Logic Riddles for Chess Players (2)

by Arne Kaehler
4/29/2020 – The last riddle was difficult, but a lot of correct solutions have been posted in a very respectful manner. Thank you and well done. Now we move on to a classic logical puzzle, which exists in various forms. Fortunately, you can once again easily implement it with a chess theme. | Photo: Fritz 17 - 3D chess board.

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Black or White?

 

As you can see, eight black rooks and eight white rooks are set up on the chess board. They have this position since a couple of years. The rooks cannot move or talk. I mean, they are rooks after all, right? But did you know that the rooks are actually able to see, listen and think? I was also a little surprised by this. As a matter of fact, each rook can see each other rook the whole time on the chess board.

A chess player wanted to find out, how smart the rooks are. He went to the chess board and announced the following to the rooks:"I can see a white rook."

Well, this might not have been the most amazing thing the rooks have ever heard, but when you understand that none of the rooks really know which colour they are, the sentence has some major significance.

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The chess player continues:"I have now connected my laptop and Fritz 17 with the chess board so you are able to communicate with the program. If you know for sure, that you are a white rook, Fritz 17 enables you to move and capture a black rook on a horizontal row. But if you are a black rook instead of a white one, and try to make a move, all the rooks will be thrown into the chess box, ending this puzzle with every piece losing. Each night I will come back here at seven PM and then you can either try to make a move, or stay where you are. Understood?"

The question is now:

When will the white rooks capture a black rook?

Let's sum this riddle up:

  • There are 16 rooks
  • Some are white and some are black, but no rook knows its own colour
  • The rooks cannot move, or talk
  • The rooks can listen, see and think
  • The rooks can see all other rooks, except themselves
  • All rooks know, that the chess player can see a white rook
  • If a rook knows, that it is a white rook, it can make a move to capture a black rook
  • If a rook is not a white rook, but a black one, and it tries to move, the game ends and all lose.
  • The chess player visit's the chess board every day at seven PM, giving the rooks a chance to make a move.

There is no trickery involved and you can really solve it in a logical way.

One version of this puzzle is from the book:

The Moves That Matter: A Chess Grandmaster on the Game of Life by Jonathan Rowson

which Grandmaster Karsten Müller suggested to me.

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The five chess boxes solution

First of all, thank you very much for your active responses, questions and answers, all in a very amicable way. Great stuff!

There were two correct solutions, which could also be mirrored, so actually four correct solutions is even more accurate. 

When thinking about the solution, you find out two critical things which push you forward to the correct answer. 

  1. The chess pieces cannot move further than box one and five.
  2. If the chess pieces start in an uneven numbered box (one, three or five) they cannot be found in the first three days.

If the chess pieces would have started in box number two or four, you can find them in three days by using this combination: 2-3-4 (or 4-3-2).

Either you find them right away (they were in box two) or not.

If not, the pieces are in box four. Now, they have "zugzwang" (Thank you for this wordplay @lajosarpad) and must move to either box three or five. If they move to box three, guess what, you get them the next day. If not, you will get them on the third day, again because of the zugzwang. They have to move to box four from box five.

This system can be stretched out by repeating the same pattern in both directions, if the chess pieces start in box one, three or five. There you go. You will find them on day 6 in the worst case scenario like this:

2-3-4-4-3-2 and 4-3-2-2-3-4 as well as 2-3-4-2-3-4 and 4-3-2-4-3-2.

Links:




Arne Kaehler, a creative thinker who is passionate about board games in general was born in Hamburg and learned how to play chess at a very young age. Through teaching chess to youth teams and creating chess content on YouTube, Arne was able to extend this passion onto others and has even made an online chess course for anyone who wants to learn how to play this game. Currently, Arne blogs for the English news page of ChessBase and focuses on creating promotional and entertaining articles.
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DKRaja DKRaja 5/10/2020 04:39
I posted 3 solutions, when mirrored, it was 6. I will re-post my entire solution. I was hoping since this is a puzzle for all chessbase members, there would be a prize or at least a winner having some freebees from Chessbase......

My solution was as follows:

1. 2 3 4 2 3 4
2. 4 3 2 4 3 2
3. 4 3 2 2 3 4
4. 2 3 4 4 3 2
5. 4 4 2 2 3 4
6. 2 2 4 4 3 2
charlesthegreat charlesthegreat 5/4/2020 05:43
On the 5 chess boxes puzzle. There technically could have more than the 4 solutions published. For example, if one considers the sequence 2-2-4-3-2-2. In this case, if you have not found the pieces by opening box 2 by day 6, then it is 100% certain that the pieces are in box 4 (try working out all the possible initial locations of the pieces). So one could just open box 4 after opening box 2, and still enjoy a game of chess on that day. Granted you open 2 boxes on day 6, but it's not a random choice.
Zurubang Zurubang 5/3/2020 10:17
@Jeydra thank you for your response.

The problem is, that none of the rooks know their own colour. They could be red, green or yellow, thus they don't know for sure if there are eight white and eight black rooks. There could be seven white rooks, and nine black rooks for example.
Once the chess player says, I can see a white rook, and offers them to participate in a riddle with a goal, they start to think perfectly logical. There is no detail missing, but this puzzle is inviting to be a bit controversial. When this riddle has been posted in a philosophers discord channel, people were debating it for weeks! There were already some good answers in the comments and also some appropriate scepticism.

@JoshuaVGreen I absolutely agree with your phrase! It is just very long and people might quit reading after the first fifteen words. I am happy you are thinking about the puzzles so much. You seem to enjoy riddles a lot.
Jeydra Jeydra 5/3/2020 02:45
I don't get the puzzle. If the rooks know there are 8 white rooks and 8 black rooks, then the problem is trivial. Each rook can count the number of rooks they see and therefore deduce their own color. They do not need the chess player to say "I can see a white rook". On the other hand if the rooks don't know the exact numbers but there are actually 8 white & 8 black, then every rook can see a white rook. The chess player saying he can see a white rook therefore says nothing, and the rooks will never move.

Pretty sure there's some detail missing.
JoshuaVGreen JoshuaVGreen 5/2/2020 01:16
@Andre67, I like your attempt to exploit the "chessiness" that was added to this classic riddle. I suppose the condition should be rephrased as "If you know for sure that you are a white rook, and if there's a black rook that you can reach via a horizontal move, Fritz 17 allows and, in fact, forces you to capture some such a black rook."
Andre67 Andre67 5/1/2020 09:50
Being able to communicate with the software each Rook learns its position on the board and the position of the other Rooks. So they all know the coordinates, especially the Rows. They understand Horizontal-Rows and Vertical-Columns. This must be important as the captures as announced by the human, will be done horizontally.
None of the Black Rooks can see any other Black Rook horizontally to capture. So they will never move.
All of the White Rooks can see one Black Rook horizontally to capture and no-other White Rook on the same Row.
So, all of the White Rooks, know that they are the only White Rook on each row, and they will capture all the Black Rooks, at 7pm, the first day allowed to, as announced by the player. Well, that's my idea...
varunkulkarni varunkulkarni 4/30/2020 06:25
On the 7th day post 7 pm, every white rook would know that they are a white rook as no one else has spoken on the 7th day at 7pm. So on the 8th day at 7 pm, any white rook can make a move and capture a black rook ?
JoshuaVGreen JoshuaVGreen 4/29/2020 11:35
"If the chess pieces start in an uneven numbered box (one, three or five) they cannot be found in the first three days."

Well, they could, but not via any of the optimal search strategies. At any rate, it's hard to see how one could determine this before completely solving the puzzle.
JoshuaVGreen JoshuaVGreen 4/29/2020 11:33
One must also assume that every Rook is reasoning this through very carefully, that every Rook knows that every Rook is reasoning this through very carefully, that every Rook knows that every Rook knows that every Rook is reasoning this through very carefully, etc. so that the inaction of other Rooks also provides information. This puzzle aims to demonstrate what can be deduced from the logical concept of "common knowledge."
JoshuaVGreen JoshuaVGreen 4/29/2020 11:09
You left out (at least) one important detail. Once a Rook realizes that it's White, it MUST capture a Black Rook (and thereby reveal its knowledge to the remaining Rooks).
DrBob64 DrBob64 4/29/2020 08:46
I find this sort of problem a bit unsettling. The argument seems convincing if there are only a few white rooks, but I find it harder to believe as more white rooks are added. But it doesn't really matter: as long as all of the rooks believe the argument, it works perfectly for them!
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