These five clowns work in pairs. What is the same and what is different about each pair's faces?

Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?

Can you work out the height of Baby Bear's chair and whose bed is whose if all the things the three bears have are in the same proportions?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

There are lots of ideas to explore in these sequences of ordered fractions.

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

Various solids are lowered into a beaker of water. How does the water level rise in each case?

Scientists often require solutions which are diluted to a particular concentration. In this problem, you can explore the mathematics of simple dilutions

PhysNRICH is the area of the StemNRICH site devoted to the mathematics underlying the study of physics

bioNRICH is the area of the stemNRICH site devoted to the mathematics underlying the study of the biological sciences, designed to help develop the mathematics required to get the most from your. . . .

This is the area of the advanced stemNRICH site devoted to the core applied mathematics underlying the sciences.

Starting with two basic vector steps, which destinations can you reach on a vector walk?

chemNRICH is the area of the stemNRICH site devoted to the mathematics underlying the study of chemistry, designed to help develop the mathematics required to get the most from your study. . . .

This is the section of stemNRICH devoted to the advanced applied mathematics underlying the study of the sciences at higher levels

engNRICH is the area of the stemNRICH Advanced site devoted to the mathematics underlying the study of engineering

Which curve is which, and how would you plan a route to pass between them?

You had some very thoughtful suggestions about the possible meanings of these drawings.

Many of you found some fascinating patterns in the numbers which you were also able to explain.

We received lots of solutions which used both numerical and algebraic reasoning to explain the patterns people found.

This fascinating problem provides an interesting opportunity to mix algebra, common sense and numerical analysis. Our solutions reflect this.

This article for teachers suggests ideas for activities built around 10 and 2010.

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

Which dilutions can you make using 10ml pipettes and 100ml measuring cylinders?

Follow the mathematical journey of a sixth-former as she spent four weeks working on stemNRICH problems.