Tip a bag of twelve pebbles onto a table and try to arrange them into a neat rectangle. You have options: two rows of six, three rows of four, or one long line of twelve if you're feeling stubborn. Now try it with eleven pebbles. You can shuffle all afternoon, but the only "rectangle" you will ever make is a single straight line.
The ancient Greeks noticed this, and they took it seriously. The followers of Pythagoras sorted numbers by the shapes their pebbles could make: square numbers, oblong numbers, triangular numbers. Later, around 300 BC, Euclid wrote the idea into the definitions at the start of Book VII of his Elements. A plane number is one made by multiplying two numbers together, and those two numbers are its sides. A prime number is one that only the unit measures, which is a formal way of saying: it has no sides except 1 and itself. In pebble language, a prime can only lie in a line.
That idea is more than two thousand years old, and it is also the entire engine of our puzzle Rectangles.
A number is a list of shapes
The rules of Rectangles are short. The grid is scattered with numbers. You split the whole grid into rectangles (squares count too) so that every rectangle holds exactly one number, and that number is its area. A 6 covers six cells, as a 2x3, a 3x2 or a straight line of six.
So the first thing a clue tells you is not where it goes but what shapes it could be, and that list is exactly the list of its divisors. Every way of writing a number as "rows times columns" is one shape. Take 12 again: 1x12, 2x6, 3x4, 4x3, 6x2 and 12x1. That's six shapes, one for each of the six divisors of 12 (1, 2, 3, 4, 6 and 12). Each divisor picks the height, and the width is whatever is left over.
A 7 has only two shapes: a line across or a line down. A 24, with eight divisors, has eight shapes on paper, though the longest ones won't fit on any of our boards. When we say a clue is "flexible", what we really mean is that it has a lot of divisors.
Why primes place themselves
Here is where Euclid starts earning his keep as a solving coach. A prime number in Rectangles is a line, full stop. And a line is a demanding thing to fit on a small board.
On a 5x5 grid, a 5 must fill an entire row or an entire column. That's it. Look along its row and column: if either one already has a clue or a drawn rectangle in the way, the other direction is forced. A 7 on a 6x6 board would be impossible, which is why you'll never see one. Our strategy notes put it plainly: prime numbers like 5, 7 and 11 can only be straight lines, which makes them easy to place on small boards.
Composite numbers have more freedom, but they still carry tells. A 4 can be a 1x4, a 4x1 or a 2x2. Pin it in a corner with neighbours close by and the long lines often won't fit, leaving the square as the only survivor. The habit to build is this: before you look at the grid, factor the clue. Knowing its shapes in advance turns a vague "where could this go?" into a short checklist.
The locker puzzle hiding in every square
Divisors have a beautiful pattern that's worth knowing, and there's a classic riddle that shows it off.
A school has 100 closed lockers and 100 students. The first student opens every locker. The second student changes every second locker (closing it if open, opening it if closed). The third changes every third locker, and so on, until the hundredth student changes only locker 100. When they're done, which lockers are open?
A locker gets touched once for every divisor of its number. It ends up open only if it was touched an odd number of times. Divisors usually come in pairs: for 12, 1 pairs with 12, 2 with 6, 3 with 4. The only way to get an odd count is if one divisor pairs with itself, and that happens exactly when the number is a perfect square, like 36 = 6 x 6. So the open lockers are 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100: ten of them.
In Rectangles, that unpaired divisor is the square shape. On paper, a 36 has nine shapes: four pairs of long-and-wide mirror images, plus one lonely 6x6 square with no partner. Every square number on the board carries exactly one shape that looks the same turned sideways.
Counting the whole board
There's one more piece of arithmetic in every Rectangles puzzle, and it's the one experienced solvers lean on without thinking. Every cell belongs to exactly one rectangle, and every rectangle's area is its number. So the numbers on the board always add up to the number of cells. A 5x5 puzzle's clues sum to 25. A 10x10's sum to 100.
There is no slack anywhere. If you place a rectangle that leaves behind a lonely cell nobody can reach, the board can't be finished, because that cell has no number to claim it. This is the heart of the harder techniques: look for a cell that only one number can reach, and that number's rectangle must cover it. Then look for cells that every possible shape of a number covers, because those cells are taken no matter which shape wins.
Mathematicians kept pushing Euclid's question for centuries. In 1915, Srinivasa Ramanujan published a long paper on highly composite numbers, the numbers with more divisors than any smaller number: 1, 2, 4, 6, 12, 24, 36, 48, 60 and onward. The bigger ones would be nightmare clues in Rectangles, a number with shapes to spare. Which tells you something about puzzle design, too: a good board mixes rigid primes that give you a foothold with flexible composites that make you work.
Thinking in shapes
The pebble trick works because it turns an abstract question ("what divides 12?") into something you can see and touch. That's a useful move well beyond the grid. Seating forty people at equal tables, cutting a sheet cake into equal pieces, laying out a photo wall: each one is secretly a question about divisors, and each one gets easier once you list the shapes first and only then start moving things around.
So next time a Rectangles clue stares back at you, don't reach for the mouse straight away. Ask Euclid's question: what are your sides? Often the answer is shorter than you'd think, and sometimes it's just one line.
