Ordering Puzzle & Score Comparison
Put a list in order, scored three ways at once — because counting correct positions gives zero to an answer that is only offset by one.
Drag with the arrows to reorder, then Check. The correct order is computed from the numbers rather than looked up, so every answer here is checkable. All three scores are shown at once, which is the point of the page. Nothing is uploaded.
The obvious score gives zero to an almost perfect answer
Shift a correct ordering along by one place. Not a single item is in its correct slot, so a score based on positions gives zero. But every pair is still in the right relative order apart from the one that wrapped round — so by pairs it scores 1 − 2/n, which at 8 items is 0.750.
| 8 items, the answer is… | All or nothing | Exact positions | Kendall tau |
|---|---|---|---|
| perfect | 1.000 | 1.000 | 1.000 |
| one adjacent swap | 0.000 | 0.750 | 0.964 |
| shifted along by one | 0.000 | 0.000 | 0.750 |
| one item moved to the end | 0.000 | 0.000 | 0.750 |
| completely reversed | 0.000 | 0.000 | 0.000 |
The middle column is the complaint. It cannot tell "offset by one" from "completely reversed" — both score zero — while by pairs one of them is three quarters right and the other is as wrong as an answer can be. Counting correct positions does not reward a near miss; it is blind to one.
All-or-nothing is worse still: it gives the same score to the best possible wrong answer and the worst, so it carries no information about the attempt at all. Of the three, only the pair score separates every row — and it puts the smallest possible mistake, a single adjacent swap, at 0.964 rather than at zero.
And what each one does to a random guess
| Items | All or nothing | Exact positions | Kendall tau | Items in the right slot |
|---|---|---|---|---|
| 4 | 0.04063 | 0.2494 | 0.4989 | 0.997 |
| 6 | 0.00165 | 0.1655 | 0.4992 | 0.993 |
| 8 | 0.00000 | 0.1240 | 0.4996 | 0.992 |
| 12 | 0.00000 | 0.0824 | 0.4999 | 0.989 |
The last column is a classic result worth seeing measured: the expected number of items in their correct slot in a random shuffle is exactly one, at every size. So the positions score falls as 1/n purely because the list got longer — a player doing equally badly on a longer puzzle is scored worse for it.
Kendall tau does not drift: a random guess averages a half whatever the length, so the same number means the same thing on a four-item puzzle and a twelve-item one. And all-or-nothing is already indistinguishable from zero by eight items — there are 40,320 orderings and one of them is right.
Both tables come out of the same functions the page scores your answer with. A fresh sample of 5,000 shuffles neither table has seen is measured at build time as a check: pairs 0.5009, items in the right slot 1.006.
How to use
- Read what the list should be ordered by.
- Use the arrows under each number to move it left or right.
- Press Check to see all three scores at once.
- Press Show the order if you want the answer.
- Press New puzzle for another.
Frequently asked questions
How should an ordering puzzle be scored?
By pairs rather than by positions. Kendall tau counts how many pairs are in the correct relative order, which is the only one of the three common scores that tells a near miss from a disaster.
What is wrong with counting correct positions?
It is blind to a near miss. Shift a correct ordering along by one place and not a single item is in its correct slot, so it scores zero — the same as a completely reversed answer. By pairs the shifted answer is 0.75 and the reversed one is 0.
What does a single mistake cost?
Almost nothing, by pairs. Swapping one adjacent pair in an eight-item list leaves 0.964 of the pairs in order, because only that one pair is affected. By positions it costs 0.25, and under all-or-nothing scoring it costs everything.
Why is all-or-nothing scoring bad?
Because it gives the same score — zero — to the best possible wrong answer and the worst, so it carries no information about the attempt. It also becomes meaningless with length: there are 40,320 orderings of eight items and exactly one is right.
How many items land in the right place by luck?
Exactly one on average, at every size. That is a classic result about random permutations, and it means a positions-based score falls as one over the length purely because the list got longer — a player doing equally badly on a longer puzzle is scored worse for it.
Does Kendall tau have that problem?
No, which is the practical reason to prefer it. A random guess averages a half whatever the length, so the same number means the same thing on a four-item puzzle and a twelve-item one.
Where do the orderings come from?
They are computed, not looked up. Each puzzle asks you to order numbers by something calculable — how many divisors they have, their digit sum, their largest prime factor — so every answer on the page can be checked rather than taken on trust. The generator also rejects any set with a tie, so the correct order is unique.
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