Nuevas posibilidades a través del PROT y la PNOT

In a 1 992 issue of Michael Ecker's REC Newsletter I posed the fol­

lowing chess problem. The queen is at the lower right corner of a rectangular field, and the king is at the upper left corner. On square rectangles the king is in check, so he moves first. Otherwise the queen goes first. Players alternate moves. On what fields can the queen force the king into the upper left corner? The problem generalizes a puzzle by Marek Penszko, of Poland, which appeared in Games magazine, the date of which I have lost.

The king clearly can't be cornered on 2 x n fields or on a 3 x 3

square. He can be cornered on certain 3 x n fields when n > 3. The smallest board of interest, therefore, is the 3 x 4 board.

Assuming the king always makes his best moves, what's the mini­

mum number of queen moves required to corner the king on the 3 x 4?

It's not as easy to solve as it seems. And what's the story on larger boards?

I conjectured in REC Newsletter that the king could not be cornered on any square board, though I had no proof. I also conjectured that the queen could win on all rectangular n x m boards, n > 2 and m > n + l.

Andy Liu, the ace problem creator and solver at the University of Alberta, Canada, was intrigued by the problem. He and I shared a byline on the following article, written by Liu:

A Royq l Pt"oblem: And Al ice Is (q ug ht in the Midd le The Red Queen was furious, as usual. Her current ire was brought on by the absence of the Red King from his Palace. On her rare visits, she expected to see whom she had come to see.

This article first appeared in Quantum (July/August 1993) .

109

"Bring the old fool back here, or else !" roared the Red Queen, who was related to the Queen of Hearts.

"Or else what?" asked Alice, but only after Her Majesty had swept radiantly out of earshot back to her side of the Palace.

"Off with your head !" Tweedledum said.

"What else?" added Tweedledee rhetorically.

"Oh, dear," said Alice, "this puts a new meaning to ten percent off the top. What shall I do? I don't even know where the Red King is."

The twins brought out a map of the land. It was the familiar 8 x 8

chessboard in Figure l .

"I bet I know where His Majesty is," said Tweedledum.

"On h6 !" exclaimed Tweedledee.

"How do you know that?" Alice asked.

"Well," said Tweedledum, "the Red King plays it safe. He never ventures out of his Kingdom into the Borderland."

"He also refuses to cross over to the Queen Side," added Twee­

dledee.

8 7 6 5 4 3 2

Queen Side King Side

a b c d e f g

Figure 1 .

Red Kingdom

I

Wbite Kingdom

b

Palace

14. Comerin9 the Kin9 ^{1 1 1}

rected Tweedledee. "He has no control over the whereabouts of Her Majesty."

"There is another problem," said Alice. "If the Red King does not want to come back to e8, how can 1 persuade him against his wishes?"

The twins thought for a while, and fought for a while just to pass the time. Then they both came up with a brilliant idea. Not surpris­

ingly, it was the same idea.

"Are you in mortal fear of the Red Queen?" Tweedledum asked Alice.

"Of course. Who isn't?"

"Of all people, who fears her the most?" asked Tweedledee.

"Hard to say," Alice replied. Then it occurred to her. "The Red King, of course."

"Right !" said Tweedledum. "He could not risk getting caught in a mating situation with the White Queen."

"So if you disguise yourself as that good lady, you can drive His Majesty back here," declared Tweedledee triumphantly.

"It is worth a try," said Alice, somewhat encouraged. "I should not waste any time by venturing outside of those twelve squares either."

"Make sure you don't corner His Majesty on h8," Tweedledum ad­

vised Alice.

"Also, do not drive him into the Borderland," said Tweedledee.

"His Majesty may find out that it is not as dangerous as he makes it out to be."

"Well, I'd better hurry and bring His Majesty back as soon as 1 can.

The Red Queen' s patience is shorter than her temper !"

Problem 1.

8 7

6

it

e f ^{g h}

Figure 2.

Alice was able to accomplish her mission, only to have the Red King slip out again. Humpty Dumpty, in his lofty position on the wall, spotted His Majesty on h4 this time.

Alice correctly deduced that the Red King still harbored no thought of crossing over to the Queen Side. While he had temporarily con­

quered his fear of the Borderland, he was not yet willing to venture into the White Kingdom.

Having lost much time in accomplishing her first mission, Alice set out immediately to reenact the drama, but on an enlarged stage.

Problem 2.

On the miniature chessboard in Figure 3, lt7Jite has a lone Queen on e8 and Red has a lone King on h4. White moves first, and wins if the Red King is driven back to e8 within 14 moves. If this is not accomplished, then Red wins. Other than what is noted above, normal chess rules apply. With perfect play, which royalty wins ?

8 7 6 5

4

it

e f g h

Fig u re 3.

14. Comerin9 the Kin9 ^{1 1 3}

Alice drove the Red King back to his Palace just in time.

"Come along," roared the Red Queen. "We have to attend a summit conference with the White Queen and her consort."

"What is the matter this time, dear?" asked the Red King timidly.

"We have been discussing the partition of the Borderland. There is too much goings-on here, especially on h4, or so 1 hear."

"I can't imagine what," murmured the Red King.

"Anyway, the White Queen and 1 have agreed to establish our bor­

ders between ranks 4 and 5 . We just meet to formalize the deal."

"If you say so, dear."

As soon as the new treaty was signed, the Red King headed for h5 , the furthest haven within his domain. Alice was dispatched after him a third time.

P�oblem 3.

On the minia ture chessboard in Figure 4 , White has a lone Queen on e8, and Red has a lone King on h5. Red moves firs t because the King is already in check. White wins if the Red King is driven back to e8 within 12 moves. If this is not accomplished, then Red wins. Other than wha t is noted above, normal chess rules apply. With perfect play, which royalty wins ?

8 7 6

5

it

e f ^{g h}

Figure 4.

Sol utiohS

ment supporting this conclusion by solving the Royal Problem for all

m x n chessboards, m ;::: n ;::: 3.

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Postsct"ipt

The general problem was completely solved by Andy Liu's amazing students. Three of them, Jesse Chan, Peter Laffin, and Da Li, all in tenth grade, gave their solution in "Martin Gardner's 'Royal Problem,'"

in Quan tum (September/October 1 993). They confirmed my conjecture that the king can't be cornered on square fields, but shot down my other conjecture, that the queen wins only on rectangular boards when

n > 2 and m > n + 1. I thought I had shown that the king wins on the

4 x 5 board, but Andy's students surprised me by finding a queen win on this board.

Perhaps I should add that all chess rules apply. If the queen is adjacent to the king, the king can take the queen, thereby winning the game. And the king wins if he is stale-mated on a square outside the upper left corner, or if he achieves perpetual check outside that corner.

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