# Frogs Investigation - look at your results and try to find a formula which gives the least number of moves needed for any number of discs x .It may help if you can count the number of hops and slides separately .

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Introduction

Name:Maame Yaa Buckman

Date:14th February 2008

Subject: Mathematics

Project title:Frogs

Submitted to:Mr. John Essuman

(Mathematics Department)

Content Page

Introduction to frogs Page 3

Example of the game Page 4

Question 1 Pages 5&6

Question 2 Pages 7, 8&9

Question 3 Pages 10, 11, 12&13

Formula 1 Page 14

Question 4 Page 15

Question 5 Pages 16&17

Question 6 Pages 18, 19 &20

Question 7 Pages 21, 22, 23&24

Formula Page 25

Conclusion Page 26

This was a game invented by the French mathematician called Lucas .He named it Frogs.

The Aim of The Game

To gain success in this interesting game you need to swap the positions of the disc so that they end up the opposite way around. (With a space in the middle)

The rules are as follows:

- A disc can slide over one square in either direction onto an empty square.
- A disc can hop over one adjacent disc of the other color provided it can land on an empty square.

Take a look at this example

A) Slide one square to the right.

B) Hops over to the left.

C) Slide one square to the right.

Conclusion: Number of hops=1

Number of slides=2

Total number of moves=3

Question 1: Try the same thing this time with 2 discs of each colour.

Step 1

Step 2

Step 3

Step 4

Middle

Step 16 –B2 hops to the left

Step 17 –B3 hops to the left

Step 18 –B4 hops to the left

Step 19 –A3 slides to the right

Step 20-A2 hops to the right

Step 21-A1 hops to the right

Step 22-B3 slides to the left

Step 23-B4 Hops to the left

Step 24-A1 slides to the right

Conclusion: Number of hops=16

Number of slides=8

Total number of moves=24

Now look at your results and try to find a formula which gives the least number of moves needed for any number of discs x .It may help if you can count the number of “hops” and “slides” separately .

Number of frogs (x) | Number of hops (x²) | Number of slides (2x) | Number of moves (X²+2x) |

1 | 1 | 2 | (1+2) = 3 |

2 | 4 | 4 | (4+4) = 8 |

3 | 9 | 6 | (9+6) = 15 |

4 | 16 | 8 | (16+8)=24 |

Therefore to find the number of moves you can apply this formula :

X²+2x |

Question 4: Try the game with a different number of discs on each side . Say 1 green and 2 blues

Step 1-A slides to the right

Step 2-B1 hops to the left

Step 3-A slides to the right

Step 4 –B2 hops to the left

Step 5-A slides to the left

Conclusion: Number of hops =2

Number of slides=3

Total number of moves =5

Question 5: Try this with 2 greens and 3 blues .

Step 1-A2 slides to the right

Step 2-B1 hops to the left

Step 3-B2 slides to the left

Step 4-A2 hops to the right

Step 5-A1 hops to the right

Step 6-B1 slides to the left

Step 7-B2 hops to the left

Step 8-B3 hops to the left

Step 9 –A2 slides to the right

Step 10 –A1 hops to the right

Step 11-B3 slides to the left

Conclusion: Number of hops=6

Number of slides=5

Total number of moves=11

Question 6: Try this for 3 green and four blues .

Step 1-A3 slides to the right

Step 2 –B1 hops to the left

Step 3-B2 slides to the left

Step 4-A3 hops to the right

Step 5-A2 hops to the right

Step 6-A1 slides to the right

Step 7 –B1 hops to the left

Step 8- B2 hops to the left

Step 9-B3Hops to the left

Step 10 –B4 slides to the left

Step 11-A3 hops to the right

Step 12-A2 hops to the right

Step 13-A1 hops to the right

Step 14-B2 slides to the left

Conclusion

Step 22-B4 hops to the left

Step 23 –B5 hops to the left

Step 24-A3 slides to the right

Step 25-A2 hops to the right

Step 26 –A1 hops to the right

Step 27 –B4 slides to the left

Step 28-B5 hops to the left

Step 29- A1 slides to the right

Conclusion: Number of hops=20

Number of slides=9

Total number of moves=29

After you have played the game with different combinations and again try to find a formula giving the number of moves for “x” discs of one colour and “y” discs of another colour.

- Let X represent one colour
- And Y Represent another colour.

Number of one colour (X) | Number Of 2nd colour (Y) | Number of hops (XY) | Number of slides (X+Y) | Total number of moves [XY+(X+Y)] |

1 | 2 | 2 | 3 | 5 |

2 | 3 | 6 | 5 | 11 |

4 | 3 | 12 | 7 | 19 |

4 | 5 | 20 | 9 | 29 |

Use the formula below to calculate the number of moves as well as the number of slides and hops.

Xy+(X+Y) |

Let us try this formula given 4 greens and 5 blues

4×5+ (4+5)

Therefore, 20+9 where 20 = number of hops.

9=number of slides.

20 +9 =29 (this is the total number of moves.)

Conclusion

At the end of this interesting game you gain a lot of intuitive about how to obtain a formula from a random game .It also teaches you how to look at patterns .If such skills are obtained it can be applied to ones daily life.

This student written piece of work is one of many that can be found in our GCSE Miscellaneous section.

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## Here's what a teacher thought of this essay

**

The general questions of the investigation are answered. To improve this investigation more algebraic manipulation is needed to verify the identified pattern. There should be more testing of all algebraic formula to prove their validity. Specific strengths and improvements have been suggested throughout.

Marked by teacher Cornelia Bruce 18/07/2013