GCSE Mathematics - Bad tomatoes
Dale Jacques 22nd June 2001
GCSE Mathematics
Bad tomatoes
Identical 'good' Tomatoes are placed in a tray.
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Tomato no. 5 is the bad tomato
Each tomato is a sphere.
Each tomato just touches all the tomatoes next to it as shown on the diagram.
Tomato 5 goes bad first.
One hour later, all the tomatoes it touches go bad (Now tomatoes 5, 1, 6 and 9 are bad). Another hour later the bad tomatoes make all the good tomatoes they touch go bad. This continues until all the tomatoes in the box are bad.
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E.g. no. 5 is the bad tomato 1-hour later tomatoes no.1, 6 and 9 are bad. Another hour later tomatoes 2, 7, 10 and 13 are bad. Another hour later tomatoes 3, 8, 11 and 14 are bad. Another hour later tomatoes 4, 12 and 15 are bad.
Another hour later tomato 16 is bad.
Hours
No of bad tomatoes
Bad tomato no.
st hour
2
, 6, 9
2nd hour
4
2, 7, 10, 13
3rd hour
4
3, 8, 11, 14
4th hour
3
4, 12, 15
5th hour
6
What would happen if tomato no.1 was the bad tomato?
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Hours
No of bad tomatoes
Bad tomato no.
st hour
2
2, 5
2nd hour
3
3, 6, 9
3rd hour
4
4, 7, 10, 13
4th hour
3
8, 11, 14
5th hour
2
2, 15
6th hour
6
What would happen if tomato no.6 was the first bad tomato?
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Hours
No of bad tomatoes
Bad tomato no.
st hour
4
2, 5, 7, 10
2nd hour
6
, 3, 8, 11, 9, 14
3rd hour
4
4, 12, 13, 15
4th hour
6
I have found out that if tomato no.5 goes bad first it would take the same amount ...
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Hours
No of bad tomatoes
Bad tomato no.
st hour
4
2, 5, 7, 10
2nd hour
6
, 3, 8, 11, 9, 14
3rd hour
4
4, 12, 13, 15
4th hour
6
I have found out that if tomato no.5 goes bad first it would take the same amount of time to make, the whole tray bad if tomato nos.2, 3, 12, 14, 15, and 9 were the 1st tomato to go bad first it would take 5 hours.
I have also found out that if tomato no.6 goes bad first it would take the same amount of time to make the whole tray bad if tomato nos.7, 10 and 11 were the 1st tomatoes to go bad first. It would take 5 hours.
Another number pattern I have discovered is that if tomato no. 1 was the first to go bad it would take the same amount of time for the whole tray to go bad if tomato nos. 4, 13 or 16 were the first tomatoes to go bad.
It would take 6 hours.
In the next example I will change the size of the tray, this might effect the time it take for the whole tray to
go bad.
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Hours
No of bad tomatoes
Bad tomato no.
st hour
2
2, 6
2nd hour
3
3, 7, 11
3rd hour
4
4, 8, 12, 16
4th hour
5
5, 9, 13, 17, 21
5th hour
4
0, 14, 18, 22
6th hour
2
20, 24
7th hour
25
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Hours
No of bad tomatoes
Bad tomato no.
st hour
4
2, 6, 8, 12
2nd hour
6
, 3, 9, 11, 13, 17
3rd hour
6
4, 10, 14, 16, 18, 22
4th hour
5
5, 15, 19, 21, 23
5th hour
2
24, 20
6th hour
25
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Hours
No of bad tomatoes
Bad tomato no.
st hour
4
3, 7, 9, 13
2nd hour
7
2, 4, 6, 10, 12, 14, 18
3rd hour
7
, 5, 11, 15, 17, 19, 23
4th hour
4
6, 20, 22, 24
5th hour
2
21, 25
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25
Bad tomato no. 13
Hours
No of bad tomatoes
Bad tomato no.
st hour
8,12,14,18
4
2nd hour
3,7,9,11,17,23,19,15
8
3rd hour
2,6,16,22,24,20,4,10
8
4th hour
,5,21,25
4
In this example I will change the size of the tray, this might affect which no. Tomato will be the last one to go bad.
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Hours
No of bad tomatoes
Bad tomato no.
st hour
2
2, 6
2nd hour
3
3, 7, 11
3rd hour
4
4, 8, 12, 16,
4th hour
5
5, 9, 13, 17, 21
5th hour
4
0, 14, 18, 22
6th hour
3
5, 19, 23
7th hour
2
20, 24
8th hour
25
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Hours
No of bad tomatoes
Bad tomato no.
st hour
3
,8,13
2nd hour
4
2,9,14,19
3rd hour
5
3,10,15,20,22
4th hour
6
4,11,16,21,26,31
5th hour
6
5,12,17,22,27,32
6th hour
6
6,18,23,25,33
7th hour
3
24,29,34
8th hour
2
30,35
9th hour
36
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Hours
No of bad tomatoes
Bad tomato no.
st hour
4
2,7,9,14
2nd hour
6
,3,10,13,15,20
3rd hour
6
4,11,16,19,21,26
4th hour
7
5,12,17,22,25,27,32
5th hour
6
6,18,23,28,31,33
6th hour
3
24,29,34
7th hour
2
30,35
8th hour
36
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Hours
No of bad tomatoes
Bad tomato no.
st hour
3
,3,8
2nd hour
4
4,7,9,14
3rd hour
5
5,10,13,15,20
4th hour
6
6,11,16,19,21,26
5th hour
6
2,17,22,25,27,32
6th hour
5
8,23,28,31,33
7th hour
3
24,29,34
8th hour
2
33,30
9th hour
36
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Hours
No of bad tomatoes
Bad tomato no.
st hour
3
2,4,9,
2nd hour
5
,5,8,10,15
3rd hour
6
6,7,11,14,16,21
4th hour
6
2,13,17,20,22,27
5th hour
6
8,19,23,26,28,33
6th hour
5
24,25,29,32,34
7th hour
3
31,30,35
8th hour
36
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Hours
No of bad tomatoes
Bad tomato no.
st hour
4
3,13,16,20
2nd hour
7
2,7,9,16,1921,26
3rd hour
8
,3,10,17,22,25,27,33
4th hour
7
4,11,18,23,28,31,33
5th hour
5
5,12,24,29,34
6th hour
2
30,35
7th hour
36
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Hours
No of bad tomatoes
Bad tomato no.
st hour
4
9,14,16,21
2nd hour
7
3,8,10,13,17,20,22,27
3rd hour
8
2,4,7,11,18,19,23,26,28,33
4th hour
7
,5,12,24,25,29,32,34,
5th hour
4
6,30,31,35
6th hour
36
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Now I will look for a pattern in my results.
For a 4x4 tray L=4 I will put the results into a table and connect the no. Of hours it takes for the whole tray to go bad. I will try and do different formulae for each diagonal.
For diagonal 1 (The corners)
L= No.
4 6
5 8 N=2L
6 10
Middle
L No.
4 5
5 7 N=2L-3
6 9
3rd diagonal
L No.
4 4
5 6 N=2L-4
6 8
2L-Diagonal+1
N=2L-(D+1)
Now I need to test my formula.
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For a 3x3 corners
L=3
D=1
So n should be
N=2L - (D+L)
=2 x 3 - 1+ 1
=6 - 2
=4