475,738)\xrightarrow{r}(1476,1476)$

This is a path to equality for $(45,90)$ and is of length 10 with final value 1476. There is no path to equality of $(45,90)$ with smaller length.

Find the unique path to equality for $(45,90)$ with smallest odd length. Enter the final value as your answer.