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dedicated to my parents,

Raymond and Marguerite

Introduction

The RayMag pieces are polymultiforms obtained by juxtaposing several copies of the 2 triangles below :

To understand how RayMag pieces are made, let’s start by taking an equilateral triangle of side 3 :

In the sequel, this triangle will be called « basic triangle ».

We can divide this triangle into 9 small equilateral triangles of side 1 :

If we take as unit of area the area of one of these small triangles, the area of the basic triangle is therefore 9.

Below, here is another cut of the basic triangle into 9 triangles of area 1 (equilateral triangles or not) :

A particular grouping of these triangles by 2 or by 3 gives this cut :

There are three triangles of area 2 (Ray) and one triangle of area 3 (Mag).

To be able to tile the plane, it’s necessary to add the symmetry of this triangle with respect to its base :

This gives :

or

Hence the following grid with a reduced scale :

For RayMag pieces, the juxtapositions must respect this grid ; therefore there cannot be :

Here are the RayMag pieces in the grid :

There are one piece of area 2 (Ray), one piece of area 3 (Mag), two pieces of area 4, one piece of area 5, one of area 6, three of area 7 and eight of area 9.

The nine pieces whose area is between 2 and 7 will be called the « small pieces ».

The eight pieces of area 9 will be called the « large pieces ».

There are area 8 pieces (see below in red) but I don’t use them for RayMag.

Here are the small pieces with their areas :

The small pieces cover an area of 45, which corresponds to 5 basic triangles : we can therefore try to reconstitute, one by one, the 4 pentiamonds using these 9 pieces.

Here are the large pieces, all of area 9 :

Note : one of the pieces, RayMag 9a, is a basic triangle.

The large pieces cover an area of 72, which corresponds to 8 basic triangles.

All the pieces of RayMag cover an area of 117, so 13 basic triangles.

Here is the hexagon of the tray :

And here it is in the tray :

With RayMag’s pieces, I offer you three kinds of puzzles :

1) The first, respecting the grid, consists in reconstituting different polyiamonds :

the monoiamond,
the diamond,
the triamond,
the 3 tetriamonds,
the 4 pentiamonds,
the 12 hexiamonds,
the 24 heptiamonds,
the 66 octiamonds,
the 160 enneiamonds,
the 448 decaiamonds,
the 1 186 hendecaiamonds,
the 3 334 dodecaiamonds,
the 9 235 tridecaiamonds,
altogether 14 475 polyiamonds.

2) The second consists in reconstituting various shapes while respecting the grid.

3) The third offers various shapes without necessarily respecting the grid.

Polyiamonds              Shapes on grid              Off-grid shapes