Counting the Solutions: 7,188 and 14,376 — and Why Both Numbers Are True

First, what counts as a solution

Quasi Puzzle has thirteen circular pieces and thirteen positions on the board. Each piece carries its own pattern of rounded bumps around the rim, and here is the detail that makes the whole puzzle tick: those bumps don't press against neighbouring pieces — they nest into the small, curved-triangle gaps that open up wherever three circles touch. Count those gaps across the whole board and you get exactly forty. Count the bumps across all thirteen pieces and you get — exactly forty.

That equality is the puzzle's secret engine. If all thirteen pieces are legally placed, forty bumps have claimed forty gaps, which means every gap is filled and the packing is perfect, automatically. A "solution", then, is any way of assigning every piece to a position and an orientation so that no two bumps ever fight over the same gap. Simple to state; wonderfully awkward to achieve by hand.

The trap in counting: pieces that look the same twice

A naive count would try every piece in every position at every rotation and mirror flip, and tally the full arrangements. But some pieces are symmetric — rotate one of them half a turn and it presents exactly the same bump pattern to the world. Counting that as two different solutions would be cheating. So before counting, we taught the computer to recognise when two orientations of a piece are genuinely indistinguishable, and to count each physically distinct arrangement once. What remains is an honest census of arrangements you could actually tell apart on the table.

The piece with a handedness

Before the totals, the puzzle's subtlest property — one you discover fastest by holding the physical pieces in your hands. Check each piece against its own mirror image and twelve of the thirteen turn out to be achiral: their reflection is just one of their rotations wearing a disguise, so "flipping" them changes nothing that a turn couldn't. Exactly one piece is different — the small one whose lone tail-bump sits off-centre beside a pair. Rotate it as long as you like and the tail stays on its side; only a true mirror flip carries it across. Geometers call such a shape chiral: it exists in a left-handed and a right-handed version, and no amount of turning converts one into the other. We confirmed this by brute force — for every other piece, the mirrored bump patterns are literally identical to rotated ones, while for the tailed piece alone, mirroring produces six genuinely new orientations.

The two honest totals

That single piece is where the two totals come from. If pieces may be rotated but never mirror-flipped — which is the world of the physical toy, whose one-sided pieces cannot be turned face-down — there are exactly 7,188 distinct solutions, with the tailed piece locked to its manufactured handedness. Allow the flip, as the digital game's F key does, and the total becomes exactly 14,376. The doubling belongs entirely to the chiral piece: mirror any solution and you obtain another valid solution in which that piece — and, meaningfully, only that piece — has changed hands, so the solutions pair off perfectly, each with its mirror twin. The twelve achiral pieces contribute nothing to the difference; permitting their flips adds no placement that rotation didn't already provide. And the exactness of the doubling carries a small bonus theorem: no solution of this puzzle is its own mirror image — a mirror-symmetric arrangement would need the one-handed piece to be both hands at once.

You may also meet a smaller figure in connection with puzzles of this family: counts in the low thousands, such as the 1,641 quoted for this design. Numbers like that aren't wrong — they're counting families of solutions, where arrangements related to each other by the symmetry of the board itself are grouped and counted once. Whether you call two mirror-image solutions "the same" is a genuine choice of definition; what matters is saying which convention you're using. We prefer the player's-eye convention: if the board in front of you looks different, it counts as different.

How we made sure the numbers are right

A number this specific deserves suspicion, so we derived it three separate ways and demanded agreement. First, a direct enumeration walked the board position by position, trying every remaining piece in every distinguishable orientation and backtracking out of dead ends. Second, we replayed placements through the game's own rule-checking code — the very functions that decide, when you drop a piece, whether it may sit there — to confirm the rules being counted are the rules being played. Third, and most elegantly, we restated the puzzle as an exact cover problem — every position used once, every piece used once, every one of the forty gaps claimed once — and solved it with a classic technique from computer science known as Algorithm X, which attacks whichever constraint is currently tightest. Three methods, three codebases, one answer: 7,188 and 14,376, to the digit.

The same machinery now guards the game permanently. Every future version of Quasi Puzzle is automatically re-tested against these counts and dozens of other recorded behaviours; if a change ever altered the rules by even one arrangement, the tests would fail loudly before players ever saw it.

What it means when you play

Fourteen thousand solutions sounds like abundance, and it is — but they are scattered through a space of wrong arrangements so much vaster that finding any one of them still takes real work. It also means your solves are genuinely yours: the game's solutions gallery fingerprints each distinct arrangement you discover, and most players never find the same one twice. If you'd like the broader mathematical backdrop — circle packings, why the gaps are triangles, and the lovely theorem that governs touching circles — it's on our mathematics page. Or skip the theory and go add one of the 14,376 to your gallery.