Point in a Polygon
For any point inside a square, the sum of the distances from this point to the four sides is half of the perimeter.
Squares in Squares
Let’s draw a smaller square inside a unit square by dividing out a fixed fraction of each side and connecting the dividing point to an opposite angle.
Mod Cycle
Starting with 3, let’s double it and add one (or 2n+1), then modulate the answer by 14 (divide by 14 and take the remainder). If we repeat these two steps, we get the first cycle.
Splitting Ladder
Starting with a natural number at the top, we can split it at each level such that the two lower neighbours always add up to the higher neighbour. Let’s repeat this process until a row can’t be split using whole numbers in the following row anymore.
Jumping Tiles
In the following puzzle, each number indicates the distance (vertically, horizontally, or a combination of both) this tile needs to move. A solution is a rearrangement of all the tiles in the grid that satisfies all the numbers where none of the tiles overlap.
Balancing Numbers
Consider the set of natural numbers 1 to 8. For this set of numbers, 6 is called the balance because the numbers before 6 add up to the same sum as the numbers after 6. In other words, 1+2+3+4+5 = 7+8.
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.
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.
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.
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.