Question

The scores of three players in a tournament have been lost. The only information available is the total of the scores for players 1 and $2,$ the total for players 2 and $3,$ and the total for players 3 and 1 . a. Show that the individual scores can be rediscovered. b. Is this possible with four players (knowing the totals for players 1 and 2,2 and 3,3 and $4,$ and 4 and 1)?

Short answer

Individual scores for three players can be found using simple algebra, but not for four players.

Step-by-step solution

Assign Variables for the Unknown Scores

Let's assign variables for the scores of the three players. Let the score of player 1 be $x$, player 2 be $y$, and player 3 be $z$.

Write Down the Given Information as Equations

From the problem statement, we know the sum of scores for pairs of players:- For players 1 and 2: $x+y=A$- For players 2 and 3: $y+z=B$- For players 3 and 1: $z+x=C$

Add All Three Equations

Add the three equations together:$$(x+y)+(y+z)+(z+x)=A+B+C$$ This simplifies to $2x+2y+2z=A+B+C$.

Solve for the Sum of Individual Scores

Divide the equation from Step 3 by 2 to find the sum of the individual scores:$$x+y+z=\frac{A+B+C}{2}$$

Solve for Each Player's Score

Using the result from Step 4, calculate each player's score:- $x=\frac{A+C-B}{2}$- $y=\frac{A+B-C}{2}$- $z=\frac{B+C-A}{2}$

Generalize to Four Players and Explain the Limitation

For four players, if you know the totals for 1 and 2, 2 and 3, 3 and 4, and 4 and 1, you end up with equations:$x+y,y+z,z+w,w+x$, similar to before. However, adding these does not yield a system solvable by individual decomposition due to lack of unique information per player, rendering the problem underdetermined.

Key concepts

Systems of Equations

When you hear about systems of equations, think about a set of equations working together. They're like teammates trying to solve a mystery together. Each equation in the system offers a piece of the puzzle.
For example, in our exercise, we had three equations concerning the scores of players:

• The score of player 1 and player 2.
• The score of player 2 and player 3.
• The score of player 3 and player 1.

We aim to find the individual score of each player.
The trick is to combine these equations to unveil a clearer picture, which is the beauty of systems of equations. When tackled as a whole, they can reveal hidden values that looked unrelated at first.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and systems of linear equations. It's a powerful toolset for dealing with various mathematical problems.
In the case of our exercise, we deal with linear equations which can be thought of as straight lines when plotted on a graph.
**Why Linear?**

• Each equation contains variables (like our player scores) that are only multiplied by a constant factor, never raised to a power or in a complex form.
• Their solution is straightforward, relying on algebraic manipulation like addition or subtraction. This simplicity is what defines them as linear.

The usefulness of linear algebra comes in its ability to solve these equations quickly and accurately. When equations are laid out, they form equivalents to vectors that we can manipulate to reveal the scores we're looking for.

Solving Equations

Solving equations is like finding the missing piece of a jigsaw. When faced with multiple equations, each tells a part of the story, and your job is to assemble these parts to see the full picture.
For the players in our exercise, we assigned each player a variable: - Player 1: $x$- Player 2: $y$- Player 3: $z$ Then, using the provided sums, we formulated equations.
**The Journey of Solving:**

• First, we combined the equations to simplify the problem. This usually involves adding, subtracting, or dividing the equations.
• We found a general result: $x+y+z=\frac{A+B+C}{2}$ which represented the total of all scores.
• Finally, through strategic algebra, each individual score was isolated and calculated using known values replaced in the equations.

Solving gives you the knowledge to unravel problems into manageable parts, revealing the unknowns in the process.