Patterns Everywhere

A tower of discs sits on one of three pegs, biggest at the bottom. You may move one disc at a time, and a bigger disc may never sit on a smaller one. How few moves would it take to move a four-disc tower to another peg?

You're running up a staircase, and each stride takes you up either one stair or two. Three stairs can be climbed in three different ways: 1 + 1 + 1, 1 + 2 or 2 + 1. How many different ways are there to climb ten stairs?

A plus sign made of dots grows at every step, as if someone keeps stretching its arms. Look at the first three steps in the picture and work out how it grows. Without drawing them all, can you say how many dots step 10 will have?

Build a triangle of numbers with ones all down both edges, where every other number is the sum of the two just above it. Now add up each row. How big do you think the total gets ten rows below the 1 at the top?

Snap four identical squares together, edge to edge, like the pieces in a falling-block game. You may turn a shape around, but you may not flip it over like a pancake. How many different shapes can you make?

Put five dots evenly around a circle and join every second dot: one pencil line draws a perfect star and ends right where it began. Now try six dots, still joining every second one. Can you still draw it without lifting your pencil?

Pick any month on a calendar and draw a box around nine dates, three across and three down. Tell me only the date in the middle, and I'll say the total of all nine before you can add them up. What's my secret shortcut?

Some cicadas live underground for years and then crawl out all together, but only every 13 or every 17 years. Imagine a hungry predator whose numbers boom every 6 years. If a cicada came out every 12 years instead, how often would it run into the predator?

Three green frogs sit on the lily pads on the left, three brown frogs on the right, and one pad in the middle is empty. Frogs only ever move forward: a hop onto the empty pad next to them, or a jump over one frog into it. Can you swap the two teams, and how few moves does it take?

A sweet-shop owner has lost all her weights but four: 1, 3, 9 and 27 grams. A customer wants exactly 2 grams of sweets, yet she smiles and says she can weigh every whole number of grams up to a surprising amount. How far can she go without skipping one?