Motivated by: Tiling an orthogonal polygon with squares, How to prove that the minimum square partition of a 3X2 rectangle has 3 squares, Minimum square partitions for 4×3 and 5×4 rectangles, What is the minimum square partition of an almost-square rectangle?. Let: An almost-square-rectangle be a rectangle that has a width $w$ and height $h=w-1$. […]

I have been reading the paper Spiral hexagonal circle packings in the plane (Alan F. Beardon, Tomasz Dubejko and Kenneth Stephenson, Geometriae Dedicata Volume 49, Issue 1, pp 39-70), which proves that “these ’coherent’ [Doyle] spirals, together with the regular hexagonal packing, give all possible hexagonal circle packings in the plane”. On the obvious reading, […]

Let $A,B,C,D$ and $E$ be five non-overlapping squares inside a unit square, of side lengths $a,b,c,d,e$, respectively. Prove that $$a+b+c+d+e \le 2.$$ (This problem is a continuation of my previous question about two squares in a box, which received an ingenious answer. If one asks for the maximum $M$ of the sum of the side […]

This question has probably been asked before but when I searched the site I could not find the answer. Suppose we have and $n$-dimensional ball with radius $R$. How many, smaller $n$-dimensional ball with radius $r$ can we fit in this ball. Let this number be denoted by $N$. I am aware that this is […]

In his accepted answer to this question, David Bevan improved my answer to show that a unit disc can be cut into three sectors which fit into a square of side $2-\varepsilon$, where $\varepsilon\approx0.0291842223$. Now, suppose we drop the requirement that the pieces are sectors, and permit any three pieces whose perimeters are made up […]

In geometrical terms, what is the smallest number $n$ such that a disc of unit radius can be cut into $n$ sectors that can be reassembled without overlapping to fit into a square of side $2-\varepsilon$, where $\varepsilon>0$ is as small as you like? The question could be set for arbitrary straight cuts; but I […]

Would I be wrong to assume that the solution to this problem: What is the length of the shortest pipe, of internal radius 50mm, that can fully contain 21 balls of radii 30mm, 31mm, …, 50mm? …involves stacking the balls, from largest to smallest, with each ball resting against the last on alternate sides of […]

A regular tetrahedron $T$ of edge-length $\sqrt{2}$ fits inside a unit cube: (Image from MathWorld.) This means that $8$ cubes of side-length $\frac{1}{2}$ can cover this regular tetrahedron $T$. Note the volume of $T$ is $$\frac{1}{12} \sqrt{2} \sqrt{2}^3 = \frac{1}{3} \;,$$ so much of the […]

Under this answer, user Bruno Joyal asks: This might be a naive question, but… how do we know there is a best possible solution? I (but that’s just me) assume that he might be thinking of a logical possibility that there is a sequence of ever better solutions, but that the limit itself is not […]

Suppose I want to pack hexagons in a circle, as on the drawing below (red indicates “packed” hexagons). I am wondering what is known about this problem. Specifically, I am interested in an approximation to how many fit (given the radius of the circle and the length of the side of the hexagon) and the […]

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