GCSE Maths Bad Tomato Investigation

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GCSE Maths Bad Tomato Investigation

A problem has been reported involving Bad tomatoes. At first there is a tray of regular tomatoes, then, all of a sudden, one of them goes bad. An hour later, all of the ones that were touching the bad tomato go bad. An hour after that all the tomatoes touching those go bad. This continues until the whole tray goes bad.

To start this investigation I am going to use a simple 4x4 box. Let's say, after one hour, a tomato goes bad. This is marked 1. After the second hour all those touching it go bad, those are marked 2 and so on.

2

3

4

5

2

3

4

2

3

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5

3

4

5

6

With a 4x4 tray, the time taken from the tomatoes being put into the tray to the last tomato going rotten is 6 hours.

Hour

No. of bad toms after that hour

No. of tomatoes that turn bad that hour

0

0

0

2

4

3

3

8

4

4

2

4

5

5

3

6

6

A pattern is beginning to form in the "No. of tomatoes that turn bad that hour" column. If we continue this investigation to larger sized trays then we will see that the pattern becomes even more prominent.

2

3

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5

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8

9

0

1

2

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5

6

7

8

9

0

2

3

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6

7

8

9

0

1

3

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6

7

8

9

0

1

2

4

5

6

7

8

9

0

1

2

3

5

6

7

8

9

0

1

2

3

4

6

7

8

9

0

1

2

3

4

5

7

8

9

0

1

2

3

4

5

6

8

9

0

1

2

3

4

5

6

7

9

0

1

2

3

4

5

6

7

8

(Numbers indicate the hour at which the tomato went bad, the colours identify the different sized boxes)

In each size box the tomato that goes bad first starts in the same position

Size (LxL)

Time taken for whole tray to go bad (T)

No. that go bad each hour

2x2

3

,2,1

3x3

4

,3,3,2

4x4

6

,3,4,4,3,1

5x5

8

,3,4,5,5,4,2,1

6x6

0

,3,4,5,6,6,5,3,2,1

7x7

2

,3,4,5,6,7,7,6,4,3,2,1

8x8

4

,3,4,5,6,7,8,8,7,5,4,3,2,1

9x9

6

,3,4,5,6,7,8,9,9,8,6,5,3,2,1

0x10

8

,3,4,5,6,7,8,9,10,10,9,7,6,5,4,3,2,1

NB - 2x2 does not work correctly because it has to start in a corner

One obvious pattern that emerges from this is that, as length increases by 1, the Time taken for the whole tray to go bad (T) goes up by 2. The formula for this is simple;

2L-2=T

The reason for this expression is as follows; when a tomato in the corner goes bad, the one in the opposite corner will be the furthest one away. In this expression the two sides on route from corner to corner are represented by 2L but there is 2 taken away from this to account for the original bad tomato and the corner tomato which the two sides share.

This formula will work for any sized square were the initial tomato is on an edge and next to a corner.

If we try to work out a general formula for the length of time it takes for a entire tray to go bad from any starting position we need to come up with a more general rule. Quite simply, the time it takes for a tray to go bad will be determined by the distance between the starting point and the furthest tomato from that point. If the starting point is changed then the previous explanation will no longer apply. I will now investigate the effect of placing the starting point in the centre of the tray. If L is odd then there will be a defiant centre, but if it is even then there will be 4 possible starting points. This doesn't really matter as the furthest tomato away will always be the same distance from any of the four starting points. If we try only even sized trays first, then we can find If this idea leads to any formula.
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4

3

4

5

3

2

3

4

2

2

3

3

2

3

4

_ = Starting Point

_ = Furthest Tomato

Obviously the furthest tomato away is the last to go bad

Size (LxL)

Time taken for whole tray to go bad (T)

2x2

3

4x4

5

6x6

7

8x8

9

0x10

1

2x12

3

4x14

5

6x16

7

8x18
...

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