1170
/www/nrich/html/content/03/05/penta2/
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2011-02-01T00:00:01
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<p>Here is an alternative and more unusual version of the "Think of a Number" game which you may have heard of before.</p>
<p class="c1" style="text-align: center;"><mdo:image alt="numbers 0-9" height="193" src="bubble.gif" width="336"></mdo:image></p>
<blockquote>
<p class="bold">Think of two whole numbers under $10$.<br></br>
Take one of them and add $1$.<br></br>
Multiply by $5$.<br></br>
Add $1$ again.<br></br>
Double your answer.<br></br>
Subtract $1$.<br></br>
Add your second number.<br></br>
Add $2$.<br></br>
Double again.<br></br>
Subtract $8$.<br></br>
Halve this number and tell me your answer.</p>
</blockquote>
<p>From your answer I can work out both your numbers very quickly. How?</p>
<p>Try with different numbers until you see how it works and then you'll be ready to amaze your friends!</p>
<p><a href="http://nrich.maths.org/public/viewer.php?obj_id=5895&part=">Click here for a poster of this problem</a>.</p>
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<p><span class="editorial">Although we had lots of answers to this problem, very few of you explained your solutions carefully. Alistair from Histon and Impington Infant School sent us this solution:</span></p>
<br></br>
I started by working out a few answers, and spotted that (if A is the first number and B is the second), A goes up and the number of the answer goes up by 10, and B goes up and the answer goes up by 1. My chart:<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td> </td>
<td style="font-weight: bold;">1</td>
<td style="font-weight: bold;">2</td>
<td style="font-weight: bold;">3</td>
<td style="font-weight: bold;">4</td>
<td style="font-weight: bold;">5</td>
</tr>
<tr>
<td style="font-weight: bold;">1</td>
<td>20</td>
<td>30</td>
<td>40</td>
<td>50</td>
<td>60</td>
</tr>
<tr>
<td style="font-weight: bold;">2</td>
<td>21</td>
<td>31</td>
<td>41</td>
<td>51</td>
<td> </td>
</tr>
<tr>
<td style="font-weight: bold;">3</td>
<td>22</td>
<td>32</td>
<td>42</td>
<td>52</td>
<td> </td>
</tr>
<tr>
<td style="font-weight: bold;">4</td>
<td>23</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td style="font-weight: bold;">5</td>
<td>24</td>
<td> </td>
<td> </td>
<td> </td>
<td>
<div> </div>
</td>
</tr>
</tbody>
</table>
<br></br>
So then I made a new chart, showing what numbers you would get at different steps of the operations:<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td>A</td>
<td>1</td>
<td>1</td>
<td>1</td>
<td>2</td>
<td>3</td>
<td>4</td>
<td>2</td>
<td> </td>
</tr>
<tr>
<td>B</td>
<td>1</td>
<td>2</td>
<td>3</td>
<td>1</td>
<td>1</td>
<td>1</td>
<td>2</td>
<td> </td>
</tr>
<tr>
<td>A+1</td>
<td>2</td>
<td>2</td>
<td>2</td>
<td>3</td>
<td>4</td>
<td>5</td>
<td>3</td>
<td>A+1</td>
</tr>
<tr>
<td>x5</td>
<td>10</td>
<td>10</td>
<td>10</td>
<td>15</td>
<td>20</td>
<td>25</td>
<td>15</td>
<td>5A+5</td>
</tr>
<tr>
<td>+1</td>
<td>11</td>
<td>11</td>
<td>11</td>
<td>16</td>
<td>21</td>
<td>26</td>
<td>16</td>
<td>5A+6</td>
</tr>
<tr>
<td>x2</td>
<td>22</td>
<td>22</td>
<td>22</td>
<td>32</td>
<td>42</td>
<td>52</td>
<td>32</td>
<td>10A+12</td>
</tr>
<tr>
<td>-1</td>
<td>21</td>
<td>21</td>
<td>21</td>
<td>31</td>
<td>41</td>
<td>51</td>
<td>31</td>
<td>10A+11</td>
</tr>
<tr>
<td>+B</td>
<td>22</td>
<td>23</td>
<td>24</td>
<td>32</td>
<td>42</td>
<td>52</td>
<td>33</td>
<td>10A+11+B</td>
</tr>
<tr>
<td>+2</td>
<td>24</td>
<td>25</td>
<td>26</td>
<td>34</td>
<td>44</td>
<td>54</td>
<td>35</td>
<td>10A+13+B</td>
</tr>
<tr>
<td>x2</td>
<td>48</td>
<td>50</td>
<td>52</td>
<td>68</td>
<td>88</td>
<td>108</td>
<td>70</td>
<td>20A+26+2B</td>
</tr>
<tr>
<td>-8</td>
<td>40</td>
<td>42</td>
<td>44</td>
<td>60</td>
<td>80</td>
<td>100</td>
<td>62</td>
<td>20A+18+2B</td>
</tr>
<tr>
<td>/2</td>
<td>20</td>
<td>21</td>
<td>22</td>
<td>30</td>
<td>40</td>
<td>50</td>
<td>31</td>
<td>10A+9+B</td>
</tr>
</tbody>
</table>
<br></br>
<br></br>
So the answer is always 10A + B + 9. If you are told the answer, you must take away nine, then the digits of the answer are the numbers that your friend thought of.<br></br>
<br></br>
<p><span class="editorial">Chris, Isobel, Wui Shen and Alex from Maadi British International School in Cairo noticed something else:</span></p>
<p>We thought about the problem and worked it out together first. When we had our answer we tried to work out how to get our first numbers.</p>
<p>Wui Shen said we could take 1 away from the tens column and + 1 to the units column.</p>
<p>Isobel said we could + the two numbers and find two numbers that equal the same number.</p>
We then each did our own sum and discovered Wui Shen's worked every time but Isobel's didn't because there was more than one answer to the sum.<br></br>
<p class="editorial">Jordan from London explained why Wui Shen's method works:</p>
10A + B + 9 gives the final answer.<br></br>
If you subtract 9 from this you are left with 10A + B.<br></br>
<br></br>
A and B are whole numbers less than 10; this means that multiplying A by 10 would give a result with 0 in the units column and A in the tens column.<br></br>
After that the addition of B to 10A would put B in place of the 0, so A would be the tens digit and B would be the units digit.<br></br>
<br></br>
When 9 is taken away from a number it is the same as subtracting 10 (the tens column decreases by 1), and then adding 1 (the units column increases by 1).<br></br>
<br></br>
Therefore decreasing the tens column value by 1 and doing the opposite to the units column value has the same effect as subtracting 9. Both leave you with A and B as the two digits.<br></br>
<br></br>
This is what Wui Shen found.<br></br>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org/1170&part=">This
problem</a> can be solved by both trial and improvement and by
using simple algebra. It is the intriguing kind of "puzzle-game"
that can be taken from the maths classroom into the playground! A
poster of this problem is available <a href="http://nrich.maths.org/5895&part=">here</a>.<br></br>
<h3>Possible approach</h3>
<div>You could introduce the
problem as it appears on the site as a printed sheet or on a
computer. Learners could first work individually to give them
'thinking time', then work in pairs to support each other and to
give an opportunity for mathematical talk, and finally there could
be a class discussion.</div>
<div> </div>
<div>A concluding plenary
could ask them to share any insights and strategies that helped
them succeed at this task.</div>
<br></br>
<h3>Key questions</h3>
<div>Have you tried with
several numbers to see what is happening?</div>
<div>What can you say about
the answer and the first number that was chosen?</div>
<div>What can you say about
the answer and the second number that was chosen?</div>
<div>Have you tried doing it
with someone else whose numbers you do not know?</div>
<div>Have you tried using two
letters in place of the two numbers?</div>
<br></br>
<h3>Possible extension</h3>
Learners could go on to <a href="http://nrich.maths.org/2922&part=">Multiply the Addition
Square</a>.<br></br>
<h3>Possible support</h3>
Suggest trying with different numbers, thus practising simple
calculation, even if the generalising is not done.<br></br>
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The tens digit tells you one number and the unit digit tells you
the other.
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2
4
0
0
1
0
0
Think of Two Numbers
Think of two whole numbers under 10. Take one of them and add 1.
Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your
second number. Add 2. Double again. Subtract 8. Halve this number
and tell me your answer. From your answer I can work out both your
numbers very quickly. How?
Numbers and the Number System
Number - generally
Algebra
Algebra - generally
Calculations and Numerical Methods
Number operations - generally
Information and Communications Technology
smartphone
Secondary Mapping Document
Equations and formulae LS
Secondary Mapping Document
DisplayCabinet