2082
/www/nrich/html/content/01/12/six3/
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2011-02-01T00:00:01
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<p>Using 3 rods of integer lengths, none longer than 10 units and
not using any rod more than once, you can measure all the lengths
in whole units from 1 to 10 units. How many ways can you do
this?</p>
<p><mdo:image src="rods2.gif" alt=""></mdo:image></p>
<p>For example with rods of lengths $3, 4, $ and $9$ the
measurements are:</p>
<p>$4-3,$ $9-4-3,$ $3,$ $4,$ $9-3,$ $9-4,$ $3+4,$ $9+3-4,$ $9,$
$9+4-3,$</p>
<p>Using 3 rods of ANY integer lengths, what is the greatest length
N for which you can measure all lengths from 1 to N units
inclusive? Can you beat 10 units? Can you beat the highest value of
N submitted to date?</p>
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<p class="editorial">There was a great deal in this problem and
James has answered part of it:</p>
<p>Using only three rods with each one not exceeding 10 you can add
and minus the following numbers which can go from 1-13.</p>
<p>The three numbers are 9, 3 and 1.</p>
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<td>1</td>
<td>=</td>
<td>1</td>
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<td>2</td>
<td>=</td>
<td>3-1</td>
</tr>
<tr>
<td>3</td>
<td>=</td>
<td>3</td>
</tr>
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<td>4</td>
<td>=</td>
<td>3+1</td>
</tr>
<tr>
<td>5</td>
<td>=</td>
<td>9-3-1</td>
</tr>
<tr>
<td>6</td>
<td>=</td>
<td>9-3</td>
</tr>
<tr>
<td>7</td>
<td>=</td>
<td>9-3+1</td>
</tr>
<tr>
<td>8</td>
<td>=</td>
<td>9-1</td>
</tr>
<tr>
<td>9</td>
<td>=</td>
<td>9</td>
</tr>
<tr>
<td>10</td>
<td>=</td>
<td>9+1</td>
</tr>
<tr>
<td>11</td>
<td>=</td>
<td>9+3-1</td>
</tr>
<tr>
<td>12</td>
<td>=</td>
<td>9+3</td>
</tr>
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<td>13</td>
<td>=</td>
<td>9+3+1!</td>
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<p class="editorial">Well done James. Is 13 the largest number and
why?</p>
<p class="editorial">Can any of you help with some of the rest of
the question:</p>
<p class="editorial">In how many ways can you make 10?</p>
<p class="editorial">What is the maximum number you can make with 4
rods?</p>
<p class="editorial">Perhaps you can send in some more ideas.</p>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This excellent <a href="http://nrich.maths.org/public/viewer.php?obj_id=2082&part=">problem</a> is so very good for number awareness, reinforcement and addition and subtraction awareness.</div>
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<h3>Possible approach</h3>
<div>Starting off in a very practical way with suitable rods would be ideal in many circumstances.</div>
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<h3>Key questions</h3>
<div>How did you get to this solution?</div>
<div>I see you've not got a (suppose - 9 when using 2,3 & 5 ) can you explain that?</div>
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<h3>Possible extension</h3>
<div>What about four rods?</div>
<div>Which combinations work/not work and why?</div>
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<h3>Possible support</h3>
<div>text</div>
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<p>There are exactly 9 different solutions for N=10 units with no
rod longer than 10 units. Can you find them? With one of these
solutions you can measure up to N=13. Can you do better if you can
choose 3 longer measuring rods?</p>
<p>If you double the lengths of all three rods in one of these
solution sets you can measure even lengths from 2 to 2N units.
Introducing a one unit rod to make up a set of four measuring rods
you can now measure all lengths from 1 to (2N+1) units. Can you do
better than 27 units with 4 rods?</p>
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Rod Measures
Using 3 rods of integer lengths, none longer than 10 units and not
using any rod more than once, you can measure all the lengths in
whole units from 1 to 10 units. How many ways can you do this?
Calculations and Numerical Methods
Addition & subtraction
Decision Mathematics and Combinatorics
Combinations