Proofs without words
Tonight's idea
Add the odd numbers in order: \(1\), \(1 + 3 = 4\), \(1 + 3 + 5 = 9\), \(1 + 3 + 5 + 7 = 16\). Squares every time. You could check more cases, but checking is not the same as knowing why.
Here is the why, as a picture. Start with one cell. Wrap an L-shape of 3 cells round it: a \(2 \times 2\) square. Wrap an L of 5 round that: \(3 \times 3\). Each new L is the next odd number, because an L round an \(n \times n\) square has \(n\) cells along the top, \(n\) down the side and one in the corner: \(2n + 1\).
\(1 + 3 + 5 + \cdots + (2n - 1) = n^2\).
A good picture does what a proof does: it shows the pattern must continue, not just that it has so far. Mathematicians call these proofs without words. The test: can you say, in a sentence, why it works for every \(n\) and not only the one drawn? Here: the \(n\)-th L always has \(2n - 1\) cells.
Watch one: two staircases
\(1 + 2 + 3 + 4\) is a staircase of cells, 4 high. Take a second copy, turn it upside down, and fit the two together. They make a rectangle 4 high and 5 wide. So \(2(1 + 2 + 3 + 4) = 4 \times 5\), and the sum is 10.
The picture works for any height. Two staircases of height \(n\) make an \(n\) by \((n+1)\) rectangle, because row \(k\) has \(k\) blue cells and \(n + 1 - k\) orange ones. So \(1 + 2 + \cdots + n = \tfrac{n(n+1)}{2}\), which is Gauss's shortcut drawn instead of written.
Your turn
- Use the L-shape picture to find \(1 + 3 + 5 + \cdots + 99\) without adding. How many odd numbers are there, and what square do they make?
- Find \(1 + 2 + \cdots + 100\) from the rectangle. Then find \(2 + 4 + 6 + \cdots + 40\).
- Check that \(1^3 + 2^3 + 3^3 + 4^3\) is a perfect square, and compare it with \((1 + 2 + 3 + 4)^2\). Assuming the pattern holds, write down \(1^3 + 2^3 + \cdots + 10^3\) without computing a single cube beyond \(4^3\).
- Draw the picture for \(1 + 2 + \cdots + (n-1) + n + (n-1) + \cdots + 2 + 1\): two staircases again, but fitted differently. What is the sum? Check it with \(n = 10\), and say in one sentence why your picture works for every \(n\).
Stretch: a third of a square
Cut a square of area 1 in half, then cut half of what is left, and so on. The pieces have areas \(\tfrac12, \tfrac14, \tfrac18, \ldots\) and fill the square, so they add to 1. Now draw a picture that proves \(\tfrac14 + \tfrac1{16} + \tfrac1{64} + \cdots = \tfrac13\). Start by cutting the square into four.
Answers
- The odd numbers from 1 to 99 are the 50 L-shapes of a \(50 \times 50\) square, so the sum is \(50^2 = 2500\).
- \(\tfrac{100 \times 101}{2} = 5050\). The even numbers are twice \(1 + 2 + \cdots + 20 = 210\), so \(2 + 4 + \cdots + 40 = 420\), which is \(20 \times 21\): the whole rectangle this time.
- \(1 + 8 + 27 + 64 = 100 = 10^2\), and \(1 + 2 + 3 + 4 = 10\), so the two agree. For 10 cubes, \(1 + 2 + \cdots + 10 = 55\), so the sum of cubes is \(55^2 = 3025\). (The picture: \(k\) copies of a \(k \times k\) square for each \(k\) tile a square of side \(1 + 2 + \cdots + n\), once one square of each even size is cut into two halves.)
- \(n^2\). A staircase of height \(n\) and one of height \(n - 1\), turned over, fit into an \(n \times n\) square: row \(k\) has \(k\) cells from one and \(n - k\) from the other, always \(n\). With \(n = 10\): \(1 + \cdots + 9 + 10 + 9 + \cdots + 1 = 45 + 10 + 45 = 100\).
Stretch. Cut the square into four equal squares. Shade one of them and leave two blank; cut the fourth into four again, shade one, leave two blank, and keep going into the corner. At every stage you add one shaded square and two blank squares of the same size. So the shaded part is always exactly one third of everything coloured so far, and the total shaded area is \(\tfrac13\). The shaded squares have areas \(\tfrac14, \tfrac1{16}, \tfrac1{64}, \ldots\)