598 /www/nrich/html/content/98/09/six5/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"><p>A church hymn book contains 700 hymns, numbered 1 to 700.</p> <p>Each Sunday the people in the church sing four different hymns.</p> <p>The numbers of the hymns are displayed to them in a frame by dropping in single-digit boards like this:</p> <p><mdo:image alt="number 2" height="35" src="number2.gif" width="29"></mdo:image></p> <p>The board for 6 may be turned upside down to serve as a 9.</p> <p>What is the minimum number of small boards that is needed to show any possible combination of four hymn numbers?</p> <p>How many of each number must there be?</p> <p><mdo:image alt="hymn board" height="245" src="hymnboard.gif" width="125"></mdo:image></p> <p> </p> <p> </p> <p><a href="http://nrich.maths.org/6829">Click here for a poster of this problem</a>.</p> <br></br></mdoxml> <mdoxml version="1.0"><br></br> <p class="editorial">Chris and Matt at Staunton and Corse School told us:</p> You would need 81 small boards made up of:<br></br> eight 0s <br></br> nine 1s <br></br> nine 2s <br></br> nine 3s <br></br> nine 4s <br></br> nine 5s <br></br> twelve 6s <br></br> eight 7s <br></br> eight 8s <br></br> <br></br> <p><span class="editorial">They offered a good explanation for this. Beh Sze at Garden International School also explained carefully how he arrived at the answer:</span></p> 0s - There can only be two zeros in each row because there is no 000 or 1000 in the hymn book.<br></br> 1-5 - There can only be one triple digit number repeating the same single digit three times (e.g. 111, 222, 333). The rest can only repeat it twice at most (e.g. 011, 110, 101, 211, 311) because the numbers 111, 222, etc. cannot be repeated. <br></br> 6s - Even though there cannot be a triple digit that repeats 6 three times, the extra 6s would be needed to be turned in to 9s (e.g. 696, 669, 699, 996). You must not forget that 6s can be turned into 9s. <br></br> 7 and 8 - There can only be two, at maximum, of this number in each row (e.g. 177, 277, 377, 477, 070, 007). There cannot be 777 or 888 because the hymns only stop at 700. <br></br> 9s - 9 would work like 7 and 8 because it too cannot be written as 999. But the 9s do not need their own numbers because the numbers for 6 can be turned around to make 9. <br></br> <br></br> <p><span class="editorial">Well done too to James and Jasmine at Old Earth Primary School who also reasoned through their solution very clearly</span>.</p> <br></br></mdoxml> <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=598&amp;part=index"> This problem</a> encourages logical thinking and, to a lesser extent, an understanding of place value.</div> <br></br> <h3>Possible approach</h3> <div>This is an ideal activity for 'think, pair, share'. Give the children a chance to work alone and then in pairs, so that they can check out their thinking with each other. You could ask them to record their solutions (just the answers) on the board. Put pairs who have arrived at different solutions together to see if they can convince each other of who is correct - and find where their reasoning differs.</div> <br></br> <div>Some children will be able to do this question with minimal recording and maximum mathematical reasoning. Others will need practical materials to get them started and will then move to abstract working. Others will need cards or other practical materials for the whole question. Try to move children on to working without materials as soon as possible so that they are beginning to generalise (what's true for 1 is true for 2, 3, 4, 5 for example).</div> <br></br> <h3>Key questions</h3> <div>How could we start this problem?</div> <div>What's the biggest number we're going to need? What's the smallest?</div> <div>Are there any numbers that we'll need more of/fewer of?</div> <div>How could we record this?</div> <br></br> <h3>Possible extension</h3> <div>What 'What if .. ?' questions could we ask?</div> <div>Perhaps we could consider numbers up to 800 or 900 - can we now find a quicker way of doing this since we've done the numbers up to 700?</div> <br></br> <h3>Possible support</h3> <div>Children who need support might choose to use printed grids like <a href="/content/98/09/six5/Wholynumbersgrid.doc">this</a> for filling in possible numbers and lots of scrap paper cut up into squares for writing numbers on.</div> <div>Or you could make the question simpler by asking them to work out what numbers would be needed for hymns up to 100.</div> <br></br> <br></br></mdoxml> <mdoxml version="1.0">What's the biggest number we're going to need? <br></br> What's the smallest? <br></br> Are there any numbers that we'll need more of/fewer of? <br></br> How could we record this? <br></br></mdoxml> <mdoxml version="1.0"><br></br> <p class="editorial">Hannah and Kelly Jane of Maidstone Girls Grammar School sent solutions here with good clear explanations.</p> <p>The four hymns are chosen with different numbers between 1 and 700. The board for 6 may be turned upside down to serve as a 9.</p> <p>Consider first the digits 1, 2, 3, 4 and 5. There would be the same number of boards needed for each of those digits, say xxx and xx6, xx7, xx8 where x = 1, 2, 3, 4 or 5). Hence 9 small boards are needed for each of these digits.</p> <p>For 6 (or 9) it must be possible to show 666, 669, 696 and 699 so 12 of these boards are needed.</p> <p>For the 7's and 8's since 777 and 888 do not occur, it is sufficient to have 8 of each of these, for example 177, 277, 377 and 477 would use them all.</p> <p>Similarly 000 does not occur so 8 zeros are needed <em>e.g.</em> for 100, 200, 300 and 400.</p> <p>Hence the total number of boards required is 45 + 12 + 24 = 81.</p> <br></br></mdoxml> 2 5 0 1 0 0 0 A church hymn book contains 700 hymns. The numbers of the hymns are displayed by combining special small single-digit boards. What is the minimum number of small boards that is needed? Mathematics Tools Digit cards Decision Mathematics and Combinatorics Combinations Numbers and the Number System Place value