The helpmates and stalemates with n pieces of the same value

Pomocné maty a paty s více kameny stejných hodnot

(Václav Kotěšovec, 15.4.2021)

1) White pieces of the same value

Jaký je minimální počet kamenů stejných hodnot, se kterými (bez vlastního krále) existuje matová (resp. patová) pozice se samotným soupeřovým králem ?
Jaký je nejdelší pomocný mat s tímto materiálem ?

What is the minimum number of pieces of the same value with which (without his own king, against bare king) there exists a checkmate position ?
What is the longest helpmate with this material?

S ortodoxními kameny je tato úloha triviální. Mnohem zajímavější jsou přeskakující kameny.

With orthodox pieces, this problem is trivial. Hoppers are much more interesting.

Notation of the fairy pieces
G = Grasshopper
RH = Rookhopper
BH = Bishophopper
NH = Nightriderhopper
KA = Kangaroo

Zatímco úlohy s jedním řešením se cvrčky a věžovými cvrčky jsou poměrně nezajímavé (pro srovnání viz třeba PDB P1378116 se 4 řešeními!), úlohy se střelcovými, tátošovými cvrčky a klokany dosahují značných délek a i hra je zajímavá.

While helpmates with one solution with grasshoppers and rookhoppers are relatively uninteresting (for comparison, see PDB P1378116 with 4 solutions!), problems with bishophoppers, nightriderhoppers and kangaroos reach considerable lengths and also the play is more interesting.

S1 Vaclav Kotesovec, original h#19 7 Bishophoppers C+ *** 1.Kb2 BHa3 2.Kc2 BHd3 3.Kb2 BHc1 4.Kc2 BHb1 5.Kd3 BHe4 6.Ke3 BHf4 7.Kd4 BHd5 8.Ke5 BHd6 9.Kf5 BHg6 10.Ke6 BHc5 11.Kd7 BHe6 12.Kc7 BHb8 13.Kb6 BHa7 14.Kc5 BHd4 15.Kd6 BHe5 16.Ke7 BHf6 17.Kf8 BHe4 18.Kg8 BHf7+ 19.Kh8 BHg7#	S2 Vaclav Kotesovec, original h=18 6 Bishophoppers C+ *** 1.Kb2 BHa3 2.Kc2 BHb1 3.Kb2 BHc1 4.Kc2 BHd3 5.Kd2 BHe3 6.Kc2 BHb1 7.Kd3 BHe4 8.Kd4 BHc5 9.Kd5 BHc6 10.Ke6 BHf5 11.Kf6 BHe7+ 12.Kg6 BHh7 13.Kf5 BHe4 14.Ke6 BHf8 15.Kd7 BHd6 16.Kc8 BHc5 17.Kb8 BHc7+ 18.Ka8 BHb6 =
S3 Vaclav Kotesovec, original h#17 7 Nightriderhoppers C+ *** 1.Ka1 NHc4 2.Kb1 NHa5 3.Ka1 NHe3+ 4.Kb1 NHg4 5.Ka1 NHg2 6.Kb1 NHe8 7.Kb2 NHh2 8.Kc3 NHe4 9.Kd4 NHb5 10.Kd3 NHe5 11.Kd4 NHf3 12.Kd3 NHg1 13.Ke2 NHc3 14.Kd2 NHb1 15.Kc1 NHd5 16.Kb2 NHc4+ 17.Ka1 NHc2#	S4 Vaclav Kotesovec, original h=15 5 Nightriderhoppers C+ *** 1.Kc2 NHa1 2.Kb3 NHc5 3.Kb4 NHa2 4.Kb3 NHa1 5.Kb4 NHc6 6.Kb3 NHc5 7.Kc4 NHd4 8.Kd5 NHb5 9.Ke5 NHg4 10.Kf4 NHe2 11.Ke5 NHc6 12.Kd6 NHf7 13.Kc7 NHa7 14.Kb7 NHe5 15.Ka8 NHd3 =
S5 Vaclav Kotesovec, original h#25 0.1.1... 8 Kangaroos C+ *** 1...KAh8 2.Ke8 KAd8 3.Kf8 KAc8 4.Kg8 KAh8 5.Kh7 KAh1 6.Kh6 KAh7 7.Kh5 KAh1 8.Kh4 KAh5 9.Kh3 KAh6 10.Kh2 KAh1 11.Kg1 KAb1 12.Kf1 KAg1 13.Kf2 KAa7 14.Ke3 KAa8 15.Kd4 KAe3 16.Kd3 KAf3 17.Ke4 KAg2 18.Kf4 KAg5 19.Kg4 KAe2 20.Kf4 KAd2 21.Kg3 KAg1 22.Kf2 KAd4+ 23.Kf1 KAd1 24.Kg1 KAf2+ 25.Kh1 KAd5#	S6 Vaclav Kotesovec, original h=33 6 Kangaroos C+ *** 1.Kc1 KAg1 2.Kb1 KAa1 3.Ka2 KAa8 4.Ka3 KAa2 5.Ka4 KAa8 6.Ka5 KAa4 7.Ka6 KAa3 8.Ka7 KAa8 9.Kb8 KAh8 10.Kc8 KAb8 11.Kd8 KAh8 12.Ke8 KAd8 13.Kf8 KAc8 14.Kg8 KAh8 15.Kh7 KAh4 16.Kh6 KAh7 17.Kg7 KAa7 18.Kf7 KAg7 19.Ke7 KAh7 20.Kf7 KAe7 21.Kf6 KAg5 22.Ke5 KAd5 23.Kf5 KAc5 24.Ke5 KAf5 25.Kf6 KAf7 26.Ke5 KAg5 27.Kd5 KAc5 28.Ke5 KAb5 29.Kd6 KAf8 30.Kc5 KAb4 31.Kc6 KAd7 32.Kb7 KAc7+ 33.Ka8 KAb7 =

2) Black pieces of the same value

Přejděme teď k opačné situaci. Bílý má jen jeden kámen (bez krále), černý má krále a několik kamenů stejných hodnot.
Jaký je minimální počet těchto kamenů aby byl možný pomocný mat s tímto materiálem a jakou má největší délku ?

Většinou existují matové nebo patové pozice i s menším počtem černých kamenů, neexistuje k nim však poslední tah. To vyžaduje přidání dalšího černého kamene.

Let us now turn to the opposite situation. White has only one piece (without a king), black has a king and several pieces of the same value.
What is the minimum number of these pieces such that to exists a helpmate with this material and what is its greatest length?

There are usually mating or stalemate positions with a smaller number of black pieces, but there is no last move. This requires the addition of next black piece.

S7 Vaclav Kotesovec, original h#19 1 Nightriderhopper 5 Grasshoppers C+ *** 1.Ke7 NHc6 2.Gh1 NHg8 3.Kf6 NHe4 4.Gd5 NHg8 5.Ge6 NHe4 6.Gf5 NHg8 7.Gc5 NHe4 8.Gb6 NHg8 9.Ge5 NHe4 10.Ge7 NHg8 11.Ke5 NHc6 12.Ge4 NHg8 13.Kd6 NHc6 14.Kc7 NHg8 15.Gb8 NHc6 16.Kb7 NHg8 17.Ka8 NHc6 18.Gb7 NHg8 19.Ga7 NHd2#	S8 Vaclav Kotesovec, original h#42 1+6 Nightriderhoppers C+ *** 1.Kf6 NHg4 2.Ke5 NHc6 3.NHb8 NHg4 4.Kf6 NHe8 5.NHh5 NHg4 6.Ke5 NHc6 7.Kd4 NHe2 8.Kc3 NHa4 9.NHd1 NHe2 10.Kd4 NHc6 11.Ke5 NHg4 12.Kf6 NHe8 13.NHd7 NHg4 14.Ke5 NHc6 15.Kd4 NHe2 16.Kc3 NHa4 17.NHb5 NHe2 18.Kd4 NHc6 19.Ke5 NHg4 20.NHf3 NHc6 21.NHh2 NHg4 22.NHf6 NHe8 23.Ke4 NHg4 24.NHf2 NHe8 25.Kd5 NHg4 26.NHb6 NHe8 27.Kc4 NHb2 28.NHd2 NHd6 29.NHb5 NHb2 30.NHh2 NHd6 31.NHd4 NHb2 32.NHb5 NHd6 33.NHf7 NHh8 34.Kd3 NHd6 35.NHb4 NHh8 36.Ke4 NHd6 37.Kf5 NHh4 38.Kg6 NHb1 39.Kh7 NHf3 40.Kh8 NHb1 41.NHh7 NHf3 42.NHg8 NHb5#
S9 Vaclav Kotesovec, original h#65 1 Nightriderhopper 5 Kangaroos C+ * 1.Kb6 NHa4 2.Kc5 NHe6 3.Kd4 NHc2 4.Ke3 NHg4 5.KAd1 NHc2 6.Kd4 NHe6 7.Kc5 NHa4 8.Kb6 NHc8 9.KAg8 NHa4 10.Kc5 NHe6 11.Kd4 NHc2 12.Ke3 NHg4 13.KAh5 NHc2 14.Kd4 NHe6 15.Kc5 NHa4 16.Kb6 NHc8 17.KAb8 NHa4 18.Kc5 NHe6 19.Kd4 NHc2 20.Ke3 NHg4 21.KAd1 NHc2 22.Kd4 NHe6 23.Kc5 NHa4 24.Kb6 NHc8 25.KAa8 NHa4 26.Kc5 NHe6 27.Kd4 NHc2 28.Ke3 NHg4 29.KAh5 NHc2 30.Kd4 NHe6 31.Kc5 NHa4 32.Kb6 NHc8 33.KAd8 NHa4 34.Kc5 NHe6 35.Kd4 NHc2 36.Ke3 NHg4 37.KAd1 NHc2 38.Kd4 NHe6 39.Kc5 NHa4 40.Kb6 NHc8 41.KAe8 NHa4 42.Kc5 NHe6 43.Kd4 NHc2 44.Ke3 NHg4 45.KAe1 NHc2 46.KAe4 NHg4 47.KAe5 NHc6 48.Kd4 NHg4 49.Kc5 NHc6 50.KAc7 NHg4 51.Kd6 NHc6 52.Ke7 NHg8 53.Kd6 NHd2 54.KAd7 NHf6 55.KAb8 NHd2 56.Kd5 NHf6 57.KAd4 NHd2 58.Kc4 NHb6 59.KAb4 NHf8 60.Kb5 NHb6 61.KAb7 NHf8 62.Ka6 NHb6 63.Ka7 NHf8 64.Ka8 NHc2 65.KAa7 NHe6# Longest helpmate without fairy conditions of all times!**	S10 Vaclav Kotesovec, original h#56 1+6 Kangaroos C+ *** 1.Ka2 KAa1 2.Kb1 KAh1 3.Kc1 KAb1 4.Kd1 KAh1 5.Ke1 KAd1 6.Kf1 KAh1 7.Kg2 KAa8 8.Kf3 KAg2 9.Ke2 KAc2 10.Kd3 KAg6 11.Kc2 KAb1 12.Kb3 KAb8 13.Kb4 KAb3 14.Kb5 KAb8 15.Kb6 KAb5 16.Kc5 KAg5 17.Kd5 KAc5 18.Ke5 KAg5 19.Kf4 KAc1 20.Kg5 KAh6 21.Kh5 KAh2 22.Kh6 KAh7 23.Kg7 KAa7 24.Kf7 KAg7 25.Ke7 KAa7 26.Kd7 KAe7 27.KAc8 KAa7 28.Kc7 KAd7 29.KAg4 KAh3 30.Kd6 KAe6 31.Ke5 KAh3 32.Kf4 KAe6 33.Kg3 KAh3 34.KAg5 KAe6 35.Kf4 KAh3 36.KAh6 KAe6 37.KAe3 KAh3 38.Kf3 KAd3 39.Ke4 KAg6 40.KAg7 KAd3 41.KAf3 KAg3 42.Kf4 KAd3 43.Kg4 KAg3 44.KAh3 KAg6 45.Kg3 KAg2 46.KAg1 KAa8 47.Kf4 KAg2 48.Ke5 KAa8 49.Ke6 KAg2 50.KAd7 KAa8 51.KAc8 KAg2 52.Kf7 KAa8 53.KAh7 KAg2 54.Kg8 KAa8 55.Kh8 KAg2 56.KAg8 KAa8# Switchback of the white kangaroo
S11 Vaclav Kotesovec, original h=20 1 Grasshopper 3 Kangaroos C+ *** 1.Kg1 Gh1 2.Kf1 Ge1 3.Ke2 Ge3 4.Kd2 Gc1 5.Kc3 Gc4 6.Kd4 Ge4 7.KAf4 Gc4 8.Kd3 Ge2 9.Ke4 Ge5 10.KAe3 Gg3 11.Kd5 Ge5 12.KAf5 Gc5 13.KAb5 Gf2 14.Kc4 Gd4 15.KAb4 Gf2 16.Kb3 Gd4 17.KAb2 Ga1 18.KAb1 Gc1 19.Ka2 Gf4 20.Ka1 Gd2 =	S12 Vaclav Kotesovec, original h=31 1+4 Nightriderhoppers C+ *** 1.Kb4 NHa6 2.Kc5 NHe4 3.Kd6 NHc8 4.Ke7 NHg6 5.NHh4 NHc8 6.Kd6 NHe4 7.Kc5 NHa6 8.Kb4 NHc2 9.NHa1 NHa6 10.Kc5 NHe4 11.NHd7 NHa6 12.NHb8 NHe4 13.NHf6 NHg8 14.Kd5 NHe4 15.NHb6 NHg8 16.Kd6 NHe4 17.NHb7 NHc8 18.Ke5 NHa4 19.NHf3 NHc8 20.Ke4 NHf2 21.Kd3 NHb4 22.NHe1 NHh7 23.NHg5 NHd5 24.NHf4 NHh7 25.NHb2 NHd5 26.Kc3 NHb1 27.NHa2 NHh4 28.Kc2 NHd2 29.Kb1 NHh4 30.Ka1 NHd2 31.NHb1 NHg8 =
S13 Vaclav Kotesovec, original h=40 1 Nightriderhopper 4 Kangaroos C+ *** 1.Kf6 NHg8 2.KAh8 NHe4 3.KAf1 NHg8 4.KAf7 NHe4 5.KAf8 NHg8 6.KAe8 NHe4 7.KAf5 NHg8 8.Ke7 NHc6 9.KAa4 NHg8 10.Kf6 NHe4 11.KAf8 NHg8 12.Ke7 NHc6 13.KAd7 NHg8 14.KAc7 NHc6 15.KAe8 NHg8 16.KAh8 NHc6 17.KAf7 NHg8 18.Kf6 NHe4 19.KAf5 NHg8 20.Ke7 NHc6 21.KAg7 NHg8 22.Kf6 NHe4 23.KAe5 NHg8 24.Ke7 NHc6 25.Kf6 NHg4 26.Kg5 NHc6 27.Kf4 NHg4 28.KAf3 NHc6 29.KAf2 NHg4 30.Ke4 NHc6 31.KAb7 NHg4 32.Kd4 NHc6 33.KAc3 NHe2 34.KAb2 NHa4 35.Kc4 NHe2 36.Kb3 NHa4 37.KAb1 NHe2 38.Ka2 NHa4 39.Ka1 NHe2 40.KAa2 NHa4 =	S14 Vaclav Kotesovec, original h=52 1+5 Kangaroos C+ *** 1.Kg2 KAg1 2.Kf1 KAc1 3.Kg1 KAh1 4.Kh2 KAh6 5.Kh3 KAh2 6.Kh4 KAh6 7.Kg5 KAd2 8.Kf4 KAg5 9.Kg4 KAg2 10.Kg5 KAg6 11.Kf5 KAb1 12.Ke4 KAf5 13.Kd3 KAb1 14.KAc3 KAe4 15.KAb3 KAb1 16.Ke4 KAf5 17.KAa4 KAb1 18.Kf5 KAg6 19.KAd1 KAb1 20.Kg6 KAh7 21.Kh6 KAh4 22.Kh7 KAh8 23.Kg7 KAb2 24.KAa4 KAh8 25.Kh6 KAh4 26.KAd1 KAh7 27.Kg6 KAb1 28.Kf5 KAg6 29.Ke4 KAb1 30.Kd3 KAe4 31.KAe3 KAb1 32.KAf3 KAe4 33.Ke2 KAe1 34.KAg4 KAe4 35.Kd3 KAb1 36.KAe2 KAe4 37.KAc3 KAe1 38.Kd4 KAe4 39.Ke5 KAe1 40.Kf5 KAe4 41.KAg6 KAh7 42.Ke4 KAd3 43.Kf3 KAg3 44.KAg2 KAd3 45.KAb3 KAa3 46.Ke3 KAd3 47.KAf3 KAg3 48.Kf2 KAa3 49.KAh2 KAg3 50.Kg1 KAa3 51.Kh1 KAg3 52.KAg1 KAa3 = Switchback of the black king

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