Numbers of Polycubes
Let’s define polycubes as three dimensional shapes made up of unit cubes connected by shared faces.
Mini Sudoku
Sudoku is a one-player game where the player tries to fill the spaces of an nxn grid using numbers 1 to n, where n is a natural number, such that every column and every row contains exactly one of each number.
Square Gluing Topology
Now let’s consider the surfaces we obtain by stretching and gluing the sides of a square together.
Variants of Sudoku
Today let’s explore one of the deviations from the standard sudoku rules that every row, column, and subgrid (excluded so we can generalize to non-perfect-square side lengths) must have exactly one of each number from 1 to n, where n is the side length of the sudoku puzzle. Let’s include the new rule that a valid solution to a sudoku puzzle must also have one of each number in each of its two diagonals.
Triangles in a Square
This problem is inspired by this geometry puzzle on Twitter. The original problem asks to find the ratio between two equilateral triangles fitted into a square. Let’s generalize it.
3D Sudoku
Imagine instead of a two dimensional grid, we are now filling a 3D grid made up of nxnxn unit cubes with numbers 1 to n such that every row, column, and depth has exactly one of each number.
Fractions of a Square
The original problem asks to cut a square by drawing straight lines between any number of midpoints of sides and corners of the square such that a resulting region is 1/5 of the area of the whole square.
Two-Colourable Triangles
We start with a polygon and a number of points inside it. We are asked to draw line segments between the points to subdivide the entire polygon into triangles.
Tiling with Rods
The original problem asks for a combination of 1x2 and 1x3 rods that can tile a 5x5 square. Tiling problems have a tendency to get complicated quickly, so let’s start small.
Subtraction Polygons
Consider the subtraction game that goes like this: Write down one whole number at each corner of the square below, then fill in each side with the positive difference the two neighbouring corners and connect the four new numbers in a smaller square. Repeat the subtraction to continue forming new squares in the middle.