Rep-Tile Puzzles: How to Cut a Shape Into Copies of Itself

A rep-tile puzzle asks you to cut a shape into a few smaller copies of itself, all the same size and all exactly the same shape as the original. To solve one, count how many copies the area allows, then fill the tightest corners first, because each corner of the big shape has to be the corner of one copy.

This guide explains which shapes can do this, gives a six-step method, works through one puzzle cell by cell and ends with a practice puzzle and its answer.

What is a rep-tile puzzle?

A rep-tile is a shape that can be split into smaller copies of itself with no gaps and no overlaps. The name is short for replicating tile. Every copy must be the same size, and each one is a scaled-down twin of the whole.

A paper published on 7 February 2020 describes a regular rep-tile as a shape whose sides and angles are all equal and which splits into ever smaller copies of itself, step after step without end ^[1]. The square and the equilateral triangle are the regular rep-tiles in the plane, and the cube is the only one in three dimensions ^[1].

Many rep-tiles are not regular at all. An L-shape made of three squares, often called the L-tromino or the chair, is a rep-tile. So are some right triangles and some shapes made of four or five squares.

In a rep-tile puzzle you are shown the big shape drawn on a grid and told what each piece must look like. Your task is to mark where every piece goes.

Which shapes are rep-tiles?

The table compares the shapes most often used in these puzzles. The middle column gives the smallest number of copies that works.

Every cut in the last five rows was checked by Riddle10 with a short program. It confirmed that the pieces are true copies, stay inside the shape and cover it exactly once, and that no row can manage with fewer copies.

The paper on regular rep-tiles also notes that a square or triangle with side length n splits into n times n smaller copies ^[1]. So a square of side 2 holds 4 small squares, and so does a triangle of side 2 ^[1].

How many pieces does a rep-tile cut need?

Area tells you almost everything. If a shape is cut into some number of equal copies, each copy has that fraction of the area. On a grid this becomes a simple count.

When a puzzle shows a shape scaled up to double size, every square of the original becomes a block of two by two squares. The big shape therefore has four times as many cells, and it needs exactly four pieces. At triple size it needs nine.

Take the L-tromino at double size as our first example. It covers twelve cells, so it needs four L-trominoes of three cells each. Our search found exactly one way to do it.

Piece B sits in the middle and turns the other way, while A, C and D sit in the three corners.

At triple size the same shape covers 27 cells and takes nine L-trominoes, and our search found exactly four different ways to cut it, counting mirror images separately. Two of those cuts are their own mirror image and the other two mirror each other, so there are three if a cut and its mirror image count as one.

What is the step-by-step method?

1. Count the cells. Divide the cells of the big shape by the cells of one piece. If the answer is not the number of copies you expect, recheck the drawing.

1. List the allowed positions. Sketch every way the piece can be turned. Check the rules to see whether pieces may also be flipped over like a page.

1. Start in the tightest corner. Pick the corner cell with the fewest open neighbours. List every placement of the piece that covers that cell.

1. Throw out placements that leave a bad pocket. A pocket is a group of empty cells walled in on every side. If its size is not a whole number of pieces, the placement is wrong.

1. Repeat from the next empty cell. Work in reading order, top row first and left to right. A forced placement often forces the next one.

1. Check the finished cut. Each piece must match the original shape, and every cell must belong to exactly one piece.

When a step leaves two choices and neither can be ruled out, pick one and carry on. If nothing fits, go back to that choice and take the other one.

How does the method cut a double-size L-tetromino?

Here is the worked example. The L-tetromino is an upright bar of three squares with one extra square at the bottom right, like a capital L. At double size it becomes the 16-cell shape below, where a filled square marks a cell of the shape.

The rule for this example: cut it into four L-tetrominoes. Pieces may be turned but not flipped over, so a backwards L, shaped like a J, is not allowed.

Step 1: count. The example has 16 cells and each piece has 4, so we need four pieces, named A to D.

Step 2: list the positions. Turning the L gives four positions. Two are tall: the normal L, and the same L turned upside down, with its extra square at the top left. Two lie flat: a row of three with a square hanging below its left end, and a row of three with a square sitting above its right end.

Step 3: the top left corner. The upper part of the shape is only two cells wide, so no flat position fits there. Only two placements cover the cell in row 1, column 1.

The normal L would take rows 1 to 3 of column 1 plus row 3 of column 2. That walls in rows 1 and 2 of column 2, a pocket of two cells, which no piece can fill. So piece A is the upside-down L: row 1 of columns 1 and 2, then rows 2 and 3 of column 2.

Step 4: the next empty cell. In reading order this is row 2, column 1. Above it is piece A and to its right is piece A. The only placement left is the normal L running down column 1 from row 2 to row 4, with its extra square in row 4, column 2. That is piece B, and the narrow top part is now full.

Step 5: the bottom block. What remains is a block two rows high and four cells wide. Its top left cell, in row 5, column 1, can only be covered by the flat position with a square hanging below the left end. That is piece C: all of row 5 except column 4, plus row 6, column 1.

Step 6: the last piece. The four cells left over are row 5, column 4 and row 6, columns 2 to 4. They form a flat row of three with a square above its right end, which is an allowed position. That is piece D, and the cut is finished.

Our exhaustive search confirms that this is the only cut into four turned copies. If flipping is allowed as well, the same search finds four cuts.

Why does flipping over matter?

Some pieces look the same in a mirror and some do not. The L-tromino looks the same when flipped, so the rule makes no difference to it. The L-tetromino and the P-pentomino change into their mirror images, and that changes the puzzle.

For the double-size L-tetromino, banning flips cut the number of answers in our search from four to one. For the double-size P-pentomino, banning flips leaves no answer at all. Always read the rule first, since it decides which positions go on your list in step 2.

How do rep-tiles link to dissection puzzles?

In a dissection puzzle the pieces of one outline are reassembled into a different outline. A rep-tile cut is a special case, in our own framing: the target is not a new outline but a set of smaller copies of the starting shape.

A paper published on 27 January 2023 recalls a theorem proved in the 19th century: any polygon can be cut into finitely many straight-edged pieces which, by sliding and turning, can be rebuilt into any other polygon with the same area ^[2]. Its proof shows that the number of pieces is finite, but not how small it can be, and using the fewest pieces became the main goal of dissection puzzles ^[2].

Rep-tile puzzles share the same idea of economy. The copies count is fixed by area, so the challenge is not how few pieces you need but where each one must sit.

A further link runs through fractals. The 2020 paper describes endless sets of tiles with infinitely detailed edges, in which identical copies fill a square or an equilateral triangle ^[1]. In its standard kind, three tiles fill the triangle and four fill the square, one piece for each side of the shape ^[1].

How does a computer check a rep-tile cut?

The checks in this guide used plain backtracking, the same idea as steps 3 to 5 done at speed. The program always looks at the first empty cell in reading order and tries every allowed position of the piece that covers it.

If a position fits, the program fills it and moves on. If nothing fits, it steps back and tries the next option. Because every branch is tried, the search gives the full number of answers, not just one.

What are the most common mistakes?

• Using the wrong number of pieces. At double size the answer is always four pieces, at triple size nine.

• Forgetting a position. Draw every turned position before you start, and the flipped ones too if they are allowed.

• Ignoring pockets. A walled-in group of empty cells that cannot hold a whole number of pieces means an earlier choice is wrong.

• Starting in the open middle. The middle has many choices. A corner usually has only one or two.

• Not checking the finished pieces. Compare each piece to the original shape before calling the cut done.

Can you solve this practice puzzle?

The P-pentomino is a two by two block with one extra square below its left side. Here it is at double size, 20 cells in all. Cut it into four P-pentominoes. This time pieces may be turned and also flipped over.

A hint: start at the bottom. The two by two foot in rows 5 and 6 is walled in on three sides, so the piece that covers it must also take one more cell above it.

Answer: the search found exactly two cuts. In both, piece D is the foot plus row 4, column 1, which is a flipped P. The upper square then splits in one of two ways.

The first cut:

The second cut:

Check any piece: it should be a two by two block with one extra square on a side. The search also confirms that without flipping, this shape has no cut at all.

Want another fitting challenge? This riddle shows a gap and several shaped pieces, and only one of them fills the hole when turned.