From pattern to rule: square and triangular numbers

A pattern becomes powerful when you can write a rule for it. A rule lets you jump straight to step 100 without drawing the first 99 steps, and it lets you run the pattern backwards to find which step gives a certain number.

Linear patterns

If a pattern grows by the same amount each time, it is linear. A row of toothpick squares uses 4, 7, 10, 13 toothpicks: three more each time. The rule is t = 3n + 1, where n is the step number and t is the number of toothpicks. The 3 is the constant increase and the 1 is the extra toothpick at the start. Plot it on a graph and the points make a straight line.

Which step uses exactly 100 toothpicks? Solve 3n + 1 = 100. Take 1 from both sides: 3n = 99. Divide by 3: n = 33. Step 33 uses 100 toothpicks. If the answer had not been a whole number, no step would use exactly 100.

Square numbers

1, 4, 9, 16, 25 are square numbers: n × n, written n². You can draw each as a square of dots. The differences between them are 3, 5, 7, 9: the odd numbers. Each new square adds a strip along two sides plus a corner, which is always an odd number of dots.

Triangular numbers

1, 3, 6, 10, 15 are triangular numbers, like the balls in a triangle rack. Take two copies of the same staircase and turn one upside down. They fit together into a rectangle n wide and n + 1 tall. The staircase is half of that rectangle, so the rule is n(n + 1) ÷ 2. The tenth triangular number is 10 × 11 ÷ 2 = 55.

Why the picture matters

Checking a rule on five steps builds confidence. But the rectangle picture does more: it shows why the rule must work for every step, even step one million. That is what mathematicians call a proof. They want both kinds of certainty: