7302
/www/nrich/html/content/id/7302/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p>We can do all sorts of things with numbers - add, subtract, multiply, divide ...<br></br>
<br></br>
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.<br></br>
<br></br>
There are a couple of rules about the number we start with:<br></br>
Rule 1 - The starting number has to have just three different digits chosen from $1, 2, 3, 4$.<br></br>
Rule 2 - The starting number must have four digits - so thousands, hundreds, tens and ones.<br></br>
For example, we could choose $2124$ or $1124$. <br></br>
<br></br>
So when we've got our starting number we'll do some counting. Here is a worked example. <br></br>
<br></br>
Starting Number <br></br>
<mdo:image alt="Start 1" height="20" src="Starting%20Nos%2C%201.jpg" width="85"></mdo:image><br></br>
We then 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.<br></br>
<br></br>
<mdo:image alt="Start 2" height="107" src="Starting%20nos%202.jpg" width="171"></mdo:image><br></br>
<br></br>
So the first count gave one $1$, one $3$ and two $4$s.<br></br>
You may see that this has continued so the third line shows that the line above had three $1$s, one $2$, one $3$ and one $4$.<br></br>
The fourth line counts the line above giving four $1$s, one $2$, two $3$s and one $4$.<br></br>
<br></br>
And so it goes on until ... until when?<br></br>
Your challenge is to start with other four digit numbers which satisfy the two rules and work on it in the way I did.<br></br>
Tell us what you notice.<br></br>
<br></br>
What happens if you have five digits in the starting number? <br></br>
</p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p class="editorial">We had quite a few responses from pupils who
had found out things by exploring this situation. Calum,
Christopher and Matthew from St. Andrews in Scotland wrote to
say:</p>
<p class="editorial"> </p>
<div>We found out that using the digits $1, 2$ and $3$ making a
four-digit number you would start doubling eventually when you
reach line $10$ every time no matter what order you put the digits
in.</div>
<div>We found out that using the digits $1, 2$ and $4$ making a
four-digit number you would start doubling eventually when you
reach line $7$ every time no matter what order you put the digits
in.</div>
<div>We found out that using the digits $1, 3$ and $4$ making a
four-digit number you would also start doubling eventually when you
reach line $7$ every time no matter what order you put the digits
in.</div>
<div>We found out that using the digits $2, 3$ and $4$ making a
four-digit number you would start doubling eventually when you
reach line $6$ every time no matter what order you put the digits
in. </div>
<br></br>
<br></br>
<p><span class="editorial">Sayeed from St. Michael's London also
sent in a well thought out reponse:-</span></p>
<p>This is my answer when starting with $4$ digits:</p>
<p>$3243, 122314, 21221314, 31321314,
31123314, 31123314$</p>
<p>It will continue as the same number forever.</p>
<p>I notice: Each row has two more digits then the previous row
until the rows have the maximum amount of digits ($8$) possible. In
each row with the maximum digits, the second digit always has a
$1$, the fourth digit always has a $2$, and the sixth digit always
has a $3$ and so on. Every row ends in $14$ excluding the starting
row. This is my answer when starting with $5$ digits:</p>
<p>$22411, 212214, 213214, 21221314,
31321314, 31123314, 31123314$</p>
<p>It will continue as the same number forever.</p>
<p>I notice the same thing that happens with the starting $4$ digit
row except the rows getting two more digits each time until there
are maximum digits. I also notice that with the starting four digit
row it takes four counts till the number continues as the same
number forever. But for the starting five digit row it takes five
counts till the number continues as the same number forever!
</p>
<p class="editorial">Thomas from Colet Court School said the
following and attached his numbers.</p>
When you start with $4$ digits the series converges to $21322314$
whatever the first $4$ digits. When you start with $5$ digits
it gives the same result.
<div> </div>
<div style="text-align: center;"> <mdo:image width="408" height="543" alt="Thomas" src="Thomas.jpg"></mdo:image></div>
<div> </div>
<p class="editorial">Finally, Miss Stanley's Numeracy group from
Greystoke Leicester wrote:-</p>
<p class="editorial"> </p>
<div>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.</div>
<div>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.</div>
<div>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.</div>
<div>Thank you, we enjoyed this challenge.</div>
<br></br>
<br></br>
<p> <span class="editorial">Well done to all the contributors,
it sounds as if you really enjoyed this. (A late arrival came from
Adam at Cypress School who noted something special about $1$ &
$4$. I am wondering if that was because it was not a usual
thing you'd find going on in many Mathematics lessons? On the last
day of the month we recieved this excellent presentation of Oscar
from Spain.</span></p>
<div>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</div>
<div>$2234, 2334, 2344$ and other possible numbers that you get
changing the order of the digits in each of those $3$
numbers.</div>
<div>As the order does not affect digit counting, those give the
same counting sequence. The counting is:</div>
<div>$A 2 B
3 C 4$</div>
<div>and A-B-C have to be $2-1-1$ (for $2234$) or $1-2-1$ (for
$2334$) or $1-1-2$ (for $2344$).</div>
<div>The next counting in all cases is $2 1 2 2 1 3 1 4$ and
sequence is:</div>
<div>$3 1 3 2 1 3 1 4$</div>
<div>$3 1 1 2 3 3 1 4$ 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
$2$.</div>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>Count the Digits</h2>
<br></br>
<p>We can do all sorts of things with numbers - add, subtract, multiply, divide ...<br></br>
<br></br>
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.<br></br>
<br></br>
There are a couple of rules about the number we start with:<br></br>
Rule 1 - The starting number has to have just three different digits chosen from $1, 2, 3, 4$.<br></br>
Rule 2 - The starting number must have four digits - so thousands, hundreds, tens and ones.<br></br>
For example, we could choose $2124$ or $1124$. <br></br>
<br></br>
So when we've got our starting number we'll do some counting. Here is a worked example. <br></br>
<br></br>
Starting Number <br></br>
<mdo:image alt="Start 1" height="20" src="Starting%20Nos%2C%201.jpg" width="85"></mdo:image><br></br>
We then 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.<br></br>
<br></br>
<mdo:image alt="Start 2" height="107" src="Starting%20nos%202.jpg" width="171"></mdo:image><br></br>
<br></br>
So the first count gave one $1$, one $3$ and two $4$s.<br></br>
You may see that this has continued so the third line shows that the line above had three $1$s, one $2$, one $3$ and one $4$.<br></br>
The fourth line counts the line above giving four $1$s, one $2$, two $3$s and one $4$.<br></br>
<br></br>
And so it goes on until ... until when?<br></br>
Your challenge is to start with other four digit numbers which satisfy the two rules and work on it in the way I did.<br></br>
Tell us what you notice.<br></br>
<br></br>
What happens if you have five digits in the starting number? <br></br>
</p>
<p style="font-style: italic;">Got a solution? Then click on Submit a Solution above.</p>
<p style="font-style: italic;"> </p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/7302&part=">This activity</a>, 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.</div>
<br></br>
<h3>Possible approach</h3>
<div>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. </div>
<div> </div>
<div>Give them plenty of time to explore their own choice of numbers before bringing them together to share findings. Depending on the age and experience of the learners, you may like to give them a separate sheet of paper simply to note down anything they notice as they work. Encourage them not to rub out as they go along so they have a record of their thoughts, to some extent. </div>
<div> </div>
<div>A whole-group discussion could focus on what they notice and what other questions they might have as a result of working on this task. Some children might be keen to try to explain their findings. Do encourage them in this, even if you are not sure of the reasons yourself. Admitting your possible uncertainty will spur them on! </div>
<br></br>
<h3>Key questions</h3>
<div>Tell me about what you see happening.</div>
<div>What will you do now?</div>
<div>Can you make any predictions before you start the next one? </div>
<br></br>
<h3>Possible extension</h3>
<div>Change the rules so that only odd numbers are available, for example:</div>
<div>Rule 1 - The starting number must have just three different digits chosen from $1, 3, 5, 7$ Rule 2 - The starting number should have four digits, so thousands, hundreds, tens and ones.</div>
<div>For example, $3155$ or $1135$.</div>
<div>Some children might like to find out about 'Golomb sequences' which are related to this task.</div>
<br></br>
<h3>Possible support</h3>
<div>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. </div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">It might help to ask someone else to check your counting!<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">21322314 comes from 1123 in 8 steps<br></br>
1223 in 8 steps<br></br>
1233 in 8 steps<br></br>
1134 in 5 steps<br></br>
1334 in 5 steps<br></br>
1344 in 5 steps<br></br>
<br></br>
31123314 comes from 1244 in 5 steps<br></br>
1124 in 5 steps<br></br>
1224 in 5 steps<br></br>
2234 in 4 steps<br></br>
2334 in 4 steps<br></br>
2344 in 4 steps <br></br></mdoxml>
5
3
1
1
0
0
0
Count The Digits
In this investigation we are going to count the number of 1s, 2s,
3s etc in numbers. Can you predict what will happen?
Using, Applying and Reasoning about Mathematics
Making and testing hypotheses
Using, Applying and Reasoning about Mathematics
Investigations
Numbers and the Number System
Counting
Admin
Lower primary mapping document
Admin
Upper primary mapping document