87 /www/nrich/html/content/01/03/bbprob1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>This is a 4 x 4 Magic Square made from the numbers 1 to 16.</p> <p style="text-align: center;"><mdo:image alt="Magic square : 15, 10, 3, 6; 4, 5, 16, 9; 14, 11, 2, 7; 1, 8, 13, 12." src="magic1.gif"></mdo:image></p> <p>In a Magic Square all the rows, columns and diagonals add to the same number. This number is called the &#39;Magic Constant&#39;.</p> <p>Here are some questions about this Magic Square.</p> <p style="margin-left: 40px;">1/What is the Magic Constant of this Magic Square?</p> <p style="font-style: italic; font-weight: bold;">This particular square is especially &#39;magic&#39; as some 2 x 2 squares within it also add to that number.</p> <p style="margin-left: 40px;">2/How many of these squares can you find?<br></br> <br></br> 3/What happens to the Magic Constant if you add 2 to each number in the square?<br></br> <br></br> 4/What happens if you double each number?<br></br> <br></br> 5/Can you make a square in which the Magic Constant is 17?<br></br>  </p> <div style="margin-left: 40px;">How did you do it?</div> <br></br> <div style="margin-left: 40px;">6/Can you make a square in which the Magic Constant is 38?<br></br>  </div> <div style="margin-left: 40px;">How did you do it?</div> <br></br> <p style="margin-left: 40px;">7/What other numbers under 100 can you make into the Magic Constant by changing all the numbers in the square in the same way?</p> <p style="margin-left: 40px;">8/Can some be made in more than one way?</p> <p style="margin-left: 40px;">9/Are there some numbers you really cannot make?</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=87&amp;part=">This investigation</a> gives a lot of opportunities for a wide range of abilities to exercise both their spatial/number awareness.</div> <br></br> <h3>Possible approach</h3> <div>It might be neccesary for the pupils to be introduced to simpler more common magic squares.</div> <br></br> <div style="text-align: center;"><mdo:image width="195" height="195" alt="" src="mgic%2087%201.jpg"></mdo:image></div> <div>If there is a problem in identifying the 2 x 2 little squares within the 4 x 4 square the first one in the bottom left hand corner could be selected.</div> <div style="text-align: center;"><mdo:image width="162" height="153" src="magic%2087%202.jpg" alt="87,2"></mdo:image></div> <br></br> <h3>Key questions</h3> <div>Can you tell me about the way you are doing this?</div> <div>What have you decided to do to the first set of numbers?</div> <br></br> <h3>Possible extension</h3> <div>Questions 7, 8 &amp; 9 act as a good set of extension activities, further ones could be suggested by the pupils. Then magic squares of a different size could be explored.</div> <br></br> <h3>Possible support</h3> <div>Some pupils will find it useful to have small square cards with the numbers on a a prepared grid to place them on.</div> <br></br> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <br></br> <br></br> After the initial search for four numbers that add to the Magic Constant in the initial given Magic Square investigation changes direction.<br></br> <br></br> The suggestion to find the Magic Constant if 2 is added to each number, and if the numbers are doubled, should give hints enough for pupils to explore ways of making different Magic Constants.<br></br> <br></br> Pupils should write down the function used to make each one then, if different ways are found, the most simple one could be chosen. Thus the idea of an 'elegant' method could be discussed.<br></br> <br></br> The given 4 x 4 Magic Square can be further explored. If, for example, the left hand column is moved entirely to right hand side, the square is still 'Magic'.<br></br> <br></br> <div style="text-align: center;"><mdo:image width="209" height="207" alt="3" src="magic%2087%203.jpg"></mdo:image></div> <br></br> <br></br> <br></br> Similar changes as this can be explored and lists of more ways to make the Magic Constant made. Do these cover all the ways of making it from four numbers from 1 to 16?<br></br> <br></br> What would the Magic Constant be for a 1 - 25 (5 x 5) Magic Square?<br></br> <br></br> Or a 1 - 36 (6 x 6) one?<br></br> <br></br> -------------------------------<br></br> OLD SOLUTION<br></br> Tom did a superb job working on this Bernard's Bag problem.<br></br> He really did explore a great number of possibilities, and Tom is wondering if there are even more investigations that he can make.<br></br> Well done, Tom.<br></br> Here is a copy of the results that Tom came up with.<br></br> <ul> <li>The magic constant is 34. There are 8 2x2 squares - each corner and each middle part of each side makes a square.</li> <li>If you add 2 to each number in the square then the magic constant becomes: +95-14+17x0-20+62 = 42</li> </ul> <br></br> <br></br> <div style="text-align: center;"><mdo:image width="226" height="226" src="MagSQSol1.jpg" alt="sol1"></mdo:image></div> <br></br> If you double each number then the magic constant doubles to equal 68.<br></br> Or, you can make a square in which the magic constant is 17 by halving each<br></br> <div style="text-align: center;"><mdo:image width="440" height="223" src="MagSQSol2.jpg" alt="sol2"></mdo:image></div> <br></br> To make a square in which the magic constant is 38 you add one to each number:<br></br> <div style="text-align: center;"><mdo:image width="226" height="226" src="MagSQSol3.jpg" alt="sol3"></mdo:image></div> <br></br> You can also make a magic square that has a magic constant of 50:<br></br> <div style="text-align: center;"><mdo:image width="226" height="226" src="MagSQSol4.jpg" alt="sol4"></mdo:image></div> <br></br> And, you can even make a magic square a magic constant of zero!<br></br> <div style="text-align: center;"><mdo:image width="226" height="226" src="MagSQSol5.jpg" alt="sol5"></mdo:image></div> <br></br> Along the diagonals the squares can have the same totals along the diagonals and still work - here is an example with the constant of 50<br></br> <div style="text-align: center;"><mdo:image width="226" height="226" src="MagSQSol6.jpg" alt="sol6"></mdo:image></div> <br></br> However, You can't make a square with a magic constant of 32 - although you could if you allowed fractions to be used!<br></br> <br></br></mdoxml> 2 4 0 1 0 0 0 In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square? Using, Applying and Reasoning about Mathematics Generalising Calculations and Numerical Methods Multiplication & division Calculations and Numerical Methods Addition & subtraction Using, Applying and Reasoning about Mathematics Investigations