Count distinct shapes built by adding congruent tiles

4.G.A.2 · take · grade 4

Archetype: Tile and Cut Figures with Congruent Pieces · step in a 5-type progression

Using $3$ identical rhombus tiles (a parallelogram whose four sides are all the same length), how many different shapes can you make? (You must join the tiles edge to edge, and two shapes that match after rotating or flipping count as the same shape.)

Show solution

Understand

I have 3 identical rhombus tiles. I join them edge to edge (whole side to whole side) into one connected shape, and I count how many different shapes are possible, treating two shapes as the same if one becomes the other by rotating or flipping.

Givens

There are 3 congruent rhombus tiles (a rhombus is a parallelogram with all four sides equal).
Tiles must be joined edge to edge, full side against full side.
Shapes that match after rotation or reflection count as one and the same shape.

Unknowns

The number of different shapes that can be built from the 3 rhombi.

Constraints

Start from the shape made by 2 rhombi joined edge to edge, then attach the third tile in every distinct edge position.
Remove duplicates that are just rotations or flips of a shape already counted.

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram#16 Count the Complement

This is a 'how many shapes' question with a tiny finite set, so the safe method is to list every way to add the third tile to the two-tile shape and then cross out repeats that are the same after turning or flipping.

Execute

#1 Draw a Diagram 4.G.A.2

Join two rhombi along one full edge. Because both tiles are identical rhombi, all the two-tile joinings give the same single starting shape (a longer rhombus / parallelogram strip). So there is exactly 1 shape made of 2 tiles.

2 \text{ tiles} \Rightarrow 1 \text{ shape}

Two equal rhombi can only line up one way up to turning, so I have a single shape to grow.

#2 Make a Systematic List 4.G.A.2

Add the third rhombus onto the open edges of the two-tile shape. Trying each open edge and removing the ones that are just rotations or flips of another, exactly three genuinely different shapes appear: a straight row of 3 rhombi, and two different 'bent' arrangements where the third tile turns a corner.

\text{straight} + \text{bend type 1} + \text{bend type 2} = 3

Listing every attachment spot and crossing out flips/turns leaves three truly different outlines.

#16 Count the Complement 4.G.A.2

Check each of the three shapes against the others by rotating and flipping: the straight strip cannot become either bent shape, and the two bent shapes are mirror-different in outline, so all three stay distinct. The count is 3.

\text{distinct shapes} = 3

Turning or flipping never merges these three outlines, so none is a repeat.

Answer: 3 shapes

Review

Three tiles can only make a short straight piece or a bent piece, so a small count like 3 is reasonable; it is far less than if we wrongly counted rotations and flips as separate.

Physically place 3 paper rhombi (tool 10) and slide the third tile to each open edge, sorting the results into piles that look the same after turning -- you end with 3 piles.

Standards · min grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Recognizing the rhombus tiles and judging when two built shapes are the same after rotation or reflection.

💡 Just list every way to add the last tile and toss out the flips and turns -- only 3 truly different shapes survive!