what is the minimal amount of questions to find the value of a value dependent probability dice

Question

Assume I have a 6 sided dice with the probability rule: for each dice value (1 to 6) the chances of getting the value $x$ on a dice roll is $\frac{x}{21}$, e.g the chances to get 6 is $\frac{6}{21}$, 5 is $\frac{5}{21}$....

I know that the entropy of the cube is 2.3983. so this sets the lower boundary of the mean amount of yes/no questions to 2.3983 (correct me if i am wrong).

given a secret cube, what is the best set of yes/no questions to ask so that the mean questions amount is minimal?

one solution i thought of is first question: is the value even? if yes then start asking about even values in decreasing order until I hit the right number (so 6 then 4 and then i am done), same manner if the answer to the first question is no (meaning odd).

Answer 1

Asking for the optimal set of yes/no questions is the same as asking for the best prefix-free binary code. And the answer is given by the Huffman code.

In our case I get (there are other alternatives, but all have the same $L$)

6 - 11
5 - 01 
4 - 00
3 - 101
2 - 1001
1 - 1000

This give an average length $L=2.42857 > 2.3983$

In words, this would correspond to:

 Ask "Is the value 4 or 5?"
    If yes: Ask if it's 5.
    If no:  Ask if it's 6.
       If no: Ask if it's 3.
           If no: Ask if it's 2.

Your proposal amounts to

6 - 10
5 - 00 
4 - 110
3 - 010
2 - 111
1 - 011

which is not bad, but suboptimal: $L=2.47619$

Answer 2

what about the following set of questions?

is the number in {1,2,3,5}?
   if not, is it 6? (yes = 6, no = 4)
   if yes, is it in {1,2,3}? (no = 5)
     if yes, is it in {1,2}? (no = 3)
       if yes, is it 1? (no = 2, yes =1)

my guiding principle is to always cut the probability space by half every step.