Self-protecting Chess (SPC)
Hello Chess Fans,
Self-protecting Chess diagrams are legal chess positions with 8 white pawns, 8 white men and no black piece at all (but one empty square, at least, must be available for a Black King: should the Black King be added there, then the diagram would become « fully » legal — according to the normal chess rules).
The SPC concept claims two things:
1) every white piece protects exactly one other white piece
2) every white piece is protected by exactly one other white piece
This can be achieved by one giant loop (like in diag.1a), where A protects only B, which protects only C, which protects only D, ... until P which protects only A (start with the Queen on a1 and check the loop):
(one giant loop)
If you put a Black King on a6 (diag. 1b), the position becomes « fully » legal (the diagram hereunder could arise from a normal chess game; the d7 pawn, for example, could have been coming from h2, after having made 4 captures. Now that the concept of « legality » of a SPC position is clear, no more Black Kings will be shown):
(one giant loop with Black King)
One could achieve the same task with more than one loop; we show here (diag.2a and 2b) a 2-loop and a 3-loop solution:
(two loops) (three loops)
Is this old hat? I’ve started to play with this idea 15 years ago -- but lost ALL my notes & diagrams (which is a shame, some of them were full of surprises).
Questions (among thousands — I’ve forgotten some of the answers):
- create a legal SPC diagram with 6 loops and one white pawn per column (a solution here)
- create a legal SPC diagram where the 16 white pieces fit in a 7x7 sub-square of the board (is this possible? I’m not sure...)
- leave a pawn aside, then build a legal SPC with the 15 remaining pieces, two of which only are on dark squares (this puzzle is difficult, one solution is here; if you can add the last pawn to the diagram, I’ll offer you 64 US$!)
This genre is great fun! If you find interesting things, please let me know [eric (dot) angelini (at) skynet (dot) be]: I’ll publish them here!
And don’t forget to triple-check your diagrams: one forgets almost always a doubly-protected piece somewhere...
January 27th, 2008 note :
Bernd Schwarzkopf, from the Retro Mailing List, writes that this is not new (at least 1961 !) :
(...) the idea is old, look:
Journal of Recreational Mathematics
wKd1, Qc1, Rg8, Rh2, Bg3, Bh1, Sa7, Sb7, Pa5,a6,b5,b6,d5,e4,e6,f7
Each man protects exactly one other man; 3 loops.
(bK could be on b3 or h5.)
Jexon J. Secker
wKe1, Qa2, Re8, Rg6, Bf1, Bg5, Sc1, Sc5, Pb2,b4,c3,d7,f3,f5,g2,g4
Each man protects exactly one other man; 1 loop.
wKg6, Qb7, Rg8, Rh7, Be6, Be7, Sc7, Sd6 (no Pawns)
Each man protects exactly one other man; 1 loop on a 7 x 3-rectangular.
wKh6, Qe7, Rf5, Rh8, Bd4, Bg6, Sd5, Se6 (no Pawns)
Each man protects exactly one other man; 1 loop on a 5 x 5-square, 23
squares unprotected (maximum).
wKg3, Qd5, Ra7, Rb1, Bd4, Bh7, Sf1, Sf4 (no Pawns)
Each man protects exactly one other man; 1 loop, 4 squares unprotected
My article in Feenschach 71, November 1984, page 475-476: "Wer deckt wen?"
("Who protects whom?") with reprints of some problems above and:
Herbert Adamsky & Bernd Schwarzkopf
wKg3, Qf1, Ra4, Rg8, Bc3, Be2, Sc7, Sd7, Pa2,b2,b3,b6,e6,g6,h2,h7
Each man protects exactly one other man; 6 loops.
(bK could be on e4.)
Herbert Adamsky & Bernd Schwarzkopf
wKa5, Qc8, Rf3, Rh6, Bd5, Be5, Sb2, Sd1, Pb4,d4,d7,e2,e4,f1,g5,h4
Each man protects exactly one other man; 7 loops, but 1 Pawn on first rank.
(bK could be on a7 or h1.)
... not new but still fun !
Thanks Bernd !
All diagrams were created on-line here.
A similar challenge involving 32 units, there.
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