Complete Scrambles
Consider trying to scramble a string of numbers, say 1, 2, 3, 4, 5, but with the requirement that none of the numbers can end up where they started. Let’s define this as a complete scramble.
Mathematics
Flattening Cubes
We want to flatten the outer shell of a cube without tearing or stretching the surface.
Cube Solids
For all questions in this problem, let’s ignore the effects of gravity, which means higher unit cubes can float in space without lower unit cubes to support them.
Triangle Average Divider
Let’s say instead of “bisecting” the triangle, we draw a line from the top vertex to the base with a length that is the average of the remaining two sides...
Triangle Bisectors
Let’s start with a right triangle with a height of 1 unit and consider two ways of dividing it into two triangles.
Covering Squares
We choose one space in a rectangular grid made of unit squares, and we count the number of squares or rectangles that cover this space.
Mirror Functions
Let’s consider all real valued functions whose inverse is the function itself. Since the graphs of these functions are their own mirror images across the line y=x, let’s define these functions as mirror functions.
Counting Rectangles Extended
Continuing from last week’s problem about counting squares, let’s look at the related problem of counting the number of rectangles in a grid, with extensions
Counting Squares Extended
Consider the widely known problem of counting the number of squares in a square grid. Today we will try to extend the problem.
Water Jugs
Consider this family of popular puzzles: You have two empty water jugs with no markings. One holds exactly 8 liters of water when full, and the other holds 3 liters. You also have a full tub with an endless supply of water and another large unmarked empty jug of unknown volume. The goal is to measure out a given volume of water to put into the large jug using the other two jugs.