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.
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.
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.
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.
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.
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.
Folding Squares
If we draw a line through the center of a square, we can fold the square in half along this line. Depending on the angle this line makes with the horizontal, as shown in the diagram, the folded square will have different looks.
Separating Numbers
Here’s one way to put a pair of 1s, a pair of 2s, and a pair of 3s in a row such that the pair of 1s have one number between them, the pair of 2s have two numbers between them, and the pair of 3s have three numbers between them:
Combination Lock
There’s a combination lock with a 3 digit code with no repeated digits, and for each of our attempts to guess the code it tells us the number of correct digits and ones in the correct places.
Nonlinear Dots
Here’s a way to put 6 dots into a 3x3 grid such that no 3 dots form a straight line.