Word Groups

Sixteen words sit in a four-by-four grid. They divide exactly into four groups of four, and nothing on the screen tells you what the groups are. Pick four words, submit them, and you are told whether they are a group. Four wrong guesses and the board is over — the groups you have not found are then shown to you, easiest first.

What makes it a puzzle rather than a vocabulary test is that the words are chosen to overlap. On one board here, BOXER sits with POODLE, TERRIER and BEAGLE while JAB, HOOK, CLINCH and BOUT sit elsewhere in the same sixteen. Both readings of BOXER are correct English; only one of them is correct on that board. Across the 44 boards there are 127 of these deliberate traps, and every board has at least one.

2,627,625. That is 16! ÷ (4!⁴ × 4!) — the number of ways to partition sixteen items into four unlabelled groups of four. Exactly one of those partitions is the answer, and you are allowed four mistakes to find it. Put like that the game sounds impossible, and the reason it is not is that you never guess a whole partition: you guess one group at a time, and each correct group shrinks the board.

Multiply the four columns together and a player guessing entirely at random finishes a board without a single mistake once in 2,627,625 attempts — the same figure, arrived at from the other end. Which is the honest way of saying that four mistakes is not a generous budget, it is roughly the amount of slack a good solver needs.

That exactly three of your four words share a group — so one word is wrong and three are right. It is far more information than it looks. Of the 1,820 four-word picks on a full board, 192 share exactly three words with some group, so a random guess comes back one away 10.5% of the time. When yours does, you are not being consoled: you are being told that a group of four exists three-quarters inside your selection, and there are only twelve candidates for the fourth slot.

The right move after one away is almost never to swap two words at once. Swap the single word you are least sure of, and if that fails, swap the next. Each attempt costs one mistake and eliminates one hypothesis, which is a fair exchange; changing two words at a time costs the same mistake and eliminates nothing you can name.

Because a generator can find four synonyms, and that is not the game. Every other word game on this site — word guess, word scramble, word ladder — builds its puzzle from the shipped dictionary, because in those games the rule itself is the puzzle. Here the rule is trivial and the craft is entirely in choosing a word that plausibly belongs to two groups at once. That is an editorial judgement, not a search.

So there are 44 boards, 176 groups and 704 word slots, filled by 671 different words — and every one of those words is in the shipped ENABLE list, the same public-domain dictionary the rest of the site's word games are built on. No board is ever generated on the fly, which is also why the set is finite and says so rather than implying an endless supply. New board deals one of the 44 at random, so you should expect the ninth board you deal to be one you have already played — the number under the grid tells you which one it is before you start.

Six kinds recur, mixed deliberately, because 44 boards of synonyms is one board played 44 times. Knowing the kinds is itself a solving technique: when three words look like a set and the fourth does not, the group is usually not a set at all — it is a category you have not thought of yet, and the fourth word is the one telling you so.