### Doodles

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

### Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

### N000ughty Thoughts

How many noughts are at the end of these giant numbers?

### Why do this problem

This problem presents an investigation which does eventually require a systematic approach. Although the generalisation is difficult for Stage 4 some of the context's structure is discernible and describable, and comparable to other similar situations. Do the problem in conjunction with Group Photo and ask learners to describe what is the same about the two situations that could explain them resulting in the same sequence of Catalan numbers.An apparent generalisation related to cubes of numbers breaks down and so the problem offers an opportunity to discuss a danger of applying inductive reasoning.

### Possible approach

One approach is to do this in conjunction with Group Photo , either following from one to the other, or dividing the class so that groups work on different problems, or why not use two classes working on the different problems. The aim would be to bring the two sets of findings together to discuss why two apparently quite different situations result in the same mathematics.

Allow plenty of time to 'play' with the problem, making sense of what is being counted and how it might be represented.
Encourage ideas that involve systematic approaches, and share them so that all learners have access to a way into the problem.
Use results from separate groups to check working.

### Key Questions

• Can you describe what is the same about the two problems that might explain the similar mathematical structure?
• What is different about and what is similar to other examples, such as One Step Two Step and Room Doubling that result in a Fibonacci sequence?

### Possible support

Group photo can be done with real people and you can start with small numbers. Spend plenty of time trying out, and considering the efficiency of, possible recording methods.

### Possible extension

Can students make connections between the structures of the two problems that may in part explain the mathematical connections?

#### Notes

$1$, $1$, $2$, $5$, $14$, $42$, $132$, $429$, $1430$, $4862$ ,...

The Catalan numbers describe things such as:
• the number of ways a polygon with n+2 sides can be cut into n triangles
• the number of ways to use n rectangles to tile a stairstep shape (1, 2, ..., n-1, n)
• the number of ways in which parentheses can be placed in a sequence of numbers to be multiplied, two at a time
• the number of planar binary trees with n+1 leaves
• the number of paths of length 2n through an n-by-n grid that do not rise above the main diagonal
They can be described by the formula $$\frac{ ^{2n}C_{n} }{(n + 1)}$$

The Catalan numbers are also generated by the recurrence relation:

$C_0=1, \qquad C_n=\sum_{i=0}^{n-1} C_i C_{n-1-i}.$

For example, $C_3=1\cdot 2+ 1\cdot 1+2\cdot 1=5$, $C_4 = 1\cdot 5 + 1\cdot 2 + 2\cdot 1 + 5\cdot 1 = 14$, etc.