Hidden Number

Compare the parity of the sum of all elements before and after one operation.

If the chosen numbers are $a,b$, the operation removes them and appends $|a-b|$. In parity,

$$|a-b|\equiv a-b\equiv a+b\pmod 2.$$

Therefore, the parity of the sum of all elements never changes.

The initial sequence is $(1,2,\cdots,2x)$, so its sum is

$$1+2+\cdots+2x=x(2x+1).$$

Since $2x+1$ is odd,

$$x(2x+1)\equiv x\pmod 2.$$

Thus, the parity of the last remaining element is equal to the parity of $x$.

So $f(x)\equiv x\pmod 2$. From $P_i\equiv r_i\pmod 2$, we get

$$K_i=1-P_i.$$

After converting the bit string $K$ to an integer, compute

$$N=C\oplus K.$$

Solution written by GPT5.5