Problem 8 - Discrete Mathematics with Applications

Question

The sum of nine integers in the range \(1-25\) is \(83 .\) Show that one of them must be at least \(10 .\)

Step-by-Step Solution

Answer

Assuming all integers are less than 10, the maximum possible sum of nine integers in the range 1-9 is 45 (9+8+7+6+5+4+3+2+1). However, the given sum is 83, which is greater than 45. Therefore, at least one of these integers must be at least 10 in order to achieve the given sum of 83.

Step 1: Assume all integers are less than 10

Let's assume that all the integers in our set are less than 10. This means we will have nine integers ranging from 1-9.

Step 2: Calculate the maximum sum of nine integers less than 10

If we want to maximize the sum, we will choose the largest available integers under 10. These integers are 9, 8, 7, 6, 5, 4, 3, 2, and 1. Now let's calculate the sum of these nine integers: Sum = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1

Step 3: Evaluate the sum

Now we will add these integers to find the maximum sum of nine integers less than 10: Sum = 45

Step 4: Compare the maximum sum with the given sum

We saw that the maximum sum we can get from nine integers less than 10 is 45. However, the given sum is 83, which is greater than our calculated maximum sum.

Step 5: Conclusion

Since our assumption that all integers are less than 10 resulted in a sum smaller than the given sum of 83, it means that at least one of these integers must be at least 10 to achieve the given sum. Therefore, we have proven that one of the nine integers must be at least 10.

Key Concepts

Discrete Mathematics

Discrete mathematics is an area of study that deals with distinct and separate mathematical structures, rather than continuous. It includes topics like logic, set theory, graph theory, and combinatorics, which all play vital roles in computer science and information theory. Within discrete mathematics, the pigeonhole principle is a powerful concept that often deals with objects being assigned to containers.

For example, if there are more pigeons than pigeonholes, and each pigeon has to go into a hole, we must conclude some pigeonhole contains more than one pigeon. This principle is regularly used to solve problems where we need to prove that something must happen a certain number of times or that a certain condition must be met, such as the presence of a minimum number in a set of integers.

Integer Sum Problem

The integer sum problem is a form of mathematical question where we are asked to consider the sum of a series of integers, often with restrictions on the values that these integers can take. These problems can sometimes be solved by direct calculation or by applying algebraic techniques, but they can also often be approached using the principles of discrete mathematics.

When dealing with an integer sum problem like the one provided, it's crucial to assess the constraints given: the range of numbers we can use and the total sum we're trying to reach. As shown in the solutions, by assuming a set of integers and calculating their potential maximum sum, we can deduce whether this assumption stands or fails given the restrictions. It hones the understanding of possible and impossible scenarios within the framework of the question.

Proof Techniques

In mathematics, proof techniques are systematic methods used to demonstrate the truth of statements. One elementary technique is direct proof, where we show that a series of logical steps lead from the hypotheses to the conclusion.

Another common strategy is proof by contradiction, such as seen in the exercise, where we assume the opposite of what we're trying to prove and show that this assumption leads to an inconsistency or impossibility. By doing so, we demonstrate that our original premise must be true. The solution we considered employs proof by contradiction effectively by assuming that all the integers are less than 10 and proving that under this assumption, the sum cannot reach the required value of 83, thus concluding that our assumption must be incorrect and that at least one integer must be 10 or greater.

This exercise also touches on the aspect of combinatorial reasoning - a subset of proof techniques that involves counting and arranging objects subject to certain conditions. Students learning proof techniques can greatly benefit from these interrelated concepts of discrete mathematics.