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See Bleicher's "Freezer" that is a human/computer interactive proof agent: you make the "rules", and it iterates over positions. (The article from 2004 is rather old, and the given examples are solvable by brute means by now.)

Probably one can write the simplest endgames (KQ vs K and KR vs K) as induction proofs, so they work on bigger boards. For KBN vs K, you can proceed by "reduction" of classes of positions (based upon opponent's king location), similar to solving a puzzle like Rubik's Cube (the KBN vs K has even been done for Kriegspiel by Ferguson).

Fortress draws are classical examples of "high-level proofs", with the KQ vs KRP example from Philidor (1777) one of the earlier ones (WQd4, WKf5, BKe8, BRe6, BPd7, though if everything is shifted a rank down, then White can spruce the Black king out, force it to d5 to stop the Re5-c5 oscillation, and win), and the above example a 7-piece variant of this. General-purpose computer programs are well-known to do poorly in solving fortress problems, as the Behting study previously demonstrated. However, there are still some problems that computers "can't solve" (without some additional methods), and a more subtle fortress example is this study by Chekhover (1947):

7r/p3k3/2p5/1pPp4/3P4/PP4P1/3P1PB1/2K5 w - - 0 1

White to move and draw. The "natural" moves (as proposed by the computer) will lose, as though White has three pawns for the exchange, this will not avail him much when the rook's activity becomes clear. But rather 1. Kd1 Rh2 2. Ke2 (or Ke1) Rxg2 3. Kf1 Rh2 4. Kg1 Rh3 5. Kg2 Rh8 6. f3, and Black can never invade, White shuffling the king between f1/f2/g1/g2 and blocking the queenside pawns as necessary. (Also to note, is that the sacrifice Re4 by Black does not work, and likely loses.)

For the second question, by now there exist a (large) number of studies whose first X moves are tricky enough so that brute force by computer is not immediately feasible, but whose last Y moves are not very understandable to humans, being just a "random" tablebase position that happens to give the desired result.

See Bleicher's "Freezer" that is a human/computer interactive proof agent: you make the "rules", and it iterates over positions. (The article from 2004 is rather old, and the given examples are solvable by brute means by now.)

Probably one can write the simplest endgames (KQ vs K and KR vs K) as induction proofs, in a manner so that they will work on bigger boards. For KBN vs K, you can proceed by "reduction" of classes of positions (say based upon opponent's king location), similar to solving a puzzle like Rubik's Cube (the KBN vs K has even been done for Kriegspiel by Ferguson).

Fortress draws are classical examples of "high-level proofs", with the KQ vs KRP example from Philidor (1777) one of the earlier ones (WQd4, WKf5, BKe8, BRe6, BPd7, though if everything is shifted a rank down, then White can spruce the Black king out, force it to d5 to stop the Re5-c5 oscillation, and win), and the above example is a 7-piece variant of this. General-purpose computer programs are well-known to do poorly in solving fortress problems, as the Behting study previously demonstrated. However, there are still some problems that computers "can't solve" (without some additional methods), and a more subtle fortress example is this study by Chekhover (1947):

7r/p3k3/2p5/1pPp4/3P4/PP4P1/3P1PB1/2K5 w - - 0 1

White to move and draw. The "natural" moves (as proposed by the computer) will lose, as though White has three pawns for the exchange, this will not avail him much when the rook's activity becomes clear. But rather 1. Kd1 Rh2 2. Ke2 (or Ke1) Rxg2 3. Kf1 Rh2 4. Kg1 Rh3 5. Kg2 Rh8 6. f3, and Black can never invade, White shuffling the king between f1/f2/g1/g2 and blocking the queenside pawns as necessary. (Also to note, is that the sacrifice Re4 by Black does not work, and likely loses.)

For the second question, by now there exist a (large) number of studies whose first X moves are tricky enough so that brute force by computer is not immediately feasible, but whose last Y moves are not very understandable to humans, being just a "random" tablebase position that happens to give the desired result.

See Bleicher's "Freezer" that is a human/computer interactive proof agent: you make the "rules", and it iterates over positions. (The article from 2004 is rather old, and the given examples are solvable by brute means by now.)

Probably one can write the simplest endgames (KQ vs K and KR vs K) as induction proofs, so they work on bigger boards. For KBN vs K, you can proceed by "reduction" of classes of positions (based upon opponent's king location), similar to solving a puzzle like Rubik's Cube (the KBN vs K has even been done for Kriegspiel by Ferguson).

Fortress draws are classical examples of "high-level proofs", with the KQ vs KRP analysis of Cheron not the first example, but maybe the most broad and mathematized analysis, and indeed the position given above is a variant of this. General-purpose computer programs are well-known to do poorly in solving fortress problems, as the Behting study previously demonstrated. However, there are still some problems that computers "can't solve" (without some additional methods), and a more subtle fortress example is this study by Chekhover (1947):

7r/p3k3/2p5/1pPp4/3P4/PP4P1/3P1PB1/2K5 w - - 0 1

White to move and draw. The "natural" moves (as proposed by the computer) will lose, as though White has three pawns for the exchange, this will not avail him much when the rook's activity becomes clear. But rather 1. Kd1 Rh2 2. Ke2 (or Ke1) Rxg2 3. Kf1 Rh2 4. Kg1 Rh3 5. Kg2 Rh8 6. f3, and Black can never invade, White shuffling the king between f1/f2/g1/g2 as necessary. (Also to note, is that the sacrifice Re4 by Black does not work, and likely loses.)

For the second question, by now there exist a (large) number of studies whose first X moves are tricky enough so that brute force by computer is not immediately feasible, but whose last Y moves are not very understandable to humans, being just a "random" tablebase position that happens to give the desired result.

See Bleicher's "Freezer" that is a human/computer interactive proof agent: you make the "rules", and it iterates over positions. (The article from 2004 is rather old, and the given examples are solvable by brute means by now.)

Probably one can write the simplest endgames (KQ vs K and KR vs K) as induction proofs, so they work on bigger boards. For KBN vs K, you can proceed by "reduction" of classes of positions (based upon opponent's king location), similar to solving a puzzle like Rubik's Cube (the KBN vs K has even been done for Kriegspiel by Ferguson).

Fortress draws are classical examples of "high-level proofs", with the KQ vs KRP example from Philidor (1777) one of the earlier ones (WQd4, WKf5, BKe8, BRe6, BPd7, though if everything is shifted a rank down, then White can spruce the Black king out, force it to d5 to stop the Re5-c5 oscillation, and win), and the above example a 7-piece variant of this. General-purpose computer programs are well-known to do poorly in solving fortress problems, as the Behting study previously demonstrated. However, there are still some problems that computers "can't solve" (without some additional methods), and a more subtle fortress example is this study by Chekhover (1947):

7r/p3k3/2p5/1pPp4/3P4/PP4P1/3P1PB1/2K5 w - - 0 1

White to move and draw. The "natural" moves (as proposed by the computer) will lose, as though White has three pawns for the exchange, this will not avail him much when the rook's activity becomes clear. But rather 1. Kd1 Rh2 2. Ke2 (or Ke1) Rxg2 3. Kf1 Rh2 4. Kg1 Rh3 5. Kg2 Rh8 6. f3, and Black can never invade, White shuffling the king between f1/f2/g1/g2 and blocking the queenside pawns as necessary. (Also to note, is that the sacrifice Re4 by Black does not work, and likely loses.)

For the second question, by now there exist a (large) number of studies whose first X moves are tricky enough so that brute force by computer is not immediately feasible, but whose last Y moves are not very understandable to humans, being just a "random" tablebase position that happens to give the desired result.

See Bleicher's "Freezer" that is a human/computer interactive proof agent: you make the "rules", and it iterates over positions. (The article from 2004 is rather old, and the given examples are solvable by brute means by now.)

See Bleicher's "Freezer" that is a human/computer interactive proof agent: you make the "rules", and it iterates over positions. (The article from 2004 is rather old, and the given examples are solvable by brute means by now.)

Probably one can write the simplest endgames (KQ vs K and KR vs K) as induction proofs, so they work on bigger boards. For KBN vs K, you can proceed by "reduction" of classes of positions (based upon opponent's king location), similar to solving a puzzle like Rubik's Cube (the KBN vs K has even been done for Kriegspiel by Ferguson).

Fortress draws are classical examples of "high-level proofs", with the KQ vs KRP analysis of Cheron not the first example, but maybe the most broad and mathematized analysis, and indeed the position given above is a variant of this. General-purpose computer programs are well-known to do poorly in solving fortress problems, as the Behting study previously demonstrated. However, there are still some problems that computers "can't solve" (without some additional methods), and a more subtle fortress example is this study by Chekhover (1947):

7r/p3k3/2p5/1pPp4/3P4/PP4P1/3P1PB1/2K5 w - - 0 1

White to move and draw. The "natural" moves (as proposed by the computer) will lose, as though White has three pawns for the exchange, this will not avail him much when the rook's activity becomes clear. But rather 1. Kd1 Rh2 2. Ke2 (or Ke1) Rxg2 3. Kf1 Rh2 4. Kg1 Rh3 5. Kg2 Rh8 6. f3, and Black can never invade, White shuffling the king between f1/f2/g1/g2 as necessary. (Also to note, is that the sacrifice Re4 by Black does not work, and likely loses.)

For the second question, by now there exist a (large) number of studies whose first X moves are tricky enough so that brute force by computer is not immediately feasible, but whose last Y moves are not very understandable to humans, being just a "random" tablebase position that happens to give the desired result.