Problem B and Solution

The Puzzler

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*.

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 \(399\)?What are three consecutive even numbers that have a sum of \(5760\)?

What are four consecutive whole numbers that have a sum of \(2022\)?

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 \(135\).

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 \(1922\).

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 \(507\).