Three Dice

6719 /www/nrich/html/content/id/6719/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Take a look at some ordinary dice. What do you notice about the way the numbers are arranged? Now look at these three dice in a row: <div style="text-align: center;"><mdo:image alt="" height="189" src="615-2.jpg" width="261"></mdo:image></div> <div> The numbers on the tops of the dice read $6$, $1$ and $5$. What do the numbers on the top add up to? </div> Can you use what you found out about the way the numbers are arranged to say what numbers are on the bottom of the dice? Were you correct? What is the sum of the numbers on the bottoms of the dice? Let's try that again. This time the numbers on the top read $1$, $4$ and $3$. <div style="text-align: center;"><mdo:image alt="" height="189" src="143-2.jpg" width="261"></mdo:image></div> <div> Can you work out the total? Can you work out the numbers on the bottom and their total?</div> Try out some arrangements yourself. Each time record the sum of the numbers on the top and the sum of the numbers on the bottom. Do you notice a relationship between the 'top sum' and the 'bottom sum'? Can you explain it? I experimented with arrangements where the top sum is a multiple of three, and find that in each case the bottom sum is also a multiple of three. Is it always true? I try to arrange the dice so that the top and bottom sums are both multiples of four, but can't seem to be able to do it. Can you? Can you explain what you find out? On the other hand, if I arrange four dice in a row it is easy to make the top and bottom sums both multiples of four. Can you arrange four dice so that the top and bottom sums are both multiples of three? Can you explain what you find out? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Bomere Heath School in England obviously looked at more than just one of this month's problems, here is what Callen, Annalise,Eddi, Alex and Holly said, We found out that $7+7+7=21$ so every time you roll a dice it will add up to seven and when you roll three dice it will add up to $21$. Noah and Guy wrote, In three dice you roll three dice and add the top score up and the boottem score together and add it together we found out that the answer was always $21$ thats because three times seven makes $21$ and whatever you get on one side add to the other side always make $7$ like $6$ and $1$ and $2$ and $5$ they both make $7$ . Sian, Mollie and Liam from Deansfield Community School sent in the following; The opposite sides always add up to $7$. When you have $3$ dice and you take the sum of the top and add the sum of the bottom the total equals $21$. This is because $3x7$ is $21$. If there are $4$ dice the sum of the top and bottom will always be $28$ because $4x7=28$ Thank you all for these interesting findings. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Three Dice</h2> Take a look at some ordinary dice. What do you notice about the way the numbers are arranged? Now look at these three dice in a row: <div style="text-align: center;"><mdo:image alt="" height="189" src="615-2.jpg" width="261"></mdo:image></div> <div> The numbers on the tops of the dice read $6$, $1$ and $5$. What do the numbers on the top add up to? </div> Can you use what you found out about the way the numbers are arranged to say what numbers are on the bottom of the dice? Were you correct? What is the sum of the numbers on the bottoms of the dice? Let's try that again. This time the numbers on the top read $1$, $4$ and $3$. <div style="text-align: center;"><mdo:image alt="" height="189" src="143-2.jpg" width="261"></mdo:image></div> <div> Can you work out the total? Can you work out the numbers on the bottom and their total?</div> Try out some arrangements yourself. Each time record the sum of the numbers on the top and the sum of the numbers on the bottom. Do you notice a relationship between the 'top sum' and the 'bottom sum'? Can you explain it? I experimented with arrangements where the top sum is a multiple of three, and find that in each case the bottom sum is also a multiple of three. Is it always true? I try to arrange the dice so that the top and bottom sums are both multiples of four, but can't seem to be able to do it. Can you? Can you explain what you find out? On the other hand, if I arrange four dice in a row it is easy to make the top and bottom sums both multiples of four. Can you arrange four dice so that the top and bottom sums are both multiples of three? Can you explain what you find out? </div> <h3>Why do this problem?</h3> <div><a href="http://nrich.maths.org/6719&part=">This problem</a> encourages children to explain observations and to generalise. It requires a good understanding of multiplication. It may also introduce the idea that opposite faces of a dice add to seven, if that is something with which learners are not already familiar.</div> <h3>Possible approach</h3> <div>Before you introduce the problem give the children an opportunity to look at and then compare several dice in order to find the connection between opposite faces. You could then introduce the problem with three large dice. Ask children to calculate the sum of the top numbers and the sum of the bottom numbers, and record the results on the board. You could try this a few times altogether, recording the sums each time. Ask the group to try more examples in pairs, then add these findings to the board. Make sure they check their calculations! Gather the group together again and invite learners to comment on anything they notice. Challenge them to explain why the sum of the top and bottom faces added together is always $21$. If they have not realised that opposite faces add to $7$, then you may have to ask a few probing questions about a single dice.</div> <div> </div> <div>Move on to introduce the second part of the problem, perhaps by arranging the dice in a particular way, for example with $1$, $2$ and $3$ on the top faces. You could remark on the fact that both the top and bottom sums are multiples of $3$ and set the children off on investigating whether, if the top sum is a multiple of $3$, then the bottom sum is always a multiple of $3$ too. </div> <div> </div> <div>After a suitable length of time, bring everyone together again to discuss their findings. This gives you a chance to talk about proof in mathematics. The first stage for children is looking for counter examples to a theory. Ask the group whether anyone has found an example when the top sum was a multiple of $3$ but the bottom sum wasn't. Is that enough for us to say that it will ALWAYS be the case? What if we just haven't found that one way of arranging the dice that gives us a multiple of $3$ on the top and a multiple of something else (not $3$) on the bottom? The justification (or proof) might be difficult for some to articulate but remind learners what we know about the total sum of the top and bottom numbers (it is $21$). What can they say about $21$?</div> <div> </div> <div>You may like to pose the final part of the problem for learners to think about over a longer period of time and perhaps devote some space on your wall for them to contribute findings. </div> <h3>Key questions</h3> <div>What do you notice about the totals of the top and bottom faces?</div> <div>How can you explain your observation?</div> <div>What do you know about opposite faces of a dice?</div> <div>How do you know that will ALWAYS be the case? </div> <h3>Possible extension</h3> <div>Encourage children to ask their own questions to investigate. They could write up their conjectures and discoveries on your maths wall.</div> <h3>Possible support</h3> <div>At least three dice per pair should be available along with miniwhiteboards or paper/books for recording. A multiplication square might be useful for the second part of the activity so that pairs can check multiples of $3$.</div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">It will help to have some dice so you can try out some different arrangements. How will you remember the totals you have made? What do you know about opposite faces of a dice? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">21 is multiple of 3, therefore if one sum is multiple of 3, other sum must be too. 21 not multiple of 4, therefore both sums can't be multiples of 4. 28 not multiple of 3 therefore not possible. Previous solutions up to Sept 2012 were: We had some extremely well-explained solutions to this problem - thank you for sending them in. Red Kites Class from Holton St Peter Primary said: We really enjoyed solving this problem. We noticed that the top number and the bottom number of any dice always added up to $7$. For example, $2$ and $5$, or $4$ and $3$. Then we looked at the sum of the top numbers and the sum of the bottom numbers. We noticed that they always added up to $21$. For example, $6+5+3=14$ on the top and $4+1+2=7$ on the bottom. $14+7=21$. We did lots of examples to make sure that we were certain. Well done! I like the way you have found lots of examples that fit your rule. Mehwish, Becky and Amber from Oakwood also tried lots of examples and found that they all made a total of $21$. When you discover a rule like that, the next stage might be to ask 'I wonder why that happens?'. Ryan from St Ebbe's Primary has started to explain for us: If you keep throwing dice, you will eventually find that the sum of the top sum and the bottom sum adds up to $21$. This is because the top and bottom number of a die is always $7$. And since there are three dice, $7\times3=21$. With four dice it is $7\times4=28$. With three dice the top and bottom sums can be a multiple of three but not four because three goes into $21$ ($21\div3=7$) but four doesn't ($21\div4=5.25$). And following on with four dice, $28\div4=7$ (that works) and $28\div3=9.3333.......$ (that doesn't work). So it pretty much depends on how many dice you have. Well done, Ryan, some clear reasoning here. Bryce, Cooper, Silas, Taylor, Aaron, Sydney, DaeLen, Dylan, Brandon and Byron from AIG also looked at having more than three dice: Brandon and Aaron realised that the number of dice affected the sum. So, as a group it was decided that the number of dice we had times seven would be the sum and it would be a multiple of $7$. We tested this theory using three, four and five dice. We then double checked it with one and two dice. We found that our ideas were correct. Krystof also sent in a good solution - well done. Stuart, Rebecca, Rhys and Eva who go to Myland Primary School clearly worked very hard on this problem too. They sent in a very full solution and looked at dice with different numbers of faces: When we throw three dice and add up the top numbers and the bottom numbers, one sum is even and the other is odd. The pairs of totals are: $3$ & $18$ $4$ & $17$ $5$ & $16$ $6$ & $15$ $7$ & $14$ $8$ & $13$ $9$ & $12$ $10$ & $11$ We noticed that the top and bottom totals all add up to $21$. Opposite sides of the dice add up to $7$. So when we add the numbers on opposite sides of three dice we get $7+7+7 = 21$. If the top numbers on three dice add up to a multiple of $3$ then the bottom numbers also add up to a multiple of $3$ because $21$ is a multiple of $3$. Totals on four ordinary dice $= 7 + 7 + 7 + 7 = 28$ Totals on five ordinary dice $= 28 + 7 = 35$ On dice the highest number is always opposite the lowest number, the next highest is opposite the next lowest. So on an eight-sided dice numbered $1$ to $8$ the opposite sides would add up to 9 and the total on three dice would be $27$. On a 100-sided dice numbered $1$ to $100$ the opposite sides would add up to $101$ and the total on three dice would be $303$. We investigated our ten-sided dice, numbered $0$ to $90$: Example 1: $90 + 80 + 70 = 240$ $0+ 10 + 20 = 30$ Total = $270$ Example 2: $40 + 50 + 60 = 150$ $50 + 40 + 30 = 120$ Total = $270$ The opposite sides of the dice add up to $90$ so the total is always $90+90+90 = 270$. What fantastic work - I love the way you have begun to ask your own questions and explored things that interested you. You also worked very sytematically to find all the possible combinations of totals on three dice, which is very helpful in answering this problem. It makes me wonder ... will a dice always have an opposite face, no matter how many faces it has? Well done everyone!</mdoxml> 2 3 0 1 0 0 0 Three Dice Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice? 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