Logic Riddles for Chess Players (8)

by Arne Kaehler
8/30/2020 – Thank you for your commentaries about the last riddle and for clarifying the riddle before finding a solution. The chess logic-riddle community is very active — thanks for your engagement! Today’s riddle fits perfectly well into the chess world. It is a little odd though, I have to admit.

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The powerful queens

The Black and White chess kingdoms were at war and the fight went back and forth. A smart tactician came up to the Black king with a brilliant idea.

“Dear Black King, I have a fantastic idea on how we can win the war over time against the dreadful White king”, he said enthusiastically.

The Black King was a bit sceptic at first:“Tactician, we have been at war with the white king for ages and ages without either of us succeeding. Your plan must be outstanding to say the least.”

“Oh, it is!”, the tactician answered with a smile. “My king, have you ever noticed that our newborn queens are far superior compared to the newborn rooks, bishops or even knights in almost every way? Your chess kingdom would be far better off with more queens than rooks, bishops or knights to win the battle over time!”

The king thought for a moment, stood up with wide open eyes: “Eureka! Why didn’t I ever think of this myself? You are absolutely right! The queens are really the strongest chess piece we have by far.”

With this, the Black king created a new law with immediate effect:

  • All piece-bearing couples must continue to bear pieces until they have a queen!
  • We have to keep in mind that the chess kingdom has its own rules though.

The chances of bearing a Queen or a Rook/Bishop/Knight is 50/50!

“But my king!”, the tactician warned, “the boards will overflow and there won’t be any squares left for pieces. We have to make a rule to avoid overpopulation and stalemates”.

“You are correct again Tactician”, the King said while he pulled his moustache back a bit. “I will make a rule adjustment”.

All piece-bearing couples must continue to bear pieces until they have a queen! Once the couples have a queen, they have to stop bearing children!

A pregnant queen

After many years have passed the King wanted to check if his plan to have more queens in his kingdom than rooks, bishops or knights was successful.

Was it though?

Solution to the “Two urns” riddle

The solutions in the comment section were of course correct.

Once the young grandmaster put only one white pawn into one of the urns, his chances for freedom were significantly bigger. They went up to 73.33%.

But the grandmaster picked the wrong urn and grabbed a black pawn and his chess career ended. This story twist was a demonstration of  the fact that even when one has good odds in life it can go downhill quickly. He should have never started to learn chess in the first place. Was this a bit too dark? Hey, I am just kidding. He got the White pawn of course! (How do I get out of this...)

Special thanks to LarsRasmussen, JoshuaVGreen, chamishavolkov, brian8871, Erdmundr, wethalon, mkkuhner, MadMox, lajosarpad, Michael Jones and melvich for contributing to the solutions, explaining and effort.

Links:

Arne Kaehler, a creative mind who is passionate about board games in general, was born in Hamburg and learned to play chess at a young age. By teaching chess to youth teams and creating chess-related videos on YouTube, Arne was able to expand this passion and has even created an online course for anyone who wants to learn how to play chess. Arne writes for the English and German news sites, but focuses mainly on content for the ChessBase media channels.

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JoshuaVGreen JoshuaVGreen 9/9/2020 12:08
I came up with a much simpler argument for "The powerful queens." Just group the children by their position in the birth order. 1/2 of the first-borns will be Queens, 1/2 of the second-borns will be Queens, 1/2 of the third-borns will be Queens, etc., hence 1/2 of all the children will be Queens.
JoshuaVGreen JoshuaVGreen 9/4/2020 01:05
@Erdmundr, nice solution to the "Two urns" riddle!

I guess now's a good time to show my work on "The powerful queens." Every couple will bear exactly one Queen. 1/2 the couples will bear no non-Queens, 1/4 will bear exactly one non-Queen, 1/8 will bear exactly two non-Queens, 1/16 will bear exactly three non-Queens, ..., 1/(2^n) will bear exactly (n - 1) non-Queens for every positive integer n. Thus, the expected number of non-Queens per family is given by:
0/2 + 1/4 + 2/8 + 3/16 + 4/32 + ... + (n - 1)/(2^n) + ...
This series can be summed in several ways; I prefer
0/2 + 1/4 + 2/8 + 3/16 + ... + (n - 1)/(2^n) + ... =
1/4 + (1/8 + 1/8) + (1/16 + 1/16 + 1/16) + (1/32 + 1/32 + 1/32 + 1/32) + ... =
(1/4 + 1/8 + 1/16 + 1/32 + ...) + (1/8 + 1/16 + 1/32 + ...) + (1/16 + 1/32 + ...) + (1/32 + ...) + ... =
1/2 + 1/4 + 1/8 + 1/16 + ... = 1.
Erdmundr Erdmundr 9/1/2020 01:41
My comment to the previous problem had the following tricky solution (and presumably illegal, since it wasn't mentioned) in mind. The grandmaster places exactly one white pawn into one of the urns. Then they are randomized. The grandmaster then shakes one of the urns. One should be able to tell whether an urn you are shaking has 15 pawns or not, compared to one. That was why I mentioned dice.
Peter B Peter B 8/31/2020 04:19
Answer: no the plan will not not successful. The key is in the 50/50. Queens and non-queens are born with equal probability, so the kingdom will have (on average) as many queens as non-queens. Incidentally (a) it is an identical problem to one where every human family keeps having children until they have a girl; and (b) in both problems, the average number of children per family will be 2.
JoshuaVGreen JoshuaVGreen 8/31/2020 03:44
Also,

"Once the young grandmaster put only one white pawn into one of the urns, his chances for freedom were significantly bigger. About 73.33% higher."

is a rather awkward phrasing. His chances for freedom were about 73.33%, but they weren't 73.33% higher than the 50% that the King was expecting.
JoshuaVGreen JoshuaVGreen 8/30/2020 08:40
Despite the new law, the expected number of Queens will still be equal to the expected number of non-Queens -- one per couple.
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