Count the digits
Problem
Count the Digits printable sheet
We can do all sorts of things with numbers - add, subtract, multiply, divide, ...
Most of us start with counting when we are very little. We usually count things, objects, people, etc. In this activity we are going to count the number of digits that are the same.
There are a couple of rules about the number we start with:
Rule 1 - The starting number has to have just three different digits chosen from $1, 2, 3, 4$.
Rule 2 - The starting number must have four digits - thousands, hundreds, tens and ones.
For example, we could choose $2124$ or $1124$.
When we've got our starting number we'll do some counting. Here is an example.
Starting Number:
| 4 | 1 | 3 | 4 |
We will count in order the number of $1$s, then the number of $2$s, then $3$s and lastly $4$s, and write it down as shown here.
| 4 | 1 | 3 | 4 | ||||
| 1 | 1 | 1 | 3 | 2 | 4 | ||
| 3 | 1 | 1 | 2 | 1 | 3 | 1 | 4 |
| 4 | 1 | 1 | 2 | 2 | 3 | 1 | 4 |
| 3 | 1 | 2 | 2 | 1 | 3 | 2 | 4 |
The first count gave one $1$, one $3$ and two $4$s.
We have continued this underneath, so the third line shows that the line above had three $1$s, one $2$, one $3$ and one $4$.
The fourth line counts the digits in the line above it, giving four $1$s, one $2$, two $3$s and one $4$.
And so it goes on until... until when?
Your challenge is to start with other four-digit numbers which satisfy the two rules and count the digits the way we did in the picture above.
What do you notice?
What happens if you have five digits in the starting number instead?
Getting Started
Student Solutions
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.
Teachers' Resources
Why do this problem?
This activity, in line with the theme for this month, offers an 'action' to perform on a group of numbers which pupils can continue and explore. Or, you could think of the writing down of the 'description' of a sequence as an action performed on that sequence. It might particularly appeal to those pupils who enjoy number work but who are perhaps not used to succeeding in this area.
Possible approach
Introduce the task by taking an example and work it through with the group/class of pupils, emphasising how careful we have to be with the simple act of counting.
Key questions
Tell me about what you see happening.
Possible extension
Change the rules so that only odd numbers are available, for example:
Possible support
Some pupils may need help in carefully counting the number of occurrences of each digit. It might, therefore, be useful for children to work in pairs so that someone else is always checking the counting.