Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, ..., gives square numbers?

Question

Homi · Accepted Answer

We can see this pattern by arranging dots in a square shape.

Step 1 — Checking the first few sums

Let us look at the first few sums.

The first sum is just 1. $1$

$\boxed{1 = 1^2}$

Next, we add 1, then 2, then 1. $1 + 2 + 1$ $= 3 + 1$ $= 4$

$\boxed{4 = 2^2}$

Now, we add 1, then 2, then 3, then 2, then 1. $1 + 2 + 3 + 2 + 1$ $= 6 + 2 + 1$ $= 8 + 1$ $= 9$

$\boxed{9 = 3^2}$

Diagram 1

Step 2 — Visualizing with a square of dots

Let us take the sum $1 + 2 + 3 + 2 + 1$ . This sum is 9. We can make a square with 9 dots. It is a $3 \times 3$ square. Imagine a square grid of dots. We can count the dots along diagonal lines. Start from the top-left corner. The first diagonal line has 1 dot. The next diagonal line has 2 dots. The main diagonal line has 3 dots. The next diagonal line has 2 dots. The last diagonal line has 1 dot. So, we have $1 + 2 + 3 + 2 + 1$ dots. This total number of dots forms a $3 \times 3$ square. The total number of dots is $3 \times 3 = 9$ .

Diagram 2

Step 3 — Seeing the general pattern

Let us take any number, say $n$ . We want to add numbers up to $n$ and then down to 1. This sum is $1 + 2 + ... + (n-1) + n + (n-1) + ... + 2 + 1$ . We can make an $n \times n$ square of dots. This square has $n \times n$ dots in total. If we count the dots along the diagonals, we see the pattern. The first diagonal has 1 dot. The next has 2 dots. This continues until the main diagonal. The main diagonal has $n$ dots. Then the number of dots in diagonals goes down. The next diagonal has $(n-1)$ dots. This continues until the last diagonal. The last diagonal has 1 dot. So, the total number of dots is $1 + 2 + ... + n + ... + 2 + 1$ . This total is always $n \times n$ . So, this sum is always a square number.

Answer

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

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