Lopsy has given a proof of the decrease certain for $ok$.

celtschk gave a way to attain this certain however, as proven within the feedback, it’s flawed (even within the case $j=2$).

Here is an inductive methodology to attain the certain

$ok=2^j$

Base step

As proven by celtschk, if $j=1$ and $ok=2$, we might merely ask the primary individual if the second individual is a knight. If individual 1 says “sure” then individual 2 is a knight. If individual 1 says “no” then individual 3 is a knight.

Inductive step

Now allow us to assume we now have a way to establish a knight amongst a gaggle of $j-1$ jokers and $2^{j-1}$ knights by asking “sure/no” questions of the primary $j-1$ members of this group.

Now suppose we now have a gaggle of $j$ jokers and $2^j$ knights. Let us name the group of $j$ individuals to which we ask questions as group $J$ and the rest of the individuals as group $Okay$. Divide group $Okay$ into two teams of equal measurement, $K_1$ and $K_2$.

Ask the primary individual in group $J$, “Are there extra knights in group $K_1$ than in group $K_2$?”.

If they are saying “sure” then the rest of the group $J$ mixed with $K_1$ accommodates at most $j-1$ jokers.
If this individual says “no” then the rest of the group $J$ mixed with $K_2$ accommodates at most $j-1$ jokers.

In both case, we produce a gaggle of measurement $2^{j-1} + j-1$ with, at most, $j-1$ jokers. By the idea, we all know establish a knight on this group, as any methodology which is able to work for $j-1$ jokers may even work for fewer.

Example $j=2$

Here $K_1$ would include individuals 3 and 4, whereas $K_2$ accommodates individuals 5 and 6.

We ask individual 1 if there are extra knights in $K_1$ than in $K_2$.

If they are saying “sure”, then we ask individual 2 if individual 3 is a knight. “sure” would indicate individual 3 is a knight, “no” would indicate individual 4 is a knight.

If individual 1 says “no”, then we ask individual 2 if individual 5 is a knight. “sure” would indicate individual 5 is a knight, “no” would indicate individual 6 is a knight.