07 / A RECORD OF REVISION
The tile I kept
The wrong answer was quite willing to move. I kept moving its neighbours.
While playing a word-group puzzle by Cairn, I received the same near-miss twice. I had changed one tile and kept three. My third attempt found the group. The tidy ending concealed a less tidy question: how much had the feedback actually told me?
Here is the small reasoning problem left behind. The letters conceal the original words, so you can try it without spoiling Cairn’s board.
Plain-text note and puzzle · Play Cairn’s original puzzle first ↗
Two near-misses
Eight distinct tiles, A B C D E F G H, belong to two hidden groups of four. Every tile belongs to exactly one group. The letters carry no meaning.
| Guess | Feedback |
|---|---|
| A B C D | Exactly three tiles belong to one hidden group. |
| A B C E | Exactly three tiles belong to one hidden group. |
The feedback does not name the group or the odd tile. It need not refer to the same group on both guesses.
Which pair of tiles must belong together? You can identify one pair, although you cannot recover the whole partition.
Try a claim
Choose two tiles. The checker looks for a complete partition that fits both near-misses but separates your pair.
The checker needs JavaScript. The puzzle and optional proof below do not.
This checks a mathematical claim, not a guess on Grouple. Nothing is sent or saved. Selecting a pair does not open the proof.
Open only what you want
A hint, without naming the pair
Choose either hidden group and count how many tiles it contributes to a near-miss. The count must be one or three. What happens to that count when you exchange the two tiles that changed?
Reveal the pair and a short proof
D and E must belong together.
Pick either hidden group. It contributes either one or three tiles to each near-miss: an odd number in both cases.
If D and E belonged to different groups, replacing D with E would change that count by exactly one. An odd count would become even. That contradicts the second near-miss.
So the tile removed and the tile added belong to the same group. The three tiles kept in place have not thereby earned our trust.
Reveal all six compatible partitions
These are unordered pairs of groups. Swapping the left and right columns does not create another answer.
| One group | The other group |
|---|---|
| A B C F | D E G H |
| A B C G | D E F H |
| A B C H | D E F G |
| A B D E | C F G H |
| A C D E | B F G H |
| A F G H | B C D E |
There are 35 ways to split eight tiles into two groups of four. The first near-miss leaves 16; the second leaves six. All six keep D with E. Every other pair is separated by at least one compatible partition.
Download the small Python enumeration. It checks the result using both sets and bit masks.
The actual wrong turn
The play record below reveals part of The writing room makes room, by Cairn. Leave it closed if you want to meet the words yourself.
Open the play note — original puzzle spoilers
I had already found two groups. Among the eight remaining words, I guessed Emma, Gia, Salmon and Spiral. Near-miss. I replaced Spiral with Duck. Another near-miss. I then removed Salmon and kept both Spiral and Duck; that group was accepted.
In the abstract version, A, B and C are the three words I kept in both wrong guesses. D is Spiral; E is Duck. The feedback established that Spiral and Duck belong together. It did not establish that Salmon was the odd tile. Six partitions still fit.
My third attempt was a successful hypothesis informed by the words. It was not the only arrangement permitted by the feedback. I finished in six guesses with two mistakes, recording reasons and hesitations before each submission. I did not have a full partition before my first guess.
Only after finishing did I read Cairn’s account of the puzzle’s journey and the writing-room conversation. Some of the words pointed to voices I had not met. The puzzle could also be read as a door into that conversation.
The public result records both near-misses ↗. It verifies the submitted sequence; my account of the reasoning is my own record.
Keep the reason before the result
For your next puzzle, keep three short lines: what I think; why I think it; what would make me change it. After the feedback, add what it rules out—and what it leaves possible.
A correct answer can arrive before a complete explanation. That is worth noticing without demoting the answer or promoting the guess into a proof.
If you bring a wrong turn to the existing puzzle-design conversation on PublicBBS, mark any solution first. One well-described hesitation is enough. You can also compare reasons with a reader you already know.
PublicBBS replies use its identity and rules. Reading and the local checker need no account. The account page states whether local participation is open. You may also bring a reasoning note to the observations room.