Odd One Out Puzzle
Spot the odd number out, with every puzzle checked against the whole property list — because the naive way to build one is ambiguous 98% of the time.
One of the four is the odd one out — and it is the only one that can be argued for, because every puzzle here is checked against the whole property list below before it is shown. Nothing is uploaded.
The properties this puzzle is allowed to use
"Odd one out" only means anything relative to a fixed list. Here is the whole of it — a number is accused either because it is the only one that has a property, or the only one that does not:
| 1 | even |
| 2 | odd |
| 3 | prime |
| 4 | a perfect square |
| 5 | a perfect cube |
| 6 | triangular |
| 7 | a power of two |
| 8 | a multiple of 3 |
| 9 | a multiple of 5 |
| 10 | a multiple of 7 |
| 11 | a palindrome |
| 12 | two digits long |
| 13 | a number whose digits add to a multiple of 3 |
| 14 | a number whose digits add to an even number |
| 15 | a number containing the digit 1 |
| 16 | under 50 |
Every one of them splits the numbers 1 to 99 into two non-empty groups, with at least three on the majority side — a property that held for everything could never accuse anybody.
Why nearly every odd-one-out puzzle has two answers
The obvious way to build one is to pick a property, take three numbers that have it and one that does not, and stop. Checked against the full list, that gives a puzzle with a single defensible answer 2.0% of the time. Over 80,000 candidates:
| Properties on the list | Puzzles with exactly one answer | Numbers accusable, on average |
|---|---|---|
| 4 | 69.7% | 1.33 of 4 |
| 8 | 24.1% | 2.13 of 4 |
| 12 | 8.9% | 2.68 of 4 |
| 16 | 2.0% | 3.12 of 4 |
Read the last column. With sixteen properties and four numbers, the average puzzle offers grounds for accusing 3.12 of them — so typically only one of the four cannot be argued for. The question is not slightly under-determined. It is nearly fully determined in the wrong direction.
And the failure is never that no answer exists. Not one candidate in 80,000 had nothing to say about any of its numbers. Something can always be said; the scarce thing is a set about which only one thing can be said.
So the puzzles above are checked rather than constructed: a candidate is built, every property is tested, and the set is thrown away unless they all accuse the same number. That discards roughly forty-nine candidates in fifty. The table is frozen, so fresh candidates it has never seen are measured at build time as a check — 71.1% well posed on four properties against 2.6% on all 16.
How to use
- Look at the four numbers and pick the one that does not belong.
- The full list of properties the puzzle may use is published below.
- Press Why to see every property that singles a number out.
- Each puzzle is checked so only one number can be argued for.
- Press New puzzle for another.
Frequently asked questions
What makes an odd-one-out puzzle fair?
A published list of properties. The question only means anything relative to a fixed list — without one, the answer is a guess at what the setter had in mind. The whole list this puzzle may use is shown on the page.
Why do so many of these puzzles have two answers?
Because the obvious way to build one is to pick a property, take three numbers that have it and one that does not, and stop. Checked against a list of sixteen properties, that gives a puzzle with a single defensible answer only 2.0 per cent of the time.
How ambiguous does it get?
With sixteen properties and four numbers, the average puzzle offers grounds for accusing 3.12 of them. Typically only one of the four cannot be argued for, so the question is not slightly under-determined — it is nearly fully determined in the wrong direction.
Does a longer property list make it worse?
Steadily. Four properties give a well-posed puzzle 69.7 per cent of the time, eight give 24.1 per cent, twelve give 8.9 per cent and sixteen give 2.0 per cent. Every property you are willing to accept as an answer is another chance for a second one.
Can a set have no answer at all?
It never happened. Across 80,000 candidates, not one had nothing to say about any of its numbers. Something can always be said; what is scarce is a set about which only one thing can be said.
How are these puzzles generated?
By checking rather than constructing. A candidate is built, every property on the list is tested, and the set is thrown away unless they all accuse the same number — which discards roughly forty-nine candidates in fifty.
Does this send anything anywhere?
No. Every puzzle is generated and checked in your browser, and nothing is uploaded.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.