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Three Cuts

The game Three Cuts was created in the year 2010. It was adjusted and became simpler in this new edition (2014).

A-chessboard and A-square
The game is played on four A-chessboards. A-chessboard is a chessboard divided into 16 squares that are called
A-squares (see picture is below).

Playing Stones

Three Cuts is played with 64 playing stones. Each of them has the same shape as A-square but in each of four fields is imprinted a number. A sum of these four numbers is zero.

Here you can download 2.1 Playing Stones.pdf

The goal of the game
The goal of the game is to place all playing stones on four A-chessboards while obeying the Three Rules.

The Three Rules
1. On each of sixty-four A-squares on four A-chessboards must be placed exactly one playing stone in such way
    in order white fields of playing stones lie on white fields of A-squares as well as dark fields of playing stones lie
    on dark fields of A-squares.
2. Playing stones must be put on A-chessboard in such way in order a sum of four integers that lie on the white
    fields in any row or any column was equal to zero (see the Three Rules Schema below).
3. Playing stones must be put on A-chessboard in such way in order a sum of four integers that lie on the dark fields
    in any row or any column was equal to zero (see the Three Rules Schema below).

Solutions and Sets of Similar Solutions
The game finished when all playing stones are placed on the board while obeying the Three Rules (a solutions of the game).

A set of similar solutions is a group of such solutions that can be reciprocally derived. For example by a replacement of rows of playing stones on one or more A-chessboards, by a replacement of columns of playing stones on one or more A-chessboards or by replacement rows by columns and columns by rows.
The game provides 3 sets of similar solutions called H-set, M-set and D-set.

Tip!
1. Everybody who knows one solution of Three Cuts can easily find remaining ones.
2. Change task complexity! Transform dark fields on all 64 playing stones in colour fields.

Four colour theorem
Let us leave aside debates if the four colour theorem is a task for computers or it is a simple logical problem. Instead that let us focus on four colours as a really powerful mathematical tool.

The Magical Set
Imagine you got a task to dye all dark-fields on all 64 playing stones using four colours. Each of playing stones must be dyed only by one colour. Each of four colours must be used for dyeing exactly 16 playing stones.

Imagine that you had a lucky hand and created the Magical Set!
What is it the Magical Set? Here is a simple explanation.
Anytime Three Cuts is played with the Magical Set and the game is successfully finished everybody can see still same scenery...each of four colours appears in any rows and columns on each of 4 A-chessboards just one times!

Be cool and create yourself the Magical Set!
2.2 Playing Stones.docx
Here you can check your work.
2.3 The Magical Set.pdf

Solutions
Below you will find three solutions. The first is a part of the D-set, the second of the M-set and the third of the H-set.
2.4 Solution H set.pdf
2.5 Solution M set.pdf
2.6 Solution D set.pdf