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The Vegetable Garden

Stage: 1 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

We had lots of correct solutions to this problem and many of you explained your thinking very clearly. Well done!

Musab from Orchard School answered the first part of the problem:

Ben might have started with a number 1 more than a multiple of 3 because it says when Ben put the garlic into groups of 3 there was 1 garlic plant left. For example Ben could have started with 7.

You're right, Musab. What other possibilities could there have been then? Pupils at St Mary's in Tetbury approached the second part of the problem in a very logical (or systematic) way. They said:

We decided to work out all the numbers it could be.

We started with 6 rows with one left over, and wrote down all the multiples of 6, add 1, up to 100.
Then we looked at 5 rows with one left over, and wrote down all the multiples of 5, add 1, up to 100.
We rubbed out all the numbers that weren't in both.

Then we looked at 4 rows with one left over, and wrote down any multiples of 4, add 1, that appeared in our lists for 5 rows and 6 rows.
Then we looked at 3 rows with one left over, and there was only one number that worked for all of them. We think the solution is 61.

PJ, Josh and Michael tried a few things out and gradually got close to the answer. This can be a very useful way of solving a problem (we call it trial and improvement):

We started by just trying numbers. Then Josh started doing multiples, and we found out some numbers that have one (in the units column?) work so we tried 41 then 51 then we found that 61 worked, and that's how we got our answer: 61.

Qiuying of Wimbledon High School explained the solution in slightly higher level maths. (Don't worry if you haven't come across these ideas yet.)

Less than 100:
The Number of garlic cloves equals the Lowest Common Multiple of 2, 3, 4, 5, 6 and then plus 1.
LCM = 60 So the Number is 60 + 1 = 61
The first one more than 100:
The numbers of garlic cloves equals 60 x 2 + 1 = 121