Write a sequence of numbers from 1 to 8192, randomly rearrange these numbers, you'll have $a_1,~~a_2,~~a_3,~~\ldots,~~a_{8190},~~a_{8191},~~a_{8192}$ Now find the difference of every adjacent two numbers: $\vert a_1-a_2\vert,~~\vert a_3-a_4 \vert,~~ \ldots,~~\vert a_{8189}-a_{8190}\vert,~~\vert a_{8191}-a_{8192} \vert$ Randomly rearrange the results and you'll have a new sequence of numbers: $b_1,~~b_2,~~b_3,~~\ldots,~~b_{4094},~~b_{4095},~~b_{4096}$ Find the difference of every adjacent two numbers: $\vert b_1-b_2\vert,~~\vert b_3-b_4 \vert,~~ \ldots,~~\vert b_{4092}-b_{4093}\vert,~~\vert b_{4094}-b_{4096} \vert$ Again rearrange the results and you'll have another new sequence of numbers: $c_1,~~c_2,~~c_3,~~\ldots,~~c_{2046},~~c_{2047},~~c_{2048}$ Repeat the process above until you get a number $x$. then is $x$ even or odd?