Push Penny

Playing the Game
You will need:
A grid of 4 cm by 4 cm squares (see Figure 2), a penny, page R 1.

Figure 2 - Push penny grid.

In this game you have to push a penny on to a square grid. If your penny falls off the grid, it is returned and you try again. You win if it lands completely inside a square. You will find out how hard this is, and decide what prize to award.

Table 1 shows the results of 20 pushes at a fair in Sheffield.

Use Table 4 on page R1 to record your 20 results like these. Your teacher will show you how.

	Tally	Total
Completely inside a square		6
Not inside a square		14
	Total	20

Table 1 - Sheffield results at push penny.

What Prize?

The stallholder keeps the penny you roll. If the penny lands completely inside the square, you win 2p. With the Sheffield results and a 2p prize, the profit to the stall is given by:

*A Simpler Game
You will need:
A penny, a sheet of centimetre square graph paper.

How likely are you to win at push penny? To find this out we start by looking at a simpler game. This is played on a board with parallel lines 4 cm apart (see Figure 3).

Figure 3 - Board for a simpler game.

You push the penny on to the board. You win if the penny lies completely between the lines.

REMEMBER A penny is a circle with 1 cm radius. This will help you decide where its centre lies.

Figure 4 - Moving the penny.

You should have a diagram like Figure 5.

Figure 5 - Winning positions.

While the penny was moving from position 1 to position 2, you had:

1 cm of losing positions, followed by 2 cm of winning positions, and then 1 cm of losing positions.

2 cm out of the 4 cm were winning positions.

When no skill is involved, the centre of the coin is equally likely to land anywhere in the 4 cm.

So the probability of winning is ²/₄ or ¹/₂

On your graph paper draw separate diagrams and shade the winning positions for parallel lines.

Work out the probability of winning when the parallel lines are:

The push penny grid is like two of these boards superimposed at right-angles to each other.

Push Penny Probabilities
You will need:
A penny, a sheet of centimetre square graph paper.

Figure 6 - Winning positions at push penny.

Look at Figure 6.

The penny does not touch the parallel lines AB or CD when its centre lies in the region shaded:

The penny does not touch the parallel lines VW or XY when its centre lies in the region shaded:

The penny will win when its centre lies in the square region shaded:

a	Draw a 4 cm square on your graph paper. Draw in the regions and . Move your penny so that its centre is inside the large square. Cheek that: When the centre is in the region , the coin wins. When the centre is anywhere else in the square, the coin loses.

When the penny lands completely inside the large square, its centre is inside the smaller square.

The area of the large square is 4 cm x 4 cm = 16 cm².
The area of the smaller square is 2 cm x 2 cm = 4 cm².

4 cm² out of the 16 cm² are winning positions. When no skill is involved, the centre of the coin is equally likely to be anywhere in the 16 cm².

So the probability of winning is ⁴/₁₆ or ¹/₄.

This means we expect a penny to land in a square one time in four. So in 100 pushes we would expect 25 pennies to land completely inside a square.

If 25 out of 100 pushes are successful, then with a prize of 2p the profit the stall would expect is given by:

Suppose that 25 out of 100 pushes are successful.

Luck or Skill?
We have assumed that pushing pennies is a matter of luck.

*Different Size Squares
You will need:
A penny, a sheet of centimetre square graph paper.

In Section C4 the shaded inner square shows where the centre of a winning penny must lie. A penny has a diameter of 2 cm. In our example the sides of the outer square are 4 cm long, and the sides of the inner square are 2 cm long. The sides of the inner square are 2 cm less than the sides of the outer square.

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