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Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

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Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum.

More Magic Potting Sheds

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

This problem requires students to work systematically and challenges them to arrive at further generalisations. It is an excellent exemplar of a context where a full understanding of one problem is really useful when applied to the next problem. It is useful as an object lesson in mathematical process skills.

The problem can be tackled on paper but access to the interactivity will speed up the process of exploring different possibilities. It is important to have paper available for students to record the results of the different trials.

Teacher's Notes

Start with Magic Potting Sheds

Once students have found several answers to this problem, bring the class together.

Suggest that if they can understand why the solutions involve multiples of $7$ and $8$ they will find it a lot easier to solve similar problems.

Ask students if they can explain why the solutions are multiples of $7$ and $8$.

When I have used this with students, I've found that a particularly effective intervention at this point is to offer the following way of picturing the situation:

Imagine Mr McGregor places his $7$ plants on three shelves:
$4$ on the bottom shelf, ready for planting the next day
$2$ on the middle shelf, ready for planting the day after
$1$ on the top shelf, ready for planting on the last day.

On the first night they become $8, 4$ and $2$ and the $8$ are planted in the first garden.

That leaves $4$ on the middle shelf and $2$ on the top shelf.

The following night they become $8$ and $4$, and the $8$ are planted in the second garden.
That leaves $4$ on the top shelf.

On the last night the $4$ become $8$, and they are planted in the third garden.

So $7$ is significant because it is the sum of $1, 2$ and $4$ (the smallest triple of numbers that have the required doubling relationships).

Equally, Mr McGregor could have placed $5, 10$ and $20$ plants on his three shelves (multiples of $\{ 1, 2, 4 \}$):
The $20$ will become $40$ after one night.
The $10$ will become $40$ after two nights (double and double again).
The $5$ will become $40$ after three nights (double and double and double again).

Students can now be set to work on More Magic Potting Sheds, using this or other insights to help them along the way. Encouraging students to reflect on the solutions to the original problem may help them tackle the follow-up questions in a more informed way.

The intention is to avoid the situation in which students just sit at the computers typing any value in until they eventually hit upon the answer. It may be a good idea to set students working on the follow-up questions with paper and pencil, and only allow them to move to the computers once they have clear ideas about what they should try.