These puzzles are from Games magazine, and I hope they forgive me for forgetting the specific issue for a complete citation. They provide a nice visual from which systems of equations can be developed. These systems tend to be underdetermined; however, there are added restrictions on the solutions--they must be integers within a certain range. Thus, these underdetermined systems have a finite number of solutions instead of infinite, specifically because equations are over a range of integers, not over the reals.
Think of these puzzles like building a mobile, where the objects on each side must balance so that the horizontal bars stay horizontal under the following rules.
- each shape represents a unique positive weight
- each weight is an integer N such that 0 < N < 100
- the right and left sides of each horizontal beam must balance
- a piece hanging directly below the fulcrum does not affect the balance between the left and right arms
- the size of the pieces has no relation to weight
- these are exercises in balancing number values and do not take into account the distance from the fulcrum
The left arm must balance the right arm:
We create an augmented matrix for this system and row reduce:
While using a matrix was not necessary for this simple problem, it will be useful for the larger systems below. So, C = 9, B is a free variable, and the general solution is
with some restrictions on B. Since 0 < A < 100,
Questions: First, solve the puzzles below, then consider these questions.
- What would it take to create one of these puzzles? Can you create a puzzle that has a unique solution?
- How would it change if we did consider distance from the fulcrum? Would it still be a linear system? If so, create a simple puzzle where distance is considered.
- What determines how many equations are required? For instance, in Puzzle 1 below how many equations are described? Don't forget that each horizontal bar needs to balance, not just the top one.