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]]>Christmas Challenge 2018 Solutions
The post Deputy First Minister’s Holiday Maths Challenge – Christmas 2018 solutions appeared first on Making Maths Count.
]]>The post Maths Week Scotland 2018 Solutions appeared first on Making Maths Count.
]]>Here are the solutions to all of this week’s puzzles.
(a) 6
(b) 12
(c) No solution because Ross steps on odd-numbered steps with his left foot and Elena steps on only even-numbered steps with her left foot.
Ross puts his left foot on steps 1, 3, 5, … and his right foot on steps 2, 4, 6, …
Elena puts her left foot on steps 2, 6, 10, … and her right foot on steps 4, 8, 12, …
Tom puts his left foot on steps 3, 9, 15, … and his right foot on steps 6, 12, 18, …
Answers: (a) 24 (b) 0, 11, 14, 15 (c) 136
(a) A 9×4 rectangle would have 24 dots inside (arranged in an 8 by 3 pattern).
(b) The dimensions of the rectangle could be 24×1,12×2,8×3 or 6×4. So the possible numbers of dots inside are 0, 11, 14, 15.
(c) If the perimeter is 54, then length + width = 27. Sharing 27 so that one number is twice the other gives 18 and 9. So the length is 18 and the width is 9. Therefore, the number of dots inside is 17×8=136.
Answer: £145
Here are two possible ways to reach the answer:
1.
The trousers and shoes cost £115.
The shirt and the shoes cost £100.
So the trousers cost £15 more than the shirt.
But the trousers and the shirt cost £75. So we need two numbers that add to 75 but which differ by 15. They are 45 and 30.
So the shirt costs £30 and the cost of all three items is £115 + £30 = £145.
2.
Imagine that Asif and his friend, let’s call him Alistair, go together and buy a shirt, a pair of trousers and a pair of shoes each. As strange as it seems, their tastes are so similar that they select exactly the same items. Together they buy:
shirt + trousers | £75 |
shirt + shoes | £100 |
trousers + shoes | £115 |
£290 |
They have paid £290 between them, which is £145 each.
Answer: Yes
When Beth sold the horses, 69 days of winter remained and she had enough hay and corn to feed 6 horses for another 24 days. She could feed the remaining 2 horses for 24×3=72 days. So there was sufficient food.
Alternatively, at the start, Beth had 6×30=180 portions of horse food.
She used 6×6=36 portions whilst she still has 6 horses, so had 144 portions left.
With 69 days to go and 2 horses to feed, she needed 138 portions. So the food proved sufficient.
Answers: 19 blocks, 146 blocks
(a) The pink layer on top has a single block.
The yellow layer has 5 blocks (one of which is hidden under the red block).
The blue layer has 1+3+5+3+1=13 blocks.
So the total number of blocks in the 3 layers of the stepped pyramid is 1+5+13=19.
(b) Layers 4, 5 and 6 (working downwards) will have the following numbers of blocks in them:
So the 6-layer stepped pyramid has 1+5+13+25+41+61=146 blocks.
Notes:
(1) Though the arithmetic required to solve this problem is straightforward, the amount of work involved is considerable. One way to handle this issue is to divide the task so that many individuals or groups work on a subset and then amalgamate the results. If this is still thought too onerous, there is a similar problem which uses the coins to 45 cents and asks for the first amount for which four coins are needed (to which the answer is 68 cents).
(2) The title of the problem invokes the name ‘Gauss’. The German mathematician, Carl Friedrich Gauss (1777–1855), is considered one of the greatest in the development of the subject. One of his results, discovered and proved when he was a teenager, was that every positive whole number can be written as the sum of at most three triangular numbers, which start:
The result is the basis for this question about coin denominations.
Answer: 95
All but two of the amounts to 99 cents are possible. The smallest amount for which four coins are needed is 95 cents and there are six different ways in which this can be done.
1 | 1 | 11 | 10,1 | 21 | 21 | 31 | 28,3 | 41 | 28,10,3 |
2 | 1,1 | 12 | 10,1,1 | 22 | 21,1 | 32 | 28,3,1 | 42 | 21,21 |
3 | 3 | 13 | 10,3 | 23 | 21,1,1 | 33 | 21,6,6 | 43 | 21,21,1 |
4 | 3,1 | 14 | 10,3,1 | 24 | 21,3 | 34 | 28,6 | 44 | 28,10,6 |
5 | 3,1,1 | 15 | 15 | 25 | 21,3,1 | 35 | 28,6,1 | 45 | 45 |
6 | 6 | 16 | 15,1 | 26 | 15,10,1 | 36 | 36 | 46 | 45,1 |
7 | 6,1 | 17 | 15,1,1 | 27 | 15,6,6 | 37 | 36,1 | 47 | 45,1,1 |
8 | 6,1,1 | 18 | 15,3 | 28 | 28 | 38 | 36,1,1 | 48 | 45,3 |
9 | 6,3 | 19 | 10,6,3 | 29 | 28,1 | 39 | 36,3 | 49 | 28,21 |
10 | 10 | 20 | 10,10 | 30 | 15,15 | 40 | 36,3,1 | 50 | 28,21,1 |
51 | 45,6 | 61 | 45,15,1 | 71 | 55,10,6 | 81 | 45,21,15 | 91 | 55,36 |
52 | 45,6,1 | 62 | 55,6,1 | 72 | 36,36 | 82 | 55,21,6 | 92 | 55,36,1 |
53 | 28,15,10 | 63 | 21,21,21 | 73 | 45,28 | 83 | 55,28 | 93 | 45,45,3 |
54 | 45,6,3 | 64 | 36,28 | 74 | 45,28,1 | 84 | 28,28,28 | 94 | 55,36,3 |
55 | 55 | 65 | 36,28,1 | 75 | 36,36,3 | 85 | 55,15,15 | 95 | |
56 | 55,1 | 66 | 45,21 | 76 | 55,21 | 86 | 55,21,10 | 96 | 45,45,6 |
57 | 55,1,1 | 67 | 45,21,1 | 77 | 55,21, 1 | 87 | 45,36,6 | 97 | 55,36,6 |
58 | 55,3 | 68 | 55,10,3 | 78 | 36,36,6 | 88 | 45,28,15 | 98 | 55,28,15 |
59 | 55,3,1 | 69 | 45,21,3 | 79 | 55,21,3 | 89 | 55,28,6 | 99 | |
60 | 45,15 | 70 | 55,15 | 80 | 55,15,10 | 90 | 45,45 |
Answer: 11.00 a.m.
Every hour, Daniel’s watch gained 1 minute and his wife’s lost 2 minutes. So an hour later they were 3 minutes apart. If in the morning they are 1 hour apart, 20 hours must have elapsed. And the time when Daniel checked his old watch was not 7.20 but actually 7.00 because it had gained 20 minutes overnight.
Daniel started the old watches going 20 hours earlier, which was 11 a.m. the previous day.
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]]>The post Maths Week Scotland – Poor Timekeeping appeared first on Making Maths Count.
]]>In cleaning out a drawer, Daniel found two old watches which he and his wife had discarded. He wound them up, and after setting them accurately, started both watches at the same time. An hour later he noticed that his old watch had gained a minute whilst his wife’s watch had lost two minutes. In fact his watch was running consistently fast and his wife’s watch was running consistently slow. Next morning, when he looked at the watches again, it was 7.20 on his old watch and 6.20 on his wife’s. What time was it when he started the watches running?
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]]>The post Maths Week Scotland – Coinage System appeared first on Making Maths Count.
]]>The imaginary country of Gaussland decides to adopt a new system of coinage. There are to be coins of ten different values (denominations). Those values (in cents) are:
The idea is that by changing to the new system it will be possible to pay for any item up to the value of 99 cents using no more than three coins. For example:
Show how no more than three coins can be used for each amount to 99 cents or, if this is not the case, find the smallest amount for which four coins are needed.
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]]>The post Maths Week Scotland – Stepped Pyramids appeared first on Making Maths Count.
]]>This stepped pyramid has been built by young children and their teacher using some of the building blocks in the classroom. (The blocks are all cubes of the same size.)
(a) How many blocks would be needed to build it?
(b) Now imagine that the children want to build a similar structure with twice as many layers. How many blocks would be needed?
Answers: 19 blocks, 146 blocks
(a) The pink layer on top has a single block.
The yellow layer has 5 blocks (one of which is hidden under the red block).
The blue layer has 1+3+5+3+1=13 blocks.
So the total number of blocks in the 3 layers of the stepped pyramid is 1+5+13=19.
(b) Layers 4, 5 and 6 (working downwards) will have the following numbers of blocks in them:
So the 6-layer stepped pyramid has 1+5+13+25+41+61=146 blocks.
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]]>The post Maths Week Scotland – Feeding the Horses appeared first on Making Maths Count.
]]>
During a very hard winter, Beth had only enough hay and corn to feed her six horses for another 30 days, and it would be another 75 days before spring would arrive. On the seventh day, before feeding time, she sold four of her horses.
Was she able to feed her remaining two horses for the rest of the winter?
Explain your reasoning.
Answer: Yes
When Beth sold the horses, 69 days of winter remained and she had enough hay and corn to feed 6 horses for another 24 days. She could feed the remaining 2 horses for 24×3=72 days. So there was sufficient food.
Alternatively, at the start, Beth had 6×30=180 portions of horse food.
She used 6×6=36 portions whilst she still has 6 horses, so had 144 portions left.
With 69 days to go and 2 horses to feed, she needed 138 portions. So the food proved sufficient.
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]]>The post Maths Week Scotland – New Outfit appeared first on Making Maths Count.
]]>Asif is shopping for a new outfit for a special occasion. He needs a shirt, a pair of trousers and a pair of shoes. Once he has picked out the items he likes and checked the prices he realises that:
But, of course, he needs the shirt, the trousers and the shoes. So what will he have to pay for all of them?
Answer: £145
Here are two possible ways to reach the answer:
1.
The trousers and shoes cost £115.
The shirt and the shoes cost £100.
So the trousers cost £15 more than the shirt.
But the trousers and the shirt cost £75. So we need two numbers that add to 75 but which differ by 15. They are 45 and 30.
So the shirt costs £30 and the cost of all three items is £115 + £30 = £145.
2.
Imagine that Asif and his friend, let’s call him Alistair, go together and buy a shirt, a pair of trousers and a pair of shoes each. As strange as it seems, their tastes are so similar that they select exactly the same items. Together they buy:
shirt + trousers | £75 |
shirt + shoes | £100 |
trousers + shoes | £115 |
£290 |
They have paid £290 between them, which is £145 each.
Don’t forget to come back tomorrow for the next Maths Week Scotland puzzle!
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]]>The post Maths Week Scotland – Tuesday appeared first on Making Maths Count.
]]>Unleash your creative side in the maths inside photo competition which looks for maths in our surroundings, with categories in my town, in the wild or through change. The competition is open for just one more day, submissions must be in on 5pm on Wednesday 12^{th} September. There will also be a prize giving ceremony on Saturday 15^{th}, winners from P1 – S6 could win amazon vouchers. Details available from the website http://mathsinside.com/
Minister for Further Education, Higher Education and Science, Richard Lochhead visited Monifeith High School today to see an enigma machine in action with Tom Briggs from the Bletchley Park team. Pupils had a great time cracking codes and finding out how to set an enigma machine.
In St Andrews, school pupils from St Leonards were able to see and discuss original texts by mathematicians through history, including Mary Somerville and letters by Isaac Newton.
The session was facilitated by the Special Collections team at St Andrews, and they brought the texts to life with stories of Mary Somerville’s struggles to be recognised as a scientist and Isaac Newton’s fight to claim discoveries as his own.
After viewing the texts, undergraduate students and school pupils all enjoyed a talk on the Mathematics of the Sun by James Threlfall. Space weather is big news, but behind the headlines teams of people including mathematicians are constantly watching the suns activity. They are working to model and predict the activity of the sun and the effect it would have on us here on earth.
Across the River Tay in Dundee, Dundee High School have a week of activities planned to include maths relays, assemblies and homing the Maths Week Scotland golf flags which flew at the Scottish Open Golf Championships at Gullane.
Maths performer Andrew Jeffrey kicked off his Maths Week Tour wowing students with the Magic of Maths in Perth. He will be continuing his maths adventures around Scotland throughout Maths Week and will be in Pitlochry tomorrow.
Across twitter we are delighted to see so many people taking part in the DFM Daily Puzzle, as well as the many other amazing puzzles and challenges on the go, and introducing maths across the curriculum; we’ve seen numeracy in Spanish classes and maths incorporated into the Daily Mile and PE lessons. There’s been bridge building, orienteering, a pharmacist and optician talking to pupils about maths in their jobs, and a forensic session with the police! Keep it up and don’t forget to tweet us #MathsWeekScot @MathsScot
Tomorrow is a busy day for Maths Week with a public lecture on Maths Coded in our Genes, statistics event for teaching practitioners whilst school pupils will be finding out about the Maths of Social Media at Heriot Watt University. Plus maths performers and workshops will be continuing to tour Scotland!
Find out more at www.nms.ac.uk/maths
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]]>The post Maths Week Scotland – Dotty Rectangles appeared first on Making Maths Count.
]]>Good luck!
Here is a 6×3 rectangle drawn on dotty paper. There are 10 dots inside.
(a) How many dots would there be inside a 9×4 rectangle drawn on dotty paper?
(b) A rectangle with area 24 (square units) is drawn on dotty paper. How many dots could there be inside? (Note that there is more than one correct answer.)
(c) A rectangle is drawn on dotty paper so that its length is twice its width. Its perimeter is 54 (units). How many dots are there inside it?
Answers: (a) 24 (b) 0, 11, 14, 15 (c) 136
(a) A 9×4 rectangle would have 24 dots inside (arranged in an 8 by 3 pattern).
(b) The dimensions of the rectangle could be 24×1,12×2,8×3 or 6×4. So the possible numbers of dots inside are 0, 11, 14, 15.
(c) If the perimeter is 54, then length + width = 27. Sharing 27 so that one number is twice the other gives 18 and 9. So the length is 18 and the width is 9. Therefore, the number of dots inside is 17×8=136.
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]]>The post Let’s Get Started! Maths Week Scotland: Monday appeared first on Making Maths Count.
]]>It’s finally, here. Maths Week Scotland 2018 is a go and what a start! All over the country we have been seeing how schools and communities are celebrating. There is still time to get involved with competitions, activities and online at www.nms.ac.uk/maths.
If you want to find out how maths is used in different careers University of Edinburgh have created a series of videos Maths Week at Work showing how maths is used in jobs such as programming as jewellery making. Each video also features a mathematical problem these people face in their day to day jobs. Watch them here https://www.maths.ed.ac.uk/school-of-mathematics/outreach/mathsweekscotland/maths-week-at-work
This morning Arbroath High School had a visit from Tom Briggs, from Bletchley Park. Students found out all about the history of Bletchley Park, and the role of mathematicians played in cracking codes during the Second World War.
Tom hadn’t just come on his own though, he brought with him a real working enigma machine. And not just any enigma machine! The machine is the same one which starred in the film The Imitation Game.
1 Enigma Machine at Arbroath High School, the one which featured in The Imitation Game film!
2 Mr Todd tries out the enigma machine
Students worked together to crack codes set by Tom and learnt how to set an enigma machine to decipher coded messages. There are 159 quintillion possible ways (that’s 159,000,000,000,000,000,000!) to set an enigma machine so we didn’t have time to go through them all this morning but it was fantastic to use logical thinking to crack the code set. Students at Arbroath High School will be continuing the code theme during their lessons throughout the week. The enigma machine will be continuing its adventures to all Angus Secondary Schools during Maths Week.
In the Highlands at Newton Park Primary School, Kjartan Poskitt shared maths fun 200 school pupils from 7 different primary schools. He will be doing 10 gigs in schools in the Highlands and Orkney, and performing at Orkney Science Festival throughout Maths Week 2018.
On social media we have seen some fantastic things happening in schools. Lots of schools have been trying out the Daily Puzzle set by John Swinney. There will be a new puzzle set each day so try it yourself!
Other schools have been incorporating maths across the curriculum and even out in the playground. We have loved seeing the range of ways people are celebrating Maths Week and the unusual places people are finding maths inspiration.
On Tuesday we will be in St Andrews looking at the Mathematics of the Sun, and scratching our heads to solve the second daily puzzle!
Follow us on @MathsScot to find out more and let us know what you are up to #MathsWeekScot
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