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12. 7 Reindeer and 5 elves partner each other at the annual ‘Dance for Elves and Reindeer’. The dances are so energetic that no reindeer or elf can manage more than 3 dances in a row, and there is only space for 4 pairs at a time. Each reindeer must dance with each elf.

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  1. 12 • 7 Reindeer and 5 elves partner each other at the annual ‘Dance for Elves and Reindeer’. • The dances are so energetic that no reindeer or elf can manage more than 3 dances in a row, and there is only space for 4 pairs at a time. • Each reindeer must dance with each elf. • How many dances must there be? • Draw up a schedule for the dances.

  2. 11 • At a Christmas gathering, Uncle Joe is demonstrating his amazing mathematical abilities. • He can multiply any two digit number by 11 quicker than anyone can work it out in their head or on a calculator. • Here are some examples: 13 x 11 = 143 34 x 11 = 374 51 x 11 = 561 • Can you spot how he does it so quickly? • Does it work for these? 72 x 11 75 x 11

  3. 10 • Baubles for Christmas trees are stacked in pyramids. • Each layer is a different sized triangle which hasone row less than the layer below it. • If there are 10 baubles on the bottom layer, how many baubles are there in the pyramid? • If there are 120 baubles to be stacked, how many layers will the pyramid have?

  4. 9 • One of the reindeer, Comet, has flown off. • Santa needs her to return home as soon as she lands in the forest. • The forest consists of huge trees planted in 9 rows of 9 in a grid formation (see next slide). • Santa sends elves out to wait for her to land. • The elves stand in the rows between the trees and can see to the end of just one tree in each direction. • How many elves are needed to monitor the whole forest?

  5. Sight line Not to scale The trees are much bigger than the gaps!

  6. 8 • Santa Claus is mixing some carrot and apple punch. • He needs to pour exactly 2 litres of apple juice and then 1 litre of carrot juice into a barrel. • Unfortunately, most of his kitchen measures are already in use, except for an old, unmarked 8 litre bucket and an equally old, unmarked 5 litre jug. He can fill and empty these as needed. • He has plenty of both apple and carrot juices, how can he make exactly the quantities he needs?

  7. 7 A red Christmassy design is made by joining each of 7 points on a circle to each other. How many lines will there be? Is it possible to draw the design without taking your pencil off the paper? Design for 5 points shown

  8. 6 • Christmas trees are planted so that there are 6 identical rows of 4 trees. • Only 12 trees are planted. • How are they arranged?

  9. 5 When wrapping Christmas presents, Santa needs some boxes shaped like an open topped cube, which have 5 square sides. The elves can make these from card. How many different nets are there for this shape?

  10. 4 • Homer, Marge, Bart and Lisa are writing cards to each other. (Maggie can’t read or write so doesn’t send or receive cards) How many do they write? • If each member of your class were to send a card to each other, how many would be sent? • They then all shake hands, how many handshakes occur? • How many would occur in your class?

  11. 3 • 3 elves were discussing the age of the current Santa. • “He said he was born in the 1900s”, said one. • “Eight years ago, he told me that his age was a square number”, said the second. • “I’m sure he said last year that his age was a prime number”, said the third. • Overhearing them , Mrs Claus said “You’re all correct, and his age will be a triangle number in six years time” • How old is he?

  12. 2 • The reindeer are asleep before The Big Night… • Rudolph wakes up, goes to the carrot pile and eats half of the carrots, plus half a carrot. • A little later Comet awakes, goes to the carrot pile and eats half of the remaining carrots, plus half a carrot. • Later, Prancer wakes and does exactly the same: eats half of the remaining carrots, plus half a carrot. • Cupid then does the same. • Santa arrives and there’s a single carrot left. • How many were there to start with?

  13. 1 In his advent calendar, Bart asked Homer to put: • $1 in the window for the 1st December • $2 in the window for the 2ndDecember • $4in the window for the 3rd December doubling the amount each time until the 24th December… until Marge made him sit down and work out how much it would cost! • When would the daily amount be more than $100? • How much would it have been on 24th? • How much would it have been in total?

  14. Teacher notes: Twelve Days of Christmas This is a selection of short activities and puzzles – many of them familiar ones with a slightly festive twist – to count down to the end of term. Also there are some ‘Simpson’s’ references to tie in with Simon Singh’s guest spot in this month’s Monthly Maths. They could be used as starter activities for a wide range of pupils in KS3 and KS4. In the teacher notes are answers for all of the problems and suggestions for additional questions for some of the problems.

  15. Teacher notes: Twelve Days of Christmas 12: There have to be a minimum of 10 dances. There are 35 pairings to get through. Elves have to dance more than Reindeer. Since only 4 pairs can be on the floor at any time, at least one elf has to sit out each time. The elf who sits out first has to follow a pattern such as: ODDDODDDOD consisting of 1xO followed by 7xD and 2xO(Dance, Out)

  16. Teacher notes: Twelve Days of Christmas 11: Separate the digits and place the total the tens and units digits between them. It always works for a two digit number, but where the total of the tens and units digits is greater than 9, the ‘tens’ digit of the total has to be added to the original ‘tens’ digit e.g. 48 x 11 would be 4_‘12’_8 which becomes 528 Additional question: why does this work? 10: 20 baubles; 8 layers.

  17. Teacher notes: Twelve Days of Christmas 9: Pupils will need to decide whether elves also patrol the outside edges of the forest. • Not including outside edges: 32 elves • Including outside edges: 50 elves Additional questions: How many elves would be needed if the forest were: 8 x 8? n x n? m x n? Answers: 40 (n+1)2÷2 (m+1)(n+1)÷2 n.b. round answers down

  18. Teacher notes: Twelve Days of Christmas 8: 2 litres of apple juice: • Fill 5 litre jug • Pour into 8 litre bucket • Fill 5 litre jug • Top up the 8 litre bucket – this will take 3 litres, leaving 2 in the jug. • 1 litre of carrot juice: • Fill the 8 litre bucket • Pour from the 8 litre into the 5 litre jug, leaving 3 litres in the bucket. • Empty the 5 litre jug • Pour the remaining 3 litres from the 8 litre bucket to the 5 litre jug • Refill the 8 litre bucket • Pour from the 8 litre bucket to top up the 5 litre jug – this will take 2 litres, leaving 6 in the bucket • Empty the 5 litre jug • Pour from the 8 litre bucket to fill the 5 litre jug • This will leave 1 litre in the bucket.

  19. Teacher notes: Twelve Days of Christmas 7: 21 lines. Yes it is possible. Additional questions: how many lines would there be if it were 8 points on the circle? (28 lines) Would it still be possible to draw it without lifting the pencil? (No, only shapes with 0 or 2 ‘odd nodes’ are traversable)

  20. Teacher notes: Twelve Days of Christmas 6:

  21. Teacher notes: Twelve Days of Christmas 5: 8 nets (not allowing reflections or rotations) Additional questions: • How many nets are there for closed cubic boxes? (11) • To minimise wastage, the elves will cut several open top boxes from each sheet of cardboard, which of the nets would be best? (All equally efficient as these shapes are pentominoes, and all pentominoes tessellate)

  22. Teacher notes: Twelve Days of Christmas 4: Cards: Simpsons 12, Class n(n-1) Handshakes: Simpsons 6, Class n(n-1)÷2 Additional questions: What if there were 100 people? (9900, 4950) Find a general expression. (as above) Note: this problem has links with the Day 7 problem 3: Santa is 72 Additional question: make up your own ‘Santa’s age’ puzzle.

  23. Teacher notes: Twelve Days of Christmas 2: 31 carrots (31 15 7 3 1) This can be determined by working backwards: starting with the single remaining carrot, add half and then double. Repeat this for each of the reindeer. 1: >$100 on 8th December 223 = $8 308 688 Total is $16 777 215 Additional question: At what point would Bart become a millionaire? in $? In £? (Day 20 in $ and Day 21 in £, assuming a conversion rate of approx. £1 to $1.6)

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