Solving Balance Puzzles with Matrices

I grew up on puzzles and games. A typical Christmas or birthday present was a puzzle book, and our family spent a lot of quality time together when the Games magazine came in the mail. Being the math nerd that I am, I would sometimes use trial and error for the puzzles, but anything I could convert into equations to solve was a bonus. So, in a way, I was introduced to linear algebra long before I took the class in undergraduate school.

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

Example 1: For the mobile at the top of the page, the total weight is 36. I have labelled each weight with a variable name, A, B and C. The total weight is 36:

A + 2B + 2C = 36.

The left arm must balance the right arm:

A + 2B = 2C or A + 2B - 2C = 0.

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

(A, B, C) = (18 - 2B, B, 9)

with some restrictions on B. Since 0 < A < 100,

0 < 18 - 2B < 100,

-18 < -2B < 82, and

9 > B > -41.

Since B > 0, we find 0 < B < 9, and thus, there are 8 possible answers, one each for B = 1, 2, 3, 4, 5, 6, 7 and 8.

Questions: First, solve the puzzles below, then consider these questions.

1. What would it take to create one of these puzzles? Can you create a puzzle that has a unique solution?
2. 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.
3. 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.

Puzzle 1. The total weight is 80. Only one shape weighs more than 9 and the cube is 1 unit heavier than the octahedron.

Puzzle 2. The total weight is 54. The three arms are equal in weight.

Puzzle 3. The total weight is 180. The three arms are equal in weight.

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A later post - December 31, 2011

I found a link with more balance puzzles.  These are different looking, but the idea is the same.  In this case you need to take the distance each weight is offset from the balance point into account.  For example, a weight of 2 units that is 1 unit from the fulcrum will balance a weight of 1 unit that is 2 units from the fulcrum.  Can you use linear algebra to solve these puzzles?  If so, how?

Here is another page of balance puzzles.http://battleships.free.fr/balance_i.pdf

Solving Number Puzzles with Matrices

The Discovery Education Classroom Resources has some great classroom tools. One of my favorites is Puzzle Master. It allows you to create custom puzzles like word searches or cryptograms. I used the Number Blocks puzzle creator for a linear algebra project on overdetermined and underdetermined systems of equations.

An example of a number block puzzle created using Puzzle Master is given above. The numbers to the right an bottom are the sums of various rows, colums and diagonals of the 3-by-3 grid of numbers. The main diagonal sums to 14; the off diagonal sums to 17. The sum of each row is to the right, and the sum of the columns is at the bottom. In this example, there are 6 unknowns and 8 equations, and so this is an overdetermined but consistent system. Puzzle Master will not create a puzzle that does not have a solution.

Questions:  We can create a system of equations for this number puzzle as shown above. There are a lot of questions that can be asked about these number puzzles and the systems that can solve them.

• Given an n-by-n grid of numbers, how many equations are needed to solve the puzzle?
• What determines how many unknows there are?
• Given an n-by-n grid of numbers, how many unknowns must there be for the system to be overdetermined? Underdetermined? Square?
• Are there equations in the system below that can be removed so that the system is not overdetermined but has an equivalent solution? If so, how many and which equations can be removed.
• If a number puzzle is underdetermined, how can a solution be found?
• Is it possible for a number puzzle to have exactly two different solutions? Exactly five different solutions? Explain.
• Is it possible for a number puzzle to have infinitely many solutions? Describe a puzzle for which this would be true, and explain how to describe the infinitely many solutions.
• Puzzle Master won't create a puzzle with no solution, but can you create a puzzle that has no solution and is overdetermined? Underdetermined? Square?
• Create a puzzle using Puzzle Master that would have infinitely many solutions. Can you make a conjecture on how Puzzle Master chose the solution that it gives?