### Teddy Town

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

### Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

### Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

# Circuit Maker

## Circuit Maker

A circuit is a set of switches, wires, bulbs and components called logic gates .

There are seven different types of logic gates, each with a technical-sounding name:
AND
OR
XOR (exclusively OR)
NAND (not AND)
NOR (not OR)
XNOR (not XOR)
NOT

One wire comes out of each logic gate and in the examples below this wire is always attached to a bulb. The bulb is on or off depending on the inputs from the switches and the type of logic gate.

Here is a circuit containing one of each type of gate wired up to a bulb.

By clicking the switches on or off (using the blue square in their top left hand corners), can you work out what each of the different logic gates does?

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Now that you are familiar with the properties of the gates you can experiment by building your own circuits with interesting properties using the interactivity below. Instructions are given at the bottom of the page. You can find some ideas and challenges in Simple Counting Machine

You can also read and learn more about logic circuits and gates and mathematical logic in the article Logic, Truth Tables and Switching Circuits .

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INSTRUCTIONS

To build a circuit follow these steps. You can delete mistakes by clicking on the cross which appears when you hover above the icon.

1. Left-click on the components to drag them on to the circuit board space.

2. FIRST click next to a red dot and drag to the right to create a wire; THEN you can stretch this wire to join with any other red dot in the circuit.

3. Click on the switches to turn them on or off. Wires 'fill' with current unless blocked by an un-triggered logic gate.

4. Click on the logic gates to change their type.

5. Left-click and drag to move the components around.

The key ideas are shown in this picture; by clicking on the triangle you can also watch a short video clip (1Mb) of the creation of the circuit.

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### Why do this problem?

This problem gives the simplest introduction into logic gates and circuits. Through experimentation with switches, students will begin to see the structure of logic gates emerge without the need for any detailed formalism. They will then see that logic gate circuits can be constructed using a more complicated approach.

### Possible approach

Put the problem on to the board. Encourage students to read the problem and decide what the circuit board means. Encourage experimentation with combinations of off /off. At each stage, encourage students to describe what they can see. How can this sensibly be recorded?

Once students feel that they understand how a gate works they should write a sentence describing the action of the gate. Do others agree that this is a clear definition? How might is be improved? Could we use it in the definition of the behaviour of the other gates?

You might like to discuss with the class how the words 'and' 'or' and 'not' are used in real life. How does this relate to logic-speak? Students might enjoy inventing logic-speak sentences such as:

'I really like eating ice cream XOR chicken curry' (meaning I like ice cream and curry, but not at the same time)
'I like tea XNOR milk' (meaning I only like tea with milk)

### Key questions

• Describe what you see.
• What are we supposed to change? What are we supposed to leave fixed?
• How might we record our findings?
• What happens if the switches in the two pairs of circuits are set to the same values?
Once students feel that they have described the gates clearly using English, usethis follow up
• How could you represent the behaviour of the gate symbolically?

### Possible extension

Once the concept of the gate is understood there are several follow up questions, such as Simple Counting Machine .

### Possible support

You might suggest focussing on the AND, OR and NOT gates to begin with