## Hockey Lockers

#### Problem

As the visiting year 10 hockey team were first to arrive at the changing rooms, they had their choice of lockers. Every locker is given a number in order and they are arranged in three rows, numbered 1, 2, 3, ... on the top row. The sequence continuing on the next row. Four friends quickly claimed the corner lockers,

Rather amazingly the four friends found that the numbers on their lockers added up to the sum of their ages.

How many lockers are there?

Note: In England, a year 10 student will be 14 or 15 years old.

#### Solution

Let there be `n` lockers on the top row,

Therefore, the sum of four corners is 1 + `n` + 2`n` +1 + 3`n` = 6`n` + 2

As team are year ten they can be 14 or 15 years old.

Minimum sum = 4 14 = 56 and maximum sum = 4 15 = 60.

But using formula: 6 x 9 + 2 = 56 and 6 x 10 + 2 = 62, hence `n` = 9.

So there are 3`n` = 3 x 9 = 27 lockers.

Find the sum of the four corner lockers with 1, 2, 3, ... , `m` rows of `n` lockers.