4903
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2011-02-01T00:00:01
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<h5>By Henry Kwok</h5>
<br></br>
<div style="text-align: center;"><mdo:image height="600" width="600" src="Sudoku.gif" alt="Sudoku Product Puzzle"></mdo:image></div>
<br></br>
<h3>The Basic Rules of "Product Sudoku"</h3>
Like the conventional Sudoku, this Sudoku variant consists of a
grid of nine rows and nine columns subdivided into nine $3 \times
3$ subgrids. Like the Sudoku Classic, it has two basic rules:<br></br>
1. Each column, each row, and each box ($3 \times 3$ subgrid) must
have the numbers 1 to 9.<br></br>
2. No column, row or box can have two squares with the same
number.<br></br>
<br></br>
The puzzle can be solved with the help of the numbers in the top
parts of certain squares. These numbers are the products of the
digits in all the squares horizontally and vertically adjacent to
the square.<br></br>
<br></br>
<h3>A Short Demonstration</h3>
<div>The square in the top left corner of this Sudoku contains the
number 20. 20 is the product of the digits in the two adjacent
squares, which therefore must contain the digits 4 and 5. The 5
cannot go in the cell below the top left hand corner because 5 is
not a factor of 96 (the product shown in the third cell down on the
left hand side of the puzzle). Therefore 5 must be entered into the
cell to the right of the cell containing 20 and 4 in the cell
below.</div>
<br></br>
A word document containing the problem can be found <a href="/content/id/4903/Product%20sudoku.doc">here</a>, with an
explanation for classroom use.<br></br></mdoxml>
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<h3>Why do this problem?</h3>
<p>This is no ordinary Sudoku because it requires mathematical knowledge in addition to logical thinking. This problem offers an engaging context in which to apply knowledge of factors, multiples and prime factor decomposition.</p>
<h3>Possible approach</h3>
<p>If your students do not know the rules of Sudoku then set aside a little time for them to become familiar with the 'standard' Sudoku.<br></br>
<br></br>
Since this product Sudoku is quite challenging, you may wish to start with <a href="http://nrich.maths.org/public/viewer.php?obj_id=5919&part=">A First Product Sudoku</a> and <a href="http://nrich.maths.org/public/viewer.php?obj_id=6434&part=">Product Doubles Sudoku</a><br></br>
<br></br>
Display the first one without explaining anything and in silence fill in two adjacent cells.<br></br>
"I'm going to fill in a few more cells but I'm not going to explain what I'm doing.<br></br>
If you can work out what I'm doing and can suggest some other numbers then put up your hand.<br></br>
Please don't spoil it for anyone else by giving away what's happening.<br></br>
The title of the problem may give you a clue."<br></br>
<br></br>
Add contributions from the class, in each case asking for a justification, and continue until you feel they have the idea. Provide printed copies for those who wish to complete the problem in their own time and offer them a chance to try Product Doubles Sudoku before they move onto the Product Sudoku.<br></br>
<br></br>
Offer pairs of students one printed copy of the Product Sudoku and one copy of the blank 9 by 9 grid - the 'journey grid'.<br></br>
<mdo:image alt="" height="600" src="4903-blank%20grid.gif" width="600"></mdo:image><br></br>
<br></br>
Warn them that this is a little more challenging! On the Sudoku they are going to write the solution. On the journey grid they are going to record the order in which they fill in the cells, from 1-81.You are expecting them to convince each other of the accuracy of their suggestions before anything gets added onto their papers.<br></br>
<br></br>
When a pair has finished, they put their journey grid on display so that everyone's journeys can be compared. As a plenary, invite comments on any similarities and differences they noticed between their own and others' solutions.</p>
<h3>Key questions</h3>
<p>Some clues have lots of possibilities and some have few. Which are which?<br></br>
Which are the most helpful clues to begin?</p>
<h3>Possible extension</h3>
<div>Can one pair take another pair's journey grid and retrace their steps?</div>
<div>You may wish to try <a href="http://nrich.maths.org/public/viewer.php?obj_id=4928&part=">Product Sudoku 2</a></div>
<br></br>
<h3>Possible support</h3>
<p>Spend longer on <a href="http://nrich.maths.org/public/viewer.php?obj_id=5919&part=">A First Product Sudoku</a><br></br>
<br></br>
Provide them with this <a href="/content/id/4903/Product%20Sudoku%20-%20Journey1.pdf">possible journey</a> through Product Sudoku and suggest they try to retrace the route.<br></br>
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<br></br>
Rosie and Zoe solved this Sudoku. Their journey can be found <a href="/content/id/4903/Product%20Sudoku%20-%20Journey1.pdf">here</a><a href="/content/id/4903/Product%20Sudoku%20-%20Journey1.doc"></a><a href="/content/id/4903/Product%20Sudoku%20-%20Journey.doc"></a>.<br></br>
<br></br>
They kept a record of the order in which they filled the Sudoku and wrote it in the boxes. This is just one possible route through the problem. <br></br>
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Product Sudoku
The clues for this Sudoku are the product of the numbers in
adjacent squares.
Using, Applying and Reasoning about Mathematics
Working systematically
Numbers and the Number System
Factors and multiples
Information and Communications Technology
smartphone
Collections
Secondary Number Play