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We received lots of good solutions to this
problem - well done everyone! Many of you spotted that Mr McGregor
should put $7$ plants in his potting shed at the beginning, and put
$8$ plants in each garden. Well done to Henry from Finton House
School, Ruth from Manchester High School for Girls, Liam from
Wilbarston School, Mel from Christ Church Grammar School, Rachel
from Beecroft Public School in Australia, Yanqing from Devenport
High School for Girls and Daniel from Junction City High School for
their detailed explanations of how they arrived at the
Henry from Finton House School
"$1 \times2 \times2 \times2 = 8$.
8 therefore seems likely to be the number in the garden. Let's try
The number in the shed at the end must be the number in the garden.
Now what number do we double to get to $8$? It must be $4$.
$4 + 8 = 12$. $12$ divided by $2 = 6$.
$6 + 8 = 14$. $14$ divided by $2 = 7$."
Liam used similar logic:
"Just work backwards from the last garden. Imagine there to be $8$
plants in each garden. (You can't have odd numbers in a garden as
the last garden must be double the whole number of plants left
after the 2nd garden was planted. I chose $8$ because it's a
conveniently sized power of 2.) There must have been $4$ plants
left after the 2nd garden was planted so before it was planted
there must have been $12$ which is double $6$. $6+8=14$. So Mr
McGregor needs to put $14/2$ or $7$ plants in his magic potting
shed at the beginning!"
Yanqing and Rachel used algebra. Here is
"First, we make the number of plants put in the shed $n$, and the
number planted each night $x$. So by the first morning, the number
has doubled to $2n$ in the shed. We plant $x$ of them, leaving
$2n-x$ in the shed overnight. By the second morning, we have
$2(2n-x)=4n-2x$ in the shed. Planting $x$ of them, we are left with
$4n-2x-x=4n-3x$ in the shed. By the third morning, there should be
$2(4n-3x)=8n-6x$ plants in the shed. There need to be $x$ plants in
the shed, as we need to plant all of them, so $8n-6x=x$ and
We can now say that the ratio of $n$ to $x$ is $7:8$, so the
smallest values for $n$ and $x$, where they are both positive whole
numbers, are obviously $7$ and $8$.
Other numbers which will work are all multiples of $7$."
Rachel also found that $8n= 7x$ and
"Now you can see that $8n$ or $7x$ could equal $56$, which
makes $n = 7$ and $x = 8$.
This works when you try it out, and if you multiply both numbers by
another number, those new numbers work too."
Daniel concluded that:
"If you want to have the same amount of plants in each garden
you must start with a multiple of $7$ plants in the shed and each
day plant the same multiple of $8$ plants in the garden."
James from C.G.S.B found such a
"Start with $35$ ... then put $40$ in each garden"
And so did Mel from Christ Church Grammar
"You start off with $301$ plants in the shed. You put $344$ in