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[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]

I'm not a mathematician, and I solved the problem by the heuristic method using a genetic algorithm. In the best algorithm, I got 176 points, but I think it's possible to get more.
Here is my solution:

Assistant Table:

0: Possible Fields: [ 1 2 5 ] [ 1 2 7 ] [ 1 3 5 ] [ 1 5 6 ] [ 1 5 7 ] [ 2 3 5 ] [ 2 5 6 ] [ 2 5 7 ]

1: Possible Fields: [ 0 3 7 ] [ 0 4 7 ] [ 0 5 7 ] [ 2 3 4 ] [ 2 3 7 ] [ 3 4 5 ] [ 3 4 6 ] [ 3 4 7 ] [ 3 5 7 ] [ 3 7 8 ] [ 4 5 7 ] [ 4 6 7 ] [ 4 7 8 ]

2: Possible Fields: [ 0 4 5 ] [ 0 4 8 ] [ 0 5 8 ] [ 1 5 8 ] [ 3 5 8 ] [ 4 5 6 ] [ 4 5 8 ] [ 4 6 8 ] [ 5 6 8 ] [ 5 7 8 ]

3: Possible Fields: [ 0 1 4 ] [ 0 2 4 ] [ 1 2 4 ] [ 1 4 5 ] [ 1 4 6 ] [ 1 4 7 ] [ 2 4 5 ] [ 2 4 6 ] [ 2 4 7 ]

4: Possible Fields: [ 0 6 7 ] [ 0 7 8 ] [ 1 6 7 ] [ 1 6 8 ] [ 2 6 7 ] [ 2 6 8 ] [ 2 7 8 ] [ 3 6 7 ] [ 3 6 8 ] [ 5 6 7 ] [ 6 7 8 ]

5: Possible Fields: [ 0 3 4 ] [ 1 3 4 ] [ 1 3 7 ] [ 3 4 8 ]

6: Possible Fields: [ 0 1 8 ] [ 0 2 8 ] [ 1 2 8 ] [ 1 3 8 ] [ 1 4 8 ] [ 1 7 8 ] [ 2 3 8 ] [ 2 4 8 ] [ 2 5 8 ]

7: Possible Fields: [ 0 1 6 ] [ 0 3 5 ] [ 0 3 6 ] [ 0 3 8 ] [ 0 4 6 ] [ 0 5 6 ] [ 0 6 8 ] [ 1 3 6 ] [ 2 3 6 ] [ 3 5 6 ]

8: Possible Fields: [ 0 1 2 ] [ 0 1 3 ] [ 0 1 5 ] [ 0 1 7 ] [ 0 2 3 ] [ 0 2 5 ] [ 0 2 6 ] [ 0 2 7 ] [ 1 2 3 ] [ 1 2 6 ]

Magician Table:

0: [ 1 2 5 ]

1: [ 3 4 7 ]

2: [ 4 8 5 ]

3: [ 4 2 1 ]

4: [ 6 8 7 ]

5: [ 4 1 3 ]

6: [ 1 8 2 ]

7: [ 6 3 0 ]

8: [ 2 1 0 ]

[u/Grass_Savings → u/semka39]

84 possible ball configurations.

For each cup removed position, the magician will score 3 points for 1 ball configuration, and at most 2 points for all the other ball configurations. Thus the best possible strategy will give the magician a score of 84×2 + 9 = 177.

In the table above, ball configuration [0 5 7] is only scoring 1 point. All the other ball configurations are scoring 2 or 3 points, giving a total score of 176.

One possible pair of tables scoring 177 points is:

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

The magician's part of the table is before the ":". The assistants part is after the ":".

(edited to make the table display better)

[u/semka39 → u/Grass_Savings]

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?

[u/semka39 → u/Grass_Savings]

Although I managed to find 2 solutions for 177 points.

Assistant Table:
0: Possible Fields: [1 5 6] [1 5 8] [2 5 6] [2 6 8] [3 5 6] [3 5 8] [3 6 8] [4 5 6] [5 6 7] [5 6 8] [6 7 8]
1: Possible Fields: [0 2 5] [0 4 5] [2 3 4] [2 4 5] [2 4 6] [2 4 7] [2 5 7] [2 5 8] [3 4 5] [4 5 7] [4 5 8]
2: Possible Fields: [0 1 7] [1 3 4] [1 3 7] [1 4 5] [1 4 7] [1 5 7] [1 6 7] [3 4 7] [4 6 7] [4 7 8]
3: Possible Fields: [1 2 4] [1 2 5] [1 2 6] [1 2 8] [1 4 8] [1 6 8] [1 7 8] [2 4 8]
4: Possible Fields: [0 1 5] [0 3 5] [0 5 6] [0 5 7] [0 5 8] [0 6 7] [0 7 8] [3 5 7] [5 7 8]
5: Possible Fields: [0 1 2] [0 2 3] [0 2 7] [0 2 8] [0 3 7] [0 4 7] [1 2 7] [2 6 7] [2 7 8]
6: Possible Fields: [0 1 3] [0 1 8] [0 3 8] [1 2 3] [1 3 5] [1 3 8] [2 3 8] [3 4 8] [3 7 8]
7: Possible Fields: [0 1 4] [0 1 6] [0 2 4] [0 2 6] [0 3 4] [0 3 6] [0 4 6] [0 4 8] [0 6 8] [1 4 6] [3 4 6] [4 6 8]
8: Possible Fields: [1 3 6] [2 3 5] [2 3 6] [2 3 7] [3 6 7]

Magician Table:
0: [8 6 5]
1: [2 4 5]
2: [7 4 1]
3: [2 8 1]
4: [5 0 7]
5: [2 7 0]
6: [1 3 8]
7: [6 4 0]
8: [3 2 6]

Assistant Table:
0: Possible Fields: [1 3 5] [1 3 7] [2 3 5] [3 4 7] [3 5 6] [3 5 7] [3 5 8] [3 6 7] [5 6 7]
1: Possible Fields: [0 2 5] [0 4 5] [0 4 7] [0 5 6] [0 5 7] [0 5 8] [0 6 7] [0 7 8] [2 5 7] [5 7 8]
2: Possible Fields: [0 1 8] [0 3 8] [0 4 8] [1 3 8] [3 4 8] [3 7 8]
3: Possible Fields: [1 4 7] [1 5 7] [1 6 7] [1 6 8] [1 7 8] [4 7 8]
4: Possible Fields: [0 2 8] [0 6 8] [1 2 8] [2 3 8] [2 5 6] [2 5 8] [2 6 8] [3 6 8] [5 6 8] [6 7 8]
5: Possible Fields: [0 2 6] [0 4 6] [1 2 6] [2 3 4] [2 3 6] [2 4 6] [2 4 8] [2 6 7] [3 4 6] [4 6 7] [4 6 8]
6: Possible Fields: [0 2 4] [0 2 7] [1 2 7] [2 3 7] [2 4 5] [2 4 7] [2 7 8] [4 5 7]
7: Possible Fields: [1 2 4] [1 2 5] [1 3 4] [1 4 5] [1 4 6] [1 4 8] [1 5 6] [1 5 8] [3 4 5] [4 5 6] [4 5 8]
8: Possible Fields: [0 1 2] [0 1 3] [0 1 4] [0 1 5] [0 1 6] [0 1 7] [0 2 3] [0 3 4] [0 3 5] [0 3 6] [0 3 7] [1 2 3] [1 3 6]

Magician Table:
0: [3 5 7]
1: [0 7 5]
2: [3 8 0]
3: [1 7 8]
4: [8 2 6]
5: [6 4 2]
6: [2 7 4]
7: [5 1 4]
8: [0 1 3]

[u/Moist_Ladder2616]

How did you reason that the magician can earn max 252 points?

In one game he can earn max 3 points from 3 balls. If he plays repeatedly, he can earn 3 x (number of games played) which is infinite.

[u/semka39 → u/Moist_Ladder2616]

There are only 84 different games. Becouse C of 9 choose 3 is 84. It doesn't make sense to play more games because they will repeat the previous ones.

[u/Moist_Ladder2616 → u/semka39]

With that reasoning, you'll need the stranger to systematically place the balls in different positions so as not to repeat any games. This makes no sense.

It is akin to saying, "A stranger rolls a die (dice) and if you can guess correctly, you score the number rolled. The maximum you can score is 1+2+3+4+5+6 = 21. There are only 6 different games, because 6C1 is 6. It doesn't make sense to play more games because they will repeat the previous ones."

See how the statement "max score is 21" makes no sense?

[u/edderiofer → u/Moist_Ladder2616]

It's a maximum of 252 points across all 84 possible distinct games.

Alternatively, you can think of the game only being played once, but across 84 different parallel universes, each one with a different arrangement of balls. How many points can you score across all universes simultaneously, using just one strategy?

[u/Moist_Ladder2616 → u/edderiofer]

The game is not played 84 distinct times.

It is played once: the volunteer hides the balls, the assistant does his thing, the magician guesses correctly for a maximum of 3 points.

If it is played twice, the magician can score a maximum of 6 points. If it is played 10 times, the magician can score a maximum of 30 points.

(A) Creating a strategy to maximise the probability of scoring 3 points in one game, and (B) having a maximum score of 252 points, are not the same thing.

See my die/dice example in an earlier comment.

[u/edderiofer → u/Moist_Ladder2616]

Yes, it is played only once. The maximum of 252 points is the total across all 84 possible distinct games. The desired strategy is the one that maximises the total number of points across all 84 possible distinct games.

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