## Board game geometry

Posted by Aapo Rantalainen on August 2, 2015

There are several board games played with square grid and TOKEN CAN MOVE DIAGONALLY. I’m not saying this is wrong or even hard to play with.

I’m describing what this mean mathematically.

Look the picture. Green token can move four steps. We can now say that each possible target point (B and C) and source point (A) has distance four.

DistanceBetween(A,B) = 4

DistanceBetween(A,C) = 4

This is true in this game-universe and with these game-rules. If we then use our physical reality and measure distances with ruler.

We get that:

DistanceBetween(A,B) = 16 centimetres

DistanceBetween(A,C) = 22 centimetres

I repeat this is not a bug in rules. It is not even paradox. But how does it look in our reality if we somehow set that “nearby points have distance of one”?

First we (mathematically) define our game board. Let say we have three types of points:

– Corner: It has three neighbourhood points (E.g. point A)

– Side: Five neighbourhood points (E.g. point B)

– Center: Eight neighbourhood points (E.g. point D)

(Where neighbourhood point = point with distance 1)

And then we have some little more complicated relation between points:

Distance(A,B) = 1

Distance(A,C) = 1

Distance(A,D) = 1

Distance(B,C) = 1

Distance(B,D) = 1

Distance(C,D) = 1

…For each point in board.

Let’s construct this:

* Starting with one corner (A) (in 0,0,0)

* Helper points X Y Z

*Lines from A to each helper point X and Y and Z. *Ball with center A and radius 1. It intersects three helper lines on points B and C and D

(Note: it is not circle but ball, therefor it looks points B and C and D are not intersecting ball)

*Lines between BC, BD and CD (length of each is 1)

*This construction is now equivalent what we wanted.

And here is a physical construction of 20 points (marked by four tokens) from corner of the board.

Because they are equivalent I suggest playing with right-side board.

## ripa said

Mistä tämä lähti liikkeelle :)?

Mutta eikös muuntamalla ruutulaudan laudaksi jonka solut ovat ympyröitä saada heti helpotettua ajattelumallia kun etäisyydet vaakaan, pystyyn ja vinoon ovat samat?