Count the Digits

We had just a few solutions sent in, here is one from Christopher, Connor and Alex from Pakuranga Heights School in New Zealand who wrote:-

A five digit would still be in thousands it would just be in the ten thousands.


Next Miss Stanley's Numeracy group from Greystoke Leicester wrote:-

We liked this challenge and worked very hard. Trying other numbers using the same rules we found that we could continue until the numbers were the same, because that number would keep repeating. We discovered some have shorter sequences and some have longer sequences until the same number repeats, but we're not sure why yet.
Caitlin and Millie found that some numbers (4122) didn't seem to have an end because we spotted the pattern that it kept repeating itself, so we decided to stop. Some of us even moved onto extending this challenge to 5, 6,and even 7 digits.
Many of us spotted that the larger the number of digits in the starting number, the shorter the sequence was to get to the end.
Thank you, we enjoyed this challenge.


Well done to all the contributors, it sounds as if you really enjoyed this. Another submission came from Adam at Cypress School who noted something special about 1 & 4. We then received this excellent presentation from Oscar from Spain.

You have to take out one digit of the 1,2,3,4 which are the possible digits to make the starting number. If you take out a number and want to get a 4 digit number, you have to repeat one of the other 3 numbers. If you take out 1, you have possible starting numbers 2234, 2334, 2344 and other possible numbers that you get changing the order of the digits in each of those 3 numbers. As the order does not affect digit counting, those give the same counting sequence. The counting is:
A    2    B    3    C    4
and A-B-C have to be 2-1-1 (for 2234) or 1-2-1 (for 2334) or 1-1-2 (for 2344).
The next counting in all cases is 2 1 2 2 1 3 1 4 and sequence is:
3 1 3 2 1 3 1 4
3 1 1 2 3 3 1 4 and this last number stays the same if you count the digits. If you take out 2 you get the number: 2 1 3 2 2 3 1 4 If you take out 3 you get the same as if you take out 1, and if you take out 4 you get the same as if you take out 2. 

Although there were not many solutions sent in they were all very interesting. We hope that you can have further thoughts about this challenge.