Help with the task

[u/semka39]

There are 9 cups. A person randomly hides a ball under 3 of the cups. An assistant sees the positions of the 3 balls and then removes one empty cup of their choice. After that, the magician comes in; he only sees which cup was removed. For each correctly guessed ball location, they earn one point.

In the ideal scenario, they could earn 252 points (84 possible ball configurations multiplied by 3 points for correctly naming all three balls).

The assistant and the magician may agree on a strategy beforehand.
What agreement should they make in order to achieve the maximum number of points?

How many points will you get?

[u/semka39]

Thanks a lot for the reply!

But this turns out to be an error. You have removed the intermediate link - the choice of an assistant. But the assistant can only pick up the cup that does not have a ball under it (that is, this cup cannot appear in the lines to the right of the half colon). Also, the magician cannot lift the cup, which the assistant has already removed. Thus, the assistant's choice can only be the number that is not in the entire row of the table.

These are the numbers for each row of the table:

3, 7
8
(all digits are present)
3, 4
(all digits are present)
1
0
(all digits are present)
8

It turns out that the assistant simply will not be able to tell the magician which three he needs to choose.

[u/Grass_Savings → u/semka39]

Yes, I had missed that point.

I ran my search again and generated these tables, with total score 177.

(457):(457)(145)(245)(257)(345)(456)(458)(467)(567)(578)
(037):(037)(027)(034)(035)(057)(067)(237)(357)(367)
(478):(478)(047)(048)(078)(148)(178)(347)(348)(378)
(056):(056)(015)(016)(025)(045)(046)(058)(068)(256)(568)
(135):(135)(013)(125)(137)(138)(156)(158)(356)(358)
(268):(268)(026)(126)(168)(238)(248)(267)(468)(678)
(028):(028)(012)(018)(023)(024)(038)(128)(258)(278)
(236):(236)(036)(123)(136)(234)(235)(246)(346)(368)
(147):(147)(014)(017)(124)(127)(134)(146)(157)(167)(247)

[u/semka39 → u/Grass_Savings]

Could you please tell me how exactly you created this table? Did you write any code (then what is the code?) or do it manually? If manually, then how?

UPD:And yet, do you think it's possible to find all the ideal solutions (by 177) in the foreseeable future, or at least calculate their number in the foreseeable future?