Up and down sums

Another way to get the square numbers: count up and then back down.

• 1 = 1
• 1 + 2 + 1 = 4
• 1 + 2 + 3 + 2 + 1 = 9
• 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
• 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25
• 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36

Look at the middle, largest number of each sum. When it is 3, the sum is 9 = 3 × 3. When it is 6, the sum is 36 = 6 × 6. The sum up to a number and back down is always that number times itself. Like climbing a hill: the higher the top, the bigger the total.

Why? Draw a square of 4 rows of 4 dots and turn it so a corner points up, like a diamond. Now read the dots row by row across: the rows hold 1, 2, 3, 4, 3, 2, 1 dots. They are the same 16 dots, so 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16.

So going up to 100 and back, 1 + 2 + … + 99 + 100 + 99 + … + 2 + 1, gives 100 × 100 = 10,000. Careful: the middle number is written only once.