1136 /www/nrich/html/content/02/10/penta3/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>I have a handful of dice:</p> <p style="text-align: center;"><mdo:image alt="5 dice" height="130" src="Dice1.gif" width="241"></mdo:image></p> <p>I throw three of them and get 5, 3 and 2:</p> <p style="text-align: center;"><mdo:image alt="3dice" height="121" src="Dice2.gif" width="159"></mdo:image></p> <p>Add the scores on the three dice. What do you get? Now multiply the scores. What do you get?</p> <p>The next time I throw four of the dice and add and multiply the score as before.<br></br> This time the product is exactly 4 times the sum. What scores could I have thrown?</p> <p>Again I throw four of the dice, adding and multiplying the scores as above. This time the product is exactly 5 times the sum. What scores could I have thrown?</p> <p>For the last go I throw five of the dice and add and multiply the scores. The product is exactly 6 times the sum. What scores could I have thrown this time?</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Clement from Sha Tin College in Hong Kong has sent in this solution for Scoring with Dice:</p> <p>For the first question, I had to find a combination using 4 numbers, and its product had to be 4 times its sum. I first tried to find a suitable number, with many factors. And it had to be big enough in order to fit the sum of the combination in.</p> <p>I thought: 40 would probably be too small to fit the sum of the combination in and 44 only had 4 factors, so I tried 48. I listed all the factors: 2*24, 3*16, 4*12 and 6*8 and tried splitting each one of them into 4 steps, such as: 2*6*4*1 or 3*4*2*2 and I eventually came up with one: 6*2*2*2, the sum being 12.</p> <p>I used the same method for the next two questions and came up with 5*3*2*2 which made 60, the sum being 12 and 3*3*2*2*2 which made 72, the sum being 12.</p> <p>Thank you for this clearly explained answer Clement.</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> <a href="http://nrich.maths.org/public/viewer.php?obj_id=1136&amp;part=index">This problem</a> will require learners to work systematically and involve trial and improvement. Learners will need to use addition and multiplication and some knowledge of factors and multiples.<br></br> <h3>Key questions</h3> <div>What number would be a good one to try first?</div> <div>Does that number have factors that you have on the dice?</div> <div>What is the highest number you could get when you add the scores of $4$ (or $5$) dice?</div> <div>What is the highest number you could get when you multiply the scores of $4$ (or $5$) dice?</div> <br></br> <h3>Possible extension</h3> Learners could throw four or five real dice and add and multiply the scores obtained.<br></br> <h3>Possible support</h3> Suggest listing the factors of suitable numbers such as $24$, $30$ and $48$ to start with.<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Try numbers which have products with suitable factors. For example $21$ would not be a good choice as one of its factors is $7$.</p> <br></br></mdoxml> 2 5 0 1 0 0 0 I throw three dice and get 5, 3 and 2. Add the scores on the three dice. What do you get? Now multiply the scores. What do you notice? Numbers and the Number System Factors and multiples Calculations and Numerical Methods Addition & subtraction Mathematics Tools Dice