Dissection Puzzles: Cutting a Triangle Into a Square

A dissection puzzle asks you to cut one shape into pieces and rearrange them, with no overlaps and no gaps, into a different shape of the same area. The best-known case turns an equilateral triangle into a square using four pieces, and a paper published on 27 January 2023 proved that three convex pieces can never manage it ^[1].

This article explains what is known about dissections, why the four-piece answer is so hard to beat, what the corners of each shape tell you, which questions are still open, and how two related kinds of assembly puzzle fit in.

What is a dissection puzzle?

A dissection puzzle gives you two shapes. You cut the first into pieces, then move and turn the pieces until they form the second. Nothing is added and nothing is thrown away, so the two shapes must have exactly the same area.

The simplest example needs one straight cut. Slice a small right triangle off one end of a slanted parallelogram, slide it across to the other end, and the parallelogram becomes a rectangle. Each piece keeps its size and shape on the way, which is the whole point: a dissection only moves material, it never stretches it.

The real challenge is economy. Finding some way to cut one shape into another is often easy. Finding a way with the fewest possible pieces is hard, and a paper on economical dissections studies exactly that question for a triangle and a square ^[1].

Can any shape be cut into any other?

For flat shapes with straight sides, yes, as long as the areas match. According to the Wallace-Bolyai-Gerwien theorem, a result from the 19th century, two polygons with equal area can always be sliced so that a finite set of pieces from one rebuilds the other ^[1].

That is a remarkable promise. It means a square can become a triangle, a long thin strip, or any outline with straight edges and the same area.

But the theorem only guarantees that the number of pieces is finite ^[1]. It says nothing about keeping that number small, and that is where dissection puzzles begin.

How does Dudeney's haberdasher dissection work?

The most famous economical dissection is Henry Dudeney's answer to his Haberdasher Puzzle: just four pieces of an equilateral triangle rebuild a square with the same area ^[1].

Four pieces is a striking result, because the two shapes look so different. The triangle has three sharp corners, and the square has four right angles. Yet a handful of carefully placed cuts turns one into the other, and the pieces fit back together with nothing left over.

A natural question follows. Could three pieces be enough? The 2023 paper took that question on directly ^[1].

Why can't three convex pieces do it?

A convex piece has no dents: a straight line between any two of its points stays inside the piece. Triangles, squares and most simple cut-outs are convex.

The paper showed that when every piece has to be convex, three pieces can never turn the triangle into the square of the same area ^[1]. Dudeney's four pieces are therefore the fewest possible under that condition ^[1].

The argument starts from a neat observation about distances. Work out the sizes and the square's diagonal turns out to be shorter than the triangle's side, a fact the paper relies on ^[1].

Now picture two corners of the triangle sitting in the same piece. A piece is rigid, so those two corners stay a full side length apart wherever the piece moves. But nothing inside the square is farther apart than its diagonal, so that piece could never fit into the square.

So no piece can hold two corners of the triangle. With three corners and only three pieces, each piece must hold exactly one corner, and the paper proves this as a lemma ^[1]. From there, the authors show that convex pieces cannot finish the job ^[1].

The lesson for anyone who designs or solves dissections is a useful one. Before you search for a clever cut, compare the longest distance in each shape. It can rule out whole groups of answers in a few lines.

What do the corner angles tell you?

Angles are the second clue. Each corner of an equilateral triangle is sixty degrees, while each corner of a square is a right angle. Moving a piece never changes its angles, so every sharp triangle corner has to land somewhere in the square where the angles around it add up properly.

There are only three kinds of place it can go. At a corner of the square, the angles meeting there must make a right angle. On a straight edge of the square, they must make a straight line. At a point inside the square, they must make a full turn.

A sixty-degree corner cannot fill a right angle on its own, so it always needs a partner piece beside it. That is one reason a triangle-to-square dissection needs pieces that fit together closely, and why each extra cut has to earn its place.

The same bookkeeping works in reverse. Every right angle of the square has to be built from angles that the pieces carried over from the triangle, including any new corners the cuts created.

What is still unknown about dissections?

Two questions about this famous pair of shapes remain open, according to the paper's concluding remarks ^[1]:

• Could three pieces work if some of them are not convex? A piece with a dent or a hook might get around the argument above, and nobody has settled whether such a three-piece dissection exists ^[1].
• Is Dudeney's four-piece dissection the only one? It is unknown whether some other set of four pieces could also turn the triangle into the square ^[1].

It is a little surprising that a shape as simple as a triangle still holds open questions. It shows how quickly fitting problems become hard once you ask for the fewest pieces rather than any pieces at all.

How can you try a dissection yourself?

Squared paper and scissors are enough. Draw both shapes at the same area, then look for a piece of one shape that could slide or turn into a gap of the other.

Start with pairs of shapes that already share a lot. A slanted parallelogram and a rectangle with the same base and height differ only by a triangle at one end, which is why a single cut is enough.

A right isosceles triangle and a rectangle make a good next step. Cut the triangle straight across, halfway up and parallel to its base. The cut meets the long slanted side exactly at its midpoint. Give the small top piece a half turn about that midpoint, and it drops into the corner beside the lower piece. The two pieces now form a rectangle as wide as the triangle's base and half as tall.

The angle rules from the previous section hold here too. The sharp tip of the top piece lands in a corner of the rectangle beside a matching sharp corner of the lower piece, and together they make a right angle. The other sharp corner of the top piece meets the wide corner of the lower piece on the top edge, and the two make a straight line.

Each time, check three things: every piece keeps its size and shape, nothing overlaps, and the outline closes with no gaps. Then try to beat your own piece count. That last step is the same question researchers ask when they hunt for economical dissections ^[1].

Harder pairs, such as a triangle and a square, rarely give way to a first guess. Sketch where each corner of one shape could end up in the other, and use distances and angles to rule out the impossible placements before you cut anything.

Which other puzzles are close relatives?

Packing puzzles turn the idea around: the pieces are fixed and only their placement is free, and a perfect area match can still fail, as with the squares of side 1 to 24, which cannot exactly fill a square of side 70 ^[2]. Colour-cube puzzles add a colour rule, asking for eight painted cubes that rebuild a ninth at double size with matching colours wherever two faces touch ^[3].

How do these puzzle types compare?

What can solvers take from the research?

Four habits carry across all of these puzzles.

• Check area or size first. A dissection needs equal areas, and a packing puzzle needs the pieces to fit the space. A failed check saves you a long search.
• Compare the longest distances. The diagonal argument behind the three-piece result shows how one measurement can rule out many answers ^[1].
• Track the corners. The three-piece argument begins by asking which piece can hold each corner of the triangle ^[1].
• Settle the edges before the middle. The packing work begins by asking which pieces could ever sit along the border ^[2].

Now count the pieces in a figure that has already been cut up. How many triangles can you find in this one?