Solutions for "Homebase" (aka "At Home") Proof Games

The first couple of problems exhibit the Pronkin theme (a promoted piece moves to the homebase of a captured piece of the same kind). The others combine Pronkin with Ceriani-Frolkin (the capture on a particular square of a piece promoted on a particular square.)

In parenthesis, am I the only one to find something poignant in the Ceriani-Frolkin theme? Something about the yearning quest for identity against the cruel backdrop of  life's short span. Sigh.

Anyway. My own guidelines for these tasks is to attempt to achieve them with the minimal number of moves, and within this constraint to attempt the minimum number of captures.

Other themes which naturally come up in this context are:
- the Phoenix theme (a piece is captured, but a pawn later promotes to that kind),
- the Donati theme (the promoted unit moves away and then returns to the promotion square),

There is (afaik) no homebase example yet of the Schnoebelen theme (the promoted unit is captured without ever moving - but has still been identified uniquely.) It is clearly impossible to have a Schnoebelen queen, but other units may be possible.

Thierry Le Gleuher has composed many of these problems, and has a different aesthetic, which is to attempt to achieve these positions with the maximum number of moves. This gets too close to "massacres" for my current tastes.

All of these are verified by Natch or Euclide. I have kept a copy of the output file for the very time-consuming ones.

See also the Pronkin & Ceriani-Frolkin analysis further down this page.

{C01} 1. h4 g5 2. hg5 Nf6 3. gf6 Bg7 4. fe7 Rf8 5. ef8=B Bb2 6. Ba3 Bc1 7. Bc1.
Minimal possible number of moves for B Pronkin, and smaller number of captures than {CP04}. (C+)

{C02} 1. e4 d5 2. ed5 Nc6 3. dc6 Qd4 4. cb7 Qb2 5. ba8=Q Qc2 6. Qf3 Qd1+ 7. Qd1.
Minimal possible number of moves for Q Pronkin. (C+)

{C03}1. b4 h5 2. b5 h4 3. b6 h3 4. ba7 hg2 5. ab8=N gh1=N 6. Nd7 Ng3 7. Ne5 Ne2 8. Nf3 Ng1 9. Ng1.
Minimal number of moves for N Pronkin + N Frolkin. Phoenix for bN. (C+)

{C04}
(a)1. f4 a5 2. f5 a4 3. f6 a3 4. fxg7 axb2 5. gxf8=B bxa1=B 6. Bxe7 Bg7 7. Bf8 Bxf8.
(b) As (a) then 5. exd8=B bxa1=Q 6. Bxc7 Qf6 7. Bd8 Qxd8
(c) As (a) then 5. exd8=N bxa1=Q 6. Nxb7 Qf6 7. Nd8 Qxd8.
(d) As (a) then 5. exf8=N bxa1=B 6. Nxd7 Bg7 7. Nf8 Bxf8. Note (d) is similar to {CP22})
These exhibit minimal length B/Q Pronkin + N/B Frolkin. Also Donati switchback & Phoenix.
(All C+.) Waiting for publication.

{C05} 1. h4 d5 2. h5 d4 3. h6 d3 4. hg7 dc2 5. gh8=Q cb1=Q 6. Qd4 Qd3 7. Qd8+ Qd8.
Minimal number of moves for Q Pronkin + Q Frolkin. Circuit for wP (d3-c2-b1-d3) (C+)

{C06} 1. g4 b5 2. g5 b4 3. g6 b3 4. gh7 ba2 5. hg8=R ab1=N 6. Rg7 Nd2 7. Rf7 Ne4 8. Rg7 Nf6 9. Rg8 Ng8. N Pronkin + R Frolkin. Length 2 Donati switchback & Phoenix. (C+)

{C07} 1. g4 d5 2. g5 d4 3. g6 d3 4. gh7 dc2 5. hg8=N cb1=B 6. Ne7 Bf5 7.Nc8 Bc8.
B Pronkin + N Frolkin
. (C+)

{C08} 1. h4 f5 2. h5 f4 3. h6 f3 4. hg7 fe2 5. gh8=Q ef1=R+ 6. Ke2 Rf2+ 7. Ke1 Rd2 8. Qd4 Rd1+ 9. Qd1.
Q Pronkin + R Frolkin
. Switchback for wK. Phoenix for bR. (C+ Euclide 0.9)

{C09} (a) 1. f4 h5 2. f5 h4 3. f6 h3  4. fxe7 hxg2 5. exf8=N gxh1=R 6. Ng6 Rxh2 7. Nxh8 Rxh8.
(b) 1. f4 h5 2. f5 h4 3. f6 h3 4. fxe7 hxg2 5. exf8=B gxh1=R 6. Bxg7 Rxh2 7. Bxh8 Rxh8.
R Pronkin + N/B Frolkin. (Both C+)

{C10} 1. b4 d5 2. b5 d4 3. b6 d3 4. ba7 dc2 5. ab8=Q cb1=B 6. Qb7 Bf5 7. Qc8 Bc8.
B Pronkin + Q Frolkin
. (C+)

{C11} 1. g4 c5 2. g5 c4 3. g6 c3 4. gh7 cb2 5. hg8=N bc1=Q 6. Ne7 Qc2 7. Nd5 Qa2 8. Nc3 Qb1 9. Nb1.
Q Pronkin + N Frolkin
. (C+)

{C12} 1. a4 g5 2. a5 g4 3. a6 g3 4. ab7 gh2 5. bc8=B hg1=R 6. Bb7 Rg2 7. Ba8 Rf2 8. Bg2 Rf1 9. Bf1
B Pronkin + R Frolkin. (C+ Natch 2.0.)

{C13} 1.f4 c5 2.f5 c4 3.f6 c3 4.fg7 cb2 5.gh8=R bc1=B 6.Rh7 Bb2 7.Rf7 Bbg7 8.Rf8+ Bf8.
B Pronkin + R Frolkin. (C+)

{C14} 1.d4 a5 2.d5 a4 3.d6 a3 4.dc7 ab2 5.cb8=R bc1=Q 6.Rb7 Qb1 7.Rd7 Qbb6 8.Rd8+ Qd8.
Q Pronkin + R Frolkin. (C+)

{C15} 1.b4 h5 2.b5 h4 3.b6 h3 4.bc7 hg2 5.cd8=B gf1=N 6.Be7 Nh2 7.Bf6 Ng4 8.Rh8 N4f6 9.Rg8 Ng8.
N Pronkin + B Frolkin. This is the only known Pronkin/Frolkin "homebase" position in which the Frolkin piece is captured before the last move.
(C+)

{C16} 1.b4 f5 2.b5 f4 3.b6 f3 4.ba7 fe2 5.ab8=R ed1=R+ 6.Ke2 Rd2+ 7.Ke1 Rc2 8.Rc8 Rca2 9.Ra8 Ra1 10.Ra1.
R Pronkin + R Frolkin.
(C+ 58 hours)

Minimal Representations of the Pronkin Theme

A lower bound on the number of moves is 4.5 + the return time of the prodigal promoted Pronkin. The queen cannot return in 1 move without checking. Nor apparently can the rook return in 1 move, so the following lengths are best possible.

Pronkin Q 6.5 {C02}
R 6.5 {CP05}
B 6.5 {C01},{CP04}
N 8.5 {CP12},{C11}{C03},{CP16}

Minimal Representations of Pronkin-Frolkin Pairs

The shortest Pronkin-Frolkin compositions can be laid out in the following diagram, according to which piece is the Pronkin promotee (row) and which piece is the Frolkin promotee (column) In none of the known examples is the Pronkin promotion Donati. In every known case, it is the Pronkin promotee which captures the Frolkin one, and in every case except {C15}, this is the final move.

  Frolkin
Pronkin   Q R B N
Q {C05} {C14} {C04b} {C04c}
R {CP7} {C16} {C09b} {C09a}
B {C10} {C13} {C04a} {C04d},{CP22},{C07}
N {C11} {C06} {C15} {C03},{CP16}

The following table shows the best values to date.

  Frolkin
Pronkin   Q R B N
Q 7.0 8.0 7.0 7.0
R 7.0 9.5? 7.0 7.0
B 7.0 8.0 7.0 7.0
N 8.5 9.0 9.0 8.5

Except possibly for NR, NB & RR, I don't think any further improvement is possible. Here's my reasoning.

The Pronkin lower bounds given before still apply, but the key issue for the Frolkin piece is that we must prove its identity before it gets captured. It can do this by moving, or by having the opposing king move near it. A Frolkin Q must move twice to prove it's identity. If a Frolkin B/N moves it leaves the back rank, so either it or its capturer must spend another move returning to the 8th rank. So 7.0 is the minimum number of moves for Frolkin Q/B/N.

The Frolkin rook is more complicated. One approach is for it to move and at some point have the opposing king remain calmly a bishop's move away from it, thus demonstrating that the piece is not a queen. On the other hand, the rook may main stationary, and the Black king may remain (i) a bishop's move away (ii) a knight's move away, thus demonstrating by elimination that the unit is a rook.

Except for situations difficult to engineer in the homebase environment, it seems that we must allow for three rook moves after promotion. This gives a minimum of 8.0 moves for the rook case.

{C17} 1. Nc3 Nf6 2. Nd5 Ne4 3. Ne7 Nc3 4. Nc6 Qg5 5. Na7 Qb5 6. Nb5 Ba3 7. Na3 Nb1 8. Nb1
Circuit, At Home (C+ Natch 2.0.)

{C18} 1. e4 Nc6 2. e5 Ne5 3. Bd3 Nd3+ 4. Kf1 Nc1 5. Qg4 Na2 6. Qb4 Nb4 7.Ra6 Na6 8.Ke1 Nb8
Circuit, At Home, Switchback (C+ Natch 2.0.)

{C19} 1. c4 a5 2. Nc3 a4 3. Na4 Ra4 4. Qc2 Rc4 5. Qh7 Rh4 6. Qh8 Rh8.
At home. Wandering rook.

And the bonus question...

{C0}
(10+10) What unit moved last? Ah, none could, so it's illegal. If any of the 18 non-king units is removed, the seal is broken, and some unit could have performed the last move. If a knight or rook is added on its starting square, the position becomes legal. Finally, if a rook pawn is added on its starting square, then that might be removed without losing illegality. Therefore this position is the unique homebase illegal cluster.


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