Build tetrads on the square grid and on the 60-degree grid.
(27 variants on square grid, 4 variants on 60-degree grid)

A 'tetrad' is an assembly of four 2-dimensional shapes which all touch each other.
A 'congruent tetrad' is a tetrad made from four identical ('congruent') shapes.

The board shows one piece of such a congruent tetrad, made from L-shaped polysquares. Click the 'change piece set' button in the tool bar to make the squares more visible.
Your task is to add three other pieces of same shape (which can be rotated or mirrored) such that each piece touches each other piece.
Select Help/ShowSolution to see the solution to this puzzle.

The other variants always show one of the four pieces of a tetrad. Your task is to complete the tetrad.
Before you have a look at the other variants, you might want to invent your own. For this purpose a 'freeplay' variant has been added.

We can ask several questions relating tetrads:

• find more congruent tetrads.
• find a congruent tetrad without a gap or hole. (A mathematician would say that such a tetrad is simply connected).
• find a congruent tetrad made from 'reptiles' (i.e., copies of this shape tiles a larger version of itself)
• find a congruent tetrad made from polytans. Polytans are shapes made from half-squares, where the square is cut along the diagonal.
• find a congruent tetrad made from shapes on the isometric grid (= 60-degree grid)
  A solution can be found in the attached 'Isolattice Light' game.
• find a 'similar' tetrad.
  A similar tetrad is a tetrad made from four similar shapes (i.e., same shape, but differently sized)
• find a similar tetrad without a gap or hole.

The variants show examples for each of these tasks.
In some variants (e.g., variant 2-10) you have to switch to the LINE DRAWING piece set (by clicking the 'change piece set' button in the tool bar) to see them properly.

Not all solutions can be drawn on the square grid. This is why this game also contains a special copy of the game ISOLATTICE LIGHT. There you can draw on a 60-degree grid, and where you can find some more examples of tetrads on this grid.

There are several alternative piece-set/board combinations available. Note that two special piece set has been included:
- one for black-and-white LINE DRAWINGS which allows to draw 45-degree angles
- one for curved line drawings. The curved piece set is only available in selected variants.

 
Newest findings:
In June 2005 found a series of new tetrads made from curved tiles, some with only a single vertex! Up to this time it was unknown have many vertices a tile must have to form a tetrad. These brand new mathematical discoveries are included in version 3.0 of 'Tetrads'.

Older research:
It has been proven by the author that no tetrad can be constructed from four congruent convex shapes. (See solution to problem 684, page 317 in the Journal of Recreational Mathematics, 1979). It seems likely that a simply-connected (i.e., gap-free) congruent tetrad cannot be convex, but it has never been proven. Similarly, a proof is still outstanding for the notion of similar convex tetrads not being convex.

Tetrads have been investigated by the author and several others. Apart from the problem cited above, two articles on this subject have appeared in the Journal of Recreational Mathematics; see Volume 10, issue 3 (?), page 297.