*Original Fairy problems**JF-2023:01.01.2023 - 31.12.2023*

☝ Definitions

No. 1765 **Shankar Ram**

India

original - 08.10.2023

white kc8 be8 sb5 pd5
black ka8 pb7d7

Mixed #3 6 variations 4+3

Solution: (click to show/hide)

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’?

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#

@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#

@Joost

This is also very good interpretation.

I also found another interesting possibility – to use word “threat” instead of “pass”.

FCCP contains the definition for “Help Free Play” (which was a direct translation of the French “Aide Libre”):

Help Free PlayA 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

Partitionof 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.@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?

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.

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’sfen-tool.: 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.Mixed #3How many solutions?

Is b6-d4-f2 (in any order) considered as solution?

There is the mate in the end…

yes!

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

If I counted the solutions correctly the total is 12.

WWW = 1 solution

WBW = 2 solutions

BBW = 6 solutions

BWW = 3 solutions

correct!

One solution with 12 variants.