If the Queen was is front of her 2 rooks and giving check, does that count ? Do 3 pieces directly have to give the check or act as helpers?
You mean the K in the corner, with 6 Rs or Qs on each of the rank and file of the K, and a blocking piece on each next to the K to prevent illegality? (But maybe he doesn't count as a "check" an X-ray "check" unless the blocking piece is also attacking). At any rate, you could add 6 Bs on the long diagonal, with a blocking piece on the diagonal square next to the K. So 18 pieces. Then, since one check is legal, remove one of the blocking pieces and add a relevant "checking" piece, or leave the arrangement but add a N checking (and mating), from say c7. So that would be 19 pieces. But as you say, that is not what is meant by checking.
Another question is, What is the maximum number of pieces that could check a king simultaneously, if one ignored the illegality of how they got there? Presumably you would want somewhere in mid board to place the K. I get a result of 16, 8 by Ns. Which raises the question, Why twice as many Ns as any other minor piece? (I say minor piece, because you could replace the 4 Rs and 4 Bs by 8 Qs, such that there would be 8 Ns and 8 Qs. But 8Ns, 4 Bs and 4 Rs are more informative.)
This is a fascinating question. Thank you @A_0123456
A triple check could very well be possible. Then again, it may not be possible. Any student of the scientific method will keep his mind tuned into "maybe" until a firm proof can be constructed.
"How do we go about constructing such a proof, Burrower?" You ask.
Mathematics comes to the rescue, as always. Using logic, we can show that the stated assumptions logically guarantee the conclusions. Let me suggest 3 methods of proof to solve this problem:
(1) Proof by contraposition
(2) Proof by construction
(3) Proof by exhaustion
I will only approach the problem through one of these, but list them for completeness. Another reader may wish to approach the problem using the two remaining methods, after I proceed to deal with one of them thoroughly.
Now, if you and Paul Erdos will forgive me for inelegance, I suggest we approach the problem through (3), proof by exhaustion. Let us start with the hypothesis that there is not a triple check. Then us take a journey through the process of arriving at a rigorous proof:
(i) First, we will define a "check" in concrete symbolic language as a mathematical object, to minimize ambiguity.
(ii) Second, we will define "triple check" in the same manner.
(iii) Third, since the number of checks as defined in (i) is finite, we will produce an exhaustive set of cases matching this definition
(iv) Fourth, we will systematically check whether any cases from the set produced in (iii) match the definition for "triple check" described in (ii)
(v) Since we have exhaustively listed all cases matching (i), then checked whether they contain case (ii), we will now have a proof as to whether or not a "triple check" as defined in (ii) is possible in chess. The original poster will then be satisfied.
Since I am a very busy individual currently working on publishing my next paper, I will require one volunteer to work at step (iii) above and report back to me. I have already done the hard work of outlining a logical process to generate the proof, you just need to do the simple (albeit brutish) work of collecting data. Please state clearly that you accept the task in a future post, if this is you. Otherwise, several individuals may begin the task simultaneously and waste time unnecessarily. We only need one person. After you have completed this step, let me know, and I will verify your work.
"Trust, but verify" - Ronald Reagan
After verification, you will be credited and we can post our findings as a follow up post in this thread.
🙏Warm regards, Burrower 🙏
@Allonautilus
@Burrower
Both of your long-winded and confused comments ignore that I already provided the proof in a short paragraph, in #7 and even more concisely in #12. I presume you ignore the proof because you like the sounds of your own voices, so to speak.
Dear @nayf I appreciate and applaud your efforts. However, I am interested in mathematical proof, not verbal arguments that you might find in a spoken debate. Yes, what you wrote is a persuasive. But mathematical proof is not about persuasion, it is about absolutes.
It means that we utilize precise natural or symbolic language that minimizes all ambiguity (meeting the usual criteria for rigorous) and undertake a process of proof via accepted rules of inference. In this case, since you have not submitted a list of cases in point (iii) of my above framework for a proof by exhaustion, then your supposed "proof" is by definition, incomplete.
🙏Warm regards, Burrower 🙏
@Burrower
Putting your bullshit aside, my proof is as mathematical as any. It rests on the theorem of plane geometry that two straight lines intersect at most at only one point. If you're a stickler for formalism, call that theorem:
Premise 1.
Now add:
Premise 2: All chess pieces attack in straight lines except N. (True by definition of each piece, i.e. analytical truth).
Intermediate deduction: it follows from 1. and 2. that
3. since there can at most be one point where the attacking lines of two non-N pieces intersect, there can be only one square that two non-N pieces attack. (Deduction from 1. and 2.)
4. It follows from 3. that there cannot be both a blocking square blocking two non-N pieces and a second square that they are both attacking. (Contrapositive of 3.)
5. It follows from 4. that there cannot be two squares, one blocking two non-N pieces, and a second one where a king would be checked by both pieces. (Deduction from 4. and rules of chess, i.e. by definition of blocking and checking).
6. It follows from 5. that no arrangement is possible whereby a piece that moves discovers in so moving a double check. (Deduction from 5. and definitions of "discover", "move", and "check").
7. A piece can check another piece only once at any one time. (Definition, rules of chess).
8. It follows from 7 and 8 that no piece can move and thereby create a third check, while discovering two non-N checks. (Deduction from 6 and 7).
9. Ns cannot be blocked. (Definition, rules of chess).
10. It follows from 9 that no piece can move and thereby discover a N check (i.e. the N would already have been checking whatever it checked in the previous move or position). (Deduction from 9 and definition of "discovered check").
11. It follows from. 8. 10. and 2. that no piece can move and render a third check while discovering two other checks. (Deduction from 8. 10. and 2.)
Q.E.D.
It is possible. Say a tournament game and white castles into a square that is attacked by black 3 times. The dumbass is checkmated. Neither player turns this in as an illegal move. The TD doesnt see it. But the game is over and the final position has white king attacked 3 times. Both score sheets are turned into the TD. Its a legal game and the position is legal.
No, the position isn't legal (but the result will stand if the players signed their scoresheets).
3.10.3 A position is illegal when it cannot have been reached by any series of legal moves.
If you like such a possibility as triple check, maybe you could invent another chess game that has mutiny. That used to be a possibility with kings and armies. Don't ask me for any further ideas, I don't have any.
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