What Makes a Deduction Puzzle Hard? We Measured 4,500 of Them

The obvious way to make a deduction puzzle harder is to add suspects. More people, more possibilities, more work. It is what almost everyone assumes, including us when we started.

It is wrong, and we have the numbers.

How we measured it

Every Red-Handed case is graded by solve depth: how many rounds of reasoning it takes before the board is fully determined. One round means applying every rule you know to everything currently on the board. If new facts appear, that is another round. A depth-2 case falls out almost immediately. A depth-5 case has a deduction that only becomes available after four earlier ones have landed.

This matters because it is the honest measure. Counting how many possible worlds a puzzle has would score every well-formed puzzle a 1, including the hardest ones ever published, because a good puzzle has exactly one answer by definition. Chain length is what actually costs you time at the table.

We generated cases across four board shapes and recorded the depth of every one that passed validation. Roughly 4,500 boards.

The result

Board	Cases sampled	Reached depth 4
6 suspects, 4 rooms	2,794	22%
7 suspects, 4 rooms	1,717	21%
7 suspects, 5 rooms	58 per 40k tried	12%
8 suspects, 5 rooms	31 per 40k tried	10%

The share of deep cases falls as the cast grows. It is not noise, it is monotonic, and the effect is large: nearly a two-to-one difference between the smallest and largest boards.

Widening the cast also made valid cases much rarer. At eight suspects we had to try 40,000 candidate boards to find 31 usable ones, against 2,794 from the same effort at six.

Why bigger is easier

Once you see the mechanism it is obvious in hindsight.

Every statement in the puzzle rules something out. "I was at the bakery" narrows where one person can be. The value of that narrowing depends on how much room there was to begin with.

Add two more suspects and two more rooms, and each statement now eliminates a smaller fraction of the possibilities. The constraints get weaker relative to the space they are constraining. Fewer facts collide, so fewer new facts fall out of the collision, so propagation runs out of steam sooner. The board is bigger and shallower at the same time.

There is a second effect. Room capacity is one of the strongest tools in the puzzle: if a two-chair room already has two confirmed occupants, everybody else is out of it. More rooms means capacity bites less often, because people are spread thinner.

The result is a board that takes longer to write down and less thinking to finish. That is the worst trade in puzzle design. It costs the solver effort without giving them anything.

What we did about it

We reshaped the ladder around depth rather than headcount.

Easy is four suspects, three rooms, two objects. Depth 2 to 3. Six placements, which fits in your head without notes.

Medium adds a fifth suspect and a third object. Depth 3 to 4. The extra object is the point: objects chain into people, so one more of them lengthens the reasoning more than one more person would.

Hard brings a sixth suspect across four rooms. Almost every case lands at depth 4.

Expert keeps Hard's board exactly. Same six suspects, same four rooms. It adds a fourth object and demands depth 5, which is the ceiling for this clue vocabulary. Every single Expert case is a depth-5 case. It is not a bigger puzzle than Hard, it is a longer one.

That last decision is the one this whole investigation produced. The intuitive move was to give Expert seven suspects. The data said that would make it easier while looking harder.

The ceiling, and what to do about it

Depth 5 is where this puzzle stops. Across all 4,500 sampled boards, depth 6 appeared zero times.

That is structural rather than bad luck. Every clue in the game reduces to a claim about which room somebody or something was in, so once roughly half the board is pinned, the chair-capacity rules finish the rest in one sweep. Five rounds is about the diameter of the constraint graph. You cannot get a sixth by asking nicely.

So the top tier could not be "Expert, one round deeper". It had to change the question instead. On Sherlock, two accounts are false rather than one. One liar was alone at the scene and did it; the other lied for reasons of their own.

That does not lengthen the chain, and we should be straight about it: Sherlock cases run at depth 3 to 4, shallower than Expert. What grows is the space you have to eliminate. Six suspects gives thirty ordered pairs to rule out instead of six candidates, and the leverage of "everyone else is telling the truth" drops from five reliable accounts to four. It is harder in a different currency: breadth and bookkeeping rather than chain length.

Whether you find that harder than Expert probably depends on how you think. Some solvers find a long chain worse. Some find thirty possibilities worse. They are genuinely different kinds of difficulty and we stopped pretending one number covers both.

What this means for you

If a puzzle feels hard because there is a lot of it, that is a different feeling from a puzzle that is hard because you cannot find the next step. Learning to tell them apart is useful.

When you are stuck on a big board, the problem is usually bookkeeping. Write more down. When you are stuck on a small one, the problem is usually that you have not tried assuming something and following it to a contradiction.

Every case here is checked before it ships to confirm the answer is reachable by reasoning rather than merely unique, so being stuck always means there is a deduction still on the table. Details of the method are in the rules guide, and how to solve who-is-lying puzzles works one case end to end.