How to Solve the Three-Piece Block
How to Solve the Three-Piece Block
Three pieces. You can hold all of them in one hand. You know exactly what you are building, because the finished shape is on the box and it is not complicated - a small pyramid of cubes.
Most people still lose an hour to it.
That gap, between a target you can picture perfectly and a solution you cannot find, is the whole design. It is a Stewart Coffin puzzle, and getting maximum difficulty out of the minimum of parts was the thing he did better than anyone.
Watch: Ten Cubes, Three Pieces
The video is the fastest way to see the problem: assembled it is clean and obvious, and the moment it comes apart the pieces stop making any sense at all.
What the Three-Piece Block Actually Is
Ten cubes, divided into three pieces. One piece has four cubes, the other two have three each.
The target is a tetrahedron - a triangular pyramid, not the square-sided kind you picture when someone says the word. It stacks in three layers:
- Six cubes on the bottom, arranged in a triangle - a row of three, a row of two behind it, then one
- Three cubes in the next layer, each sitting in a hollow between the cubes below
- One cube on top
Six, three, one. Ten cubes, and every one of them accounted for.
Why the Obvious Method Fails
Here is the thing that catches everybody, and it happens before you have consciously decided anything.
You assume a square grid. Every cube puzzle you have ever met - every polycube, every Soma-style block, every Tetris shape you have rotated in your head - joins cubes face to face on a rectangular lattice. It is such a strong habit that you do not notice you are applying it.
This puzzle does not live on that grid. When a cube sits in the hollow between the cubes underneath it, it is not resting on a face. It is bridging a gap, meeting its neighbours edge to edge, offset by half a cube in two directions at once. The pieces are built from cubes fused at exactly those offsets, which is why they look wrong in your hand. They are not wrong. They are shaped for a stack you are not picturing.
So the pieces refuse to combine into anything tidy, you conclude you have the orientation wrong, and you start turning them faster. That is the hour going.
How to Solve It: Start With the Final Shape
Every hint worth having here is a hint about the shape you are building, not about the pieces in your hand. Get the target right and the pieces stop being mysterious.
1. Learn the shape before you touch anything. Six, three, one. If you cannot describe the finished pyramid layer by layer with your eyes shut, you are not solving it yet - you are trying combinations and hoping.
2. The base is a triangle, not a square. Six cubes on the bottom, but arranged three, then two behind them, then one. It is easy to build a squarish block of six without noticing, and once the base is wrong nothing above it will ever seat.
3. Nothing above the base rests on a face. Every cube in the upper layers drops into the hollow between the cubes below it, so it meets its neighbours edge to edge, sitting half a cube across in two directions at once. This is the fact the whole puzzle turns on, and it is why the angles in the pieces look like mistakes until you see it.
4. Do the arithmetic on the layers. The layers hold six, three and one. The pieces hold four, three and three. Notice that the top two layers hold exactly four cubes between them - the same count as your largest piece. Working out whether that piece fills them or not is the most productive question available, because either answer forces most of what follows.
None of that hands you the solution. It gets you asking the right question, which is the thing this puzzle is actually testing.
Tips for Getting Unstuck
Put it down for ten minutes. The square-grid assumption is sticky, and the reliable way to shake it is to stop looking. Greg has solved this one more times than he can count and still says he struggles to find it first time.
Assemble two pieces and leave them. If two pieces make a stable partial stack that matches part of the six-three-one shape, that pairing is probably right. Test the third against it instead of taking everything apart again.
Check your triangle. A bottom layer of six in a triangle is easy to mis-build as a square-ish blob of six. If the base is wrong, nothing above it will ever seat.
Go and solve something else for a few minutes. The 2 Piece Pyramid is a very different puzzle - two identical pieces, low difficulty, and all of it orientation rather than working out which piece goes where. That is exactly what makes it a good break. It keeps your hands on a pyramid without letting you keep chewing on this one.
Frequently Asked Questions
How many pieces does the Three-Piece Block have?
Three pieces, made of ten cubes in total - one piece of four cubes and two pieces of three. The low piece count is the point, not a shortcut: Coffin built his reputation on puzzles where there is nothing to grind through and no bulk to work around.
What shape do the pieces make?
A tetrahedron - a triangular pyramid. Six cubes form a triangle on the bottom, three sit in the hollows above them, and one goes on top. It is not a square-based pyramid, which is the shape most people picture and one of the reasons the puzzle resists them.
Is there a two-piece pyramid puzzle as well?
Yes, though it is a very different puzzle rather than an easier edition of this one. The 2 Piece Pyramid uses two identical pieces, so there is no piece to identify and no order to work out - all of the difficulty is in orientation, and it is rated low. Worth owning alongside this one rather than instead of it.
Who designed the Three-Piece Block?
Stewart Coffin, the most influential designer in mechanical puzzling. He was an engineer who spent five years at MIT's Lincoln Laboratory before turning to puzzles, and he wrote the two books the field still works from. There is more on him and the rest of his designs in our profile of his work.
Why is a three-piece puzzle so hard?
Because the difficulty is irreducible. There is no long move sequence to discover and no structure to read from the outside - the only thing standing between you and the solution is understanding the geometry, and you either have it or you do not. Puzzles with more pieces often give you more to work with, not less.
How long does it take to solve?
Thirty to sixty minutes for most people on the first attempt, and it is rated Medium fairly generously. Once the stacking geometry clicks it becomes quick to rebuild, which is what makes it a good desk puzzle rather than a one-time challenge.
What should I solve after the Three-Piece Block?
The Multi-Grain Burr, which takes this exact idea to the next level. Another Coffin design and another pure geometry problem - four solid pieces that have to occupy one space - and despite the name there is no burr in it, no notched sticks and nothing that interlocks. If the Three-Piece Block clicked for you, that is the next step.
Is it a good puzzle for beginners?
It depends what you want out of it. It is a fair puzzle - no trick, no hidden mechanism, nothing you could not reason out - so nobody gets stuck through bad luck. But it will not fall apart in your hands and hand you a clue, so a complete beginner who wants an early win is better off starting with the 6-Piece Star and coming back to this one.
Want More Puzzles Like This?
The Three-Piece Block sits in the Geometry Puzzles collection, alongside the other designs where the whole challenge is spatial rather than mechanical.
If this is the kind of thing you want more of, the Stewart Coffin collection runs from the 6-Piece Star at the gentle end through to Pieces Of 8, which is eight pieces into a tray and the hardest of the group. Our profile of Coffin explains the three rules he designed by and why so few pieces beat so many people.
For the wider range, everything lives in the mechanical puzzles collection.