The Ideas Behind It

The Mathematics of Quasi Puzzle

Underneath a handful of plastic discs sits a surprising amount of real geometry — circle packing, tangent circles, and a search space large enough to keep you busy for a very long time.

It starts with packing circles

Circle packing asks a deceptively simple question: how tightly can you arrange non-overlapping circles? For equal circles filling the infinite plane, the answer has been known for a long time — the densest arrangement is the hexagonal (or triangular) packing, in which every circle is surrounded by exactly six others, like oranges settling into a tray. That arrangement covers

π ⁄ √12 ≈ 0.9069the densest possible fraction of the plane covered by equal circles

of the plane — a result argued by Axel Thue and proved rigorously by László Fejes Tóth in the twentieth century. Quasi Puzzle's frame is a finite patch of exactly this lattice: a cluster of equal, mutually touching discs arranged in short rows, with the outer discs forming the frame and the inner thirteen becoming the pieces you move.

Tangent circles and their curvatures

The study of touching circles is ancient — it goes back to Apollonius of Perga (around 200 BC) and his problem of finding a circle tangent to three given ones. Where four circles are all mutually tangent, their sizes are locked together by a beautifully compact relationship, Descartes' Circle Theorem. Writing each circle's curvature as k = 1/r (one over its radius):

(k₁ + k₂ + k₃ + k₄)² = 2 (k₁² + k₂² + k₃² + k₄²)Descartes' Circle Theorem — four mutually tangent circles

Keep filling the gaps between tangent circles with ever-smaller tangent circles and you produce the famous Apollonian gasket. Our puzzle doesn't go to that infinite extreme, but it lives in the same world of circles that meet edge-to-edge.

From a packing to a set of pieces

Here is the step that turns geometry into a puzzle. Take the hexagonal packing and look at each point where two circles touch. Instead of a clean tangent point, the design lets the circles overlap very slightly in a lens-shaped sliver. That sliver is added to one of the two pieces as a convex bump and removed from the other as a concave notch of identical shape. Because the added lens and the removed lens are congruent, they cancel exactly: the pieces interlock like a jigsaw, yet together they still cover the frame with no gaps and no overlaps. Every "tab" you see is one half of a shared lens; its matching "blank" is on the neighbour.

Why this connects to deeper mathematics

Circle packings are not just a curiosity. In 1936 Paul Koebe proved that any planar network of connections can be drawn as a circle packing, with touching circles standing for linked nodes — a result later rediscovered by William Thurston and extended by Oded Schramm, and now a genuine tool in the geometry of surfaces and in discrete versions of conformal mapping. The tidy little tray of discs in front of you is a hands-on relative of an idea that reaches into serious modern geometry.

How many solutions are there?

The feature that sets this puzzle apart from a jigsaw is that it has not one solution but many. The figure traditionally quoted for this puzzle is 1,641 distinct solutions. That number deserves a small asterisk: exactly how you count depends on whether you treat rotations and reflections of the whole arrangement as the same solution or as different ones, and on the precise set of pieces. A blind computer search that ignores those symmetries turns up thousands of raw arrangements; folding the symmetries back in collapses them toward the classic count. Either way the point stands — where a jigsaw has a single right answer, this puzzle has a whole landscape of them.

Why so many?

Each piece can be turned into six rotations and flipped over, and several differently-shaped pieces can sit in several different cells of the frame. That multiplies into an enormous number of ways to attempt the puzzle. What keeps it solvable — and finite — is the strict bump-meets-notch rule at every contact, which quietly rejects the overwhelming majority of arrangements and leaves exactly the valid packings behind. Solving by hand is really a small, tactile search through that space.

Try to find one →