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From Year $4$ at Queen Edith School, Cambridge we had the following rather good idea sent in.

After we had got the idea of following a number on its journey, we split up the work of checking out lots of numbers. Some of us began with numbers in the $30$s, some with numbers in the $40$s, and so on.
We found that all the numbers we tried ended up on one of three journeys:
$2, 4, 8, 16, 14, 10, 2, 4, 8,$ ... which we called the "red" journey
$6, 12, 6, 12,$ ... which we called the "green" journey
$18, 18, 18,$ ... which we called the "blue" journey 
Next, we used a $100$ square on the Smartboard, and coloured the numbers to match their journeys. After we had coloured a few of the numbers, some of us spotted patterns beginning to show, like the blue diagonal from $81$ up to $9$. We predicted that other numbers on the diagonal would also be blue and checked them out. We also saw green squares along diagonals and made more predictions.
 100 sq

Finally, we made a display using the 100 square and some of our work to challenge other children to predict the journeys for some of the squares we had not coloured.
Can you predict a journey and then check if you were right?

From Krystof in  Prague and  Matthew from Hamworthy Middle School we had had similar results.  From Karin in West Acton in London we had a clever further idea sent in.

  My rule for "Follow the Numbers" is to work out the difference between the $2$ digits and add $5$ to the difference.
Here is some of my "Follow the Numbers"
Starting number:$24      24,07,12,06,11,05,10,06...$
Starting number:$39      39,11,05,10,06,11...$
Starting number:$83      83,10,06,11,05,10...$
Starting number:$63      63,08,13,07,12,06,11,05,10,06... $
On my "Follow the Numbers", most of my numbers had a pattern of $08,13,07,12,06,11,05,10,06.$

Well done Karin, I like this very much, others of you could try your own rules.