Festive Math Challenges - Logic Problems and Coin Conundrums

2. Festive Challenges Sometimes, to set a good logic problem, we have to start off by assuming that something quite unbelievable has happened, such as finding yourself in a darkened room with only a block of ice for company, or that you’re going to do something for no apparent (or highly unlikely) reason. Some of these problems require just such use of your imagination; never let reality stand in the way of a good mathematical problem!

3. Festive Challenge 1 You have two bottles of very expensive, identical looking pills, and every day you need to take exactly one of each. (No idea why!) In your hand you know you have one pill from bottle A and two pills from bottle B… but you can’t tell them apart. How can you take the right amount without wasting any pills?

4. Festive Challenge 2 You have a glass of orange juice and a glass of apple juice, each has the same amount of liquid in it. You take a teaspoonful of orange juice and put it into the apple juice. You then take a teaspoonful of the apple juice (now with a little orange in it), and put it into the orange juice. Is there now more apple in the orange juice or more orange in the apple juice?

5. Festive Challenge 3 Matchstick Moves For each of the following problems you have a specified number of matchsticks arranged in a certain way. (Gaps are included to show how the matches are placed, but matches should touch.) You have to move, add or remove the specified number of matches to meet the requirements. How many can you solve?

6. Matchstick Moves 1 Can you: Add exactly three matches to this square to divide it into two congruent parts?

7. Matchstick Moves 2 Can you: Move four matches to make exactly three squares?

8. Matchstick Moves 3 Can you: Move two matches to make exactly four congruent squares?

9. Matchstick Moves 4 Can you: Add three matches to make exactly four congruent triangles? You need to think a bit differently about this one.

10. Festive Challenge 4 It’s late on Christmas Eve and Santa is getting ready to go out. He obviously needs to wear a matching pair of socks, so he rummages in his sock drawer to find a pair. He doesn’t want to put the light on as it will wake Mrs Claus, and she’ll be very cross if he does this yet again… Santa knows there are 12 red socks and 12 green socks in the drawer. How many should he take to be sure of getting a matching pair?

11. Festive Challenge 5 Four of the reindeer, Dancer, Blitzen, Comet and Vixen are discussing their favourite Christmas treats, which are: carrots, apples, mince pies and turnips. The reindeer are not all the same age, in fact, one of them is 12, one is 13, one is 14 and one is 15. Can you work out from the clues which reindeer likes which treat and how old each one is?

12. Festive Challenge 5 • Dancer can’t wait to start munching turnips. • The reindeer who snaffles Santa’s mince pies is a year younger than Comet. • Vixen is younger than the reindeer who loves turnips. • The carrot fiend is two years older than Comet.

13. Festive Challenge 6 It’s approaching the big night and Santa is restless. He opens his eyes to look at the big digital clock outside and notices that its reflection in the elves’ ice rink reads exactly the same as on the screen. • How many other times does this happen within 24 hours: • Using 12 hour clock? • Using 24 hour clock? 11:31 11:31

14. Festive Challenge 7 Mrs Claus is wrapping tinsel around the ‘barrel’ of some Christmas Crackers. She always makes 4 complete turns, but sometimes she wraps it uniformly and sometimes she doesn’t (shown below). How much tinsel does the ‘uniform’ one use? Does the ‘irregular’ one use the same, more or less tinsel than the uniform one? 12cm Circumference4cm

15. Festive Challenge 8 Coin Conundrums There are many problems involving moving coins (or counters) around, adhering to certain rules and restrictions. How many of these can you solve?

16. Festive Challenge 8 Set four coins on the vertices of a square. Move just two to form a new, smaller square.

17. Festive Challenge 8 Arrange 9 coins in a square as shown. There are 8 straight lines shown, each containing 3 coins. Move exactly 2 coins so that there are 10 straight lines which each contain 3 coins.

18. Festive Challenge 8 Starting with six coins set out as a pyramid, move one coin at a time to obtain the hexagon. When moving a coin, in its new position it must touch (at least) two other coins.

19. Festive Challenge 9 What are the next three numbers in this series?4, 6, 12, 18, 30, 42, 60, 72, 102, 108, ?, ?, ?

20. Festive Challenge 10 A festive design is created from two equilateral triangles each with side length 6cm. What’s the radius of the circle? What’s the area of the star?

22. Teacher notes This year’s festive offering is a collection of 10 puzzles and challenges to keep you and your students amused. There are a range of difficulties so that teachers should find several suitable for each class. Some ‘old favourites’ with a festive twist have also been included.

23. Teacher notes Festive challenge 1: take a pill out of bottle A, break each of the four pills now in your hand in half and take half of each one today and the other half of each one tomorrow. Festive challenge 2: they have the same amount of the other in them. Since they both are still at the same level as when you started, whatever’s missing from the orange juice must be in the apple juice and the missing amount of orange juice has been replaced by an identical amount of apple juice.

24. Matchstick Moves Answers 1 2 3 4 This one is 3d; it’s a tetrahedron

25. Teacher notes Festive challenge 4: just 3. Festive challenge 5:

26. Teacher notes Festive challenge 6 The digits that look the same in the ice rink as they do on the screen are 0, 1, 3 and 8, so all combinations of these are needed. 12 hour clock • Hours that ‘reflect’ are 00, 01, 03, 08, 10, 11 (6 of them) • Minutes that ‘reflect’ are 00, 01, 03, 08, 10, 11, 13, 18, 30, 31, 33, 38. (12 of them) Each of the hours can be combined with each of the minutes and this happens twice in 24 hours. 6 x 12 x 2 = 144 24 hour clock • Hours that ‘reflect’ are: 00, 01, 03, 08, 10, 11, 13, 18 (8 of them) • Minutes that ‘reflect’ are 00, 01, 03, 08, 10, 11, 13, 18, 30, 31, 33, 38. (12 of them) 12 x 8 = 96

27. Teacher notes Festive challenge 7 For the uniform wrapping, open out the barrel. It is then ‘obvious’ to use Pythagoras’ theorem. 4 x 5cm = 20cm The ‘irregular wrapping’ question can be solved either by thinking about specific examples, and it will becomes clear that more tinsel is required, or by geometric reasoning, as on the next slide.

28. Teacher notes Festive challenge 7 Dividing into 4 equal sections ensures that the direct route along the diagonal is taken. Dividing into 4 unequal sections means that the tinsel route is indirect and therefore longer. Using 4 turns, the tinsel must ‘cover’ 4 times the length of the circumference and one times the length of the barrel.

29. Teacher notes Festive challenge 8

30. Teacher notes Festive challenge 9 These are numbers which fall between pairs of consecutive primes 3 4 5 5 6 7 11 12 13 17 18 19 29 30 31 etc. So 138, 150 and 180 are the next ones in the series.

31. Teacher notes Festive challenge 10 DB is 2cm Triangle DBC is equilateral, so BC is 2cm. AB is 1cm Using Pythagoras’ theorem, AC is therefore The star is made up of 12 smaller equilateral triangles, congruent to DBC. The area of DBC is , so the area of the star is

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