# Problem of the Week Problem B and Solution The Puzzler

## Problem

The Puzzler is the world’s latest superhero. He uses his immense brain to win all battles by solving a series of math problems. He needs your help to solve the following problems.

Use a calculator to help when needed. You may also want to look up words like consecutive and sum.

1. The numbers $$3$$, $$5$$, and $$7$$ are three consecutive odd numbers that have a sum of $$3 + 5 + 7 = 15$$.
What are three consecutive odd numbers that have a sum of $$399$$?

2. What are three consecutive even numbers that have a sum of $$5760$$?

3. What are four consecutive whole numbers that have a sum of $$2022$$?

## Solution

1. The sum of the three consecutive odd numbers $$3$$, $$5$$, and $$7$$ is $$3 + 5 + 7 = 15$$. We notice that $$15 = 3 \times 5$$ and $$5$$ is the middle number. It seems that to find the middle of three consecutive odd numbers with a certain sum, we may divide that sum by $$3$$.

Let’s try using this to solve the problem. We note that $$399 \div 3 = 133$$. Therefore, the middle number could be $$133$$. Then the first number would be $$131$$ and the third number would be $$135$$. The sum of these numbers is indeed $$131+133+135 = 399$$. Therefore, the three consecutive odd numbers are $$131$$, $$133$$, and $$135$$.

2. We will use a process like in (a). Noting that $$5760 \div 3 = 1920$$, we see that three consecutive even numbers could be $$1918$$, $$1920$$, and $$1922$$. The sum of these numbers is indeed $$1918 + 1920 + 1922 = 5760$$. Therefore, the three consecutive even numbers are $$1918$$, $$1920$$, and $$1922$$.

3. Using a similar process, when we divide $$2022$$ by $$4$$ we get $$505.5$$. Since $$505$$ and $$506$$ are the closest whole numbers to $$505.5$$, they may be the two middle numbers. The four consecutive numbers may be $$504$$, $$505$$, $$506$$, and $$507$$. The sum of these numbers is indeed $$504 + 505 + 506 + 507 = 2022$$. Therefore, the four consecutive numbers are $$504$$, $$505$$, $$506$$, and $$507$$.