Julia's Fairies

No.1765 (NSR)

Shankar Ram

Original Fairy problems
01.01.2023 - 31.12.2023

☝ Definitions

No. 1765 Shankar Ram
original - 08.10.2023

white kc8 be8 sb5 pd5 black ka8 pb7d7
Mixed #3            6 variations            4+3

Solution: (click to show/hide)

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Joost de Heer
Joost de Heer
October 9, 2023 09:49

So a mixed #n is basically ‘black and white toss a dice to see who does a move for the first n-1 moves, the nth is always played by white, in which he delivers mate’?

Georgy Evseev
Georgy Evseev
October 9, 2023 11:01
Reply to  Joost de Heer

No, I think black can either do any move or just pass (which is not count as a move) and the last is really always played by white. So, I think the solution should be written in the following way.
pass 1.Bxd7 pass 2.Bc6 3.Bxb7# (or) 2.b6 3.Bc6#
1.b6 2.d6 3.Bc6# (or) pass 2.Bxd7 3.Bc6# (do we have stalemate in this variation?)
1.d6 2.b6 3.Bc6# (or) pass 2.Bc6 3.Bxb7#

Joost de Heer
Joost de Heer
October 10, 2023 09:20
Reply to  Georgy Evseev

@Georgy ‘pass’ is also available for white, and ‘pass’ gives an air of voluntarity, so I think my dice analogy is better.

Full solution in ‘dice’ notation:
D1=B 1. d6

D2=B 2. b6 3. Bc6#

D2=W 2. Bc6 3. Bb7#

D1=B 1. b6

D2=B 2. d6 3. Bc6#

D2=W 2. Bxd7 3. Bc6#

D1=W 1. Bd7

D2=B 2. b6 3. Bc6#

D2=W 2. Bc6 3. Bb7#

Last edited 6 months ago by Joost de Heer
Georgy Evseev
Georgy Evseev
October 10, 2023 10:25
Reply to  Joost de Heer

This is also very good interpretation.
I also found another interesting possibility – to use word “threat” instead of “pass”.

shankar ram
shankar ram
October 9, 2023 18:07

FCCP contains the definition for “Help Free Play” (which was a direct translation of the French “Aide Libre”):
Help Free Play
A series of moves, of a length equal to the indicated number of moves, some played by White, some by Black, is made in order that after the last move, the aim is reached by White.

While extending the concept to HS# and Direct#, I have used the term “Mixed”.
What Joost and Georgy have stated is mostly equivalent.
In a mixed stipulation of N moves, the Nth move will always be by a particular side. The remaining N-1 moves can be played as any combination of White and Black moves or series. Any check being the last move of a series.
Series movers have also been defined as one side “passing”, so Georgy’s definition is valid.
Joost’s definition is related to combinatorics. The total number of combinations being 2^(N-1). Some of these combinations can be duplicated, like BWW and BBW here. And in some problems, some combinations may not be present, unlike here.
Another related mathematical concept is the Partition of integers: the different ways of writing an integer as the sum of non-negative integers. For N = 2, the combinations are 2 and 1+1. Adding colours gives 2(W), 2(B), 1(W)+1(B) and 1(B)+1(W).
Kabe Moen (a math professor!) commented: “…there might be intriguing mathematical/combinatorial aspects to explore in the general form., in his post(5) on the MatPlus forum about this type of stipulation.

Last edited 6 months ago by shankar ram
Joost de Heer
Joost de Heer
October 10, 2023 09:21
Reply to  shankar ram

@Shankar Ram
What happens if a side can’t move? E.g. wPd5 is on d6. Is there no mixed #3 or is the option ‘BB’ ignored as black can’t make a second move?

shankar ram
shankar ram
October 10, 2023 15:57
Reply to  Joost de Heer

Only the first 3 “variations” are possible:
Bxd7 Bc6 Bxb7#
Bxd7 b6 Bc6#
b6 Bxd7 Bx6#
It is not necessary that all the combinations be playable.

shankar ram
shankar ram
November 13, 2023 15:01

An extended version of 1765 is attached. A greeting problem for the Indian festival of lights (Diwali) yesterday! The emojis are courtesy Mu Tsun Tsai’s fen-tool.
Mixed #3: White and Black play a (adversarial) sequence of 3 moves in any order or combination (WWW, WBW, BBW, BWW) so that W mates on the 3rd move.
How many solutions?

Georgy Evseev
Georgy Evseev
November 14, 2023 10:52
Reply to  shankar ram

Is b6-d4-f2 (in any order) considered as solution?
There is the mate in the end…

shankar ram
shankar ram
November 15, 2023 07:58
Reply to  Georgy Evseev

but only two moves can be played as the third is required for the mate by white.

Last edited 4 months ago by shankar ram
Kenneth Solja
Kenneth Solja
November 14, 2023 16:57
Reply to  shankar ram

If I counted the solutions correctly the total is 12.
WWW = 1 solution
WBW = 2 solutions
BBW = 6 solutions
BWW = 3 solutions

shankar ram
shankar ram
November 15, 2023 08:02
Reply to  Kenneth Solja


Joost de Heer
Joost de Heer
November 24, 2023 15:40
Reply to  shankar ram

One solution with 12 variants.

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