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The Answers Black Magic Egrun, the wild green witch of the Wirral, roars along its lanes at nightfall astride her wood-burning Kawasaki broomstick. One night, she rides from the former Cadbury’s factory in Moreton at midnight to visit the Wizard of Oswestry for a short spell. But when she gets to Chester, her faithful catnav, Subordinate Claws, reminds her that she has forgotten the chocolate. So she goes all the way back for it. It’s 20 miles from Moreton to Chester and 30 miles from Chester to Oswestry. Egrun always rides at a steady 120mph and nothing EVER gets in her way. At what time does she arrive in Oswestry? Black Magic Moreton 00:45 00:20 00:10 Chester Oswestry Pick and Mix Robert and Miriam need chocolates as prizes for a quiz: they decide on four bags of Revels, five Mars Bars and five Twix Bars. However, they don’t coordinate their shopping! Robert buys three bags of Revels, two Mars Bars and four Twix Bars for £5.85; Miriam separately purchases two bags of Revels, four Mars Bars and three Twix Bars for £5.60. They return the excess goodies to the shop, receiving £2.60 back. How much does each item cost? Pick and Mix This is best considered as three simultaneous equations: 3r + 2m + 4t = 585 [1] 2r + 4m + 3t = 560 [2] r + m + 2t = 260 [3] 2r + 2m +4t = 520 [4] = [3] × 2 r = 65 [5] = [1] – [4] Pick and Mix Sub into [3] → m + 2t = 195 [6] sub into [2] → 4m + 3t = 430 [7] 4m + 8t =780 [8] = [6] × 4 5t = 350, so t = 70 [8] – [7] Sub r = 65 and t = 70 into [3] → m = 55 So Revels cost 65p, Twix cost 70p, Mars cost 55p Who’ll Nut? Fred and George are to share a chocolate bar made up of an 8 by 6-square rectangular array. The top-left & top-right squares each contain a visible nut. They take it in turns, starting with Fred, to snap the chocolate into 2 pieces along a line between 2 rows or 2 columns, & then eat 1 of the 2 pieces. Neither Fred nor George wants a nut. Which person can guarantee that they won’t get the nut? Is this true for any size of bar? Who’ll Nut? All Because The Ladies Love… Ingrid, Jean, Karen, Philippa and Shirley are sharing a box of chocolates. There are 5 chocolates left in the box: a Coffee Cream, an Orange Fondant, a Hazelnut Praline, a Toffee Crunch & a Nougat Supreme. They agree to have one chocolate each. Ingrid likes only Orange Fondants and Toffee Crunches; Philippa likes only Orange Fondants and Hazelnut Pralines; Jean and Shirley both like Coffee Creams, Hazelnut Pralines and Nougat Supremes; and Karen likes only Toffee Crunches and Nougat Supremes. List all the possible ways that the chocolates can be shared such that all of the ladies get one that they like? All Because The Ladies Love… Ingrid Coffee Cream Jean Orange Fondant Karen Hazelnut Praline Philippa Shirley Toffee Crunch Nougat Supreme The Trouble With Truffles There are 9 chocolates in a selection box: 3 white, 3 milk & 3 dark chocolates. There are 3 soft centres, 3 truffles & 3 pralines. Each chocolate is different. Using the clues below, arrange the chocolates in a 3 x 3 grid: The 3 truffles are in the central column; there is 1 of each colour in each column; there are no 2 soft centres are next to each other horizontally, vertically or diagonally; the same applies for the pralines; the white praline is in the bottom-right corner; the 3 whites are arranged along a diagonal line; there are no dark chocolates in the bottom row. The Trouble With Truffles The 3 truffles are in the central column; there is 1 of each colour in each column; there are no 2 soft centres are next to each other horizontally, vertically or diagonally; the same applies for the pralines; the white praline is in the bottom-right corner; the 3 whites are arranged along a diagonal line; there are no dark chocolates in the bottom row. Be Twix and Between How many different ways can you arrange the letters of the word CHOCOLATE? There are 9 choices for the first letter, 8 for the second and so on. This gives us 9×8×7×6×5×4×3×2×1 = 362,880 = 9! However there are 2 ‘C’s and 2 ‘O’s, so we need to halve this number for each of them 362,880 ÷ 2 ÷ 2 = 90,720 Chocolate Brownies Seven Brownies and three Leaders have baked 35 chocolate cupcakes and eaten them all. One Leader ate one cupcake, another Leader ate two cupcakes, and the third Leader ate three cupcakes. Only whole cupcakes may be eaten. Show that at least one (greedy) Brownie ate at least 5 cupcakes. Chocolate Brownies Brown Owl Barn Owl Tawny Owl Brownie 1 Brownie 2 Brownie 3 Brownie 4 Brownie 5 Brownie 6 Brownie 7 13 20 27 34 35 1 3 6 Cupcakes Cupcake The Answers Fry’s Delight Miss Fry was given a box containing a variety of chocolates. Although she likes chocolates, she is not greedy, so she decided to share her chocolates and make them last. Her method of consumption was to eat one on the first day, and give away 10% of the remainder, to eat 2 on the second day and give 10% of the remainder away, eat three on the third day and give away 10% of the remainder, and continue in this way until no chocolates were left. How many chocolates were in the box and how many days did they last? Fry’s Delight We begin with the assumption that she doesn’t eat or give away parts of chocolates. This means that the minimum she can give away is 1. 1 is 10% of 10, which means she would have 9 left to eat, which would make it the 9th day. Working backwards gives: Fry’s Delight Day 9 8 7 6 5 4 3 2 1 No at start 9 18 27 36 45 54 63 72 81 Eat Leaving 9 8 7 6 5 4 3 2 1 0 10 20 30 40 50 60 70 80 Give Away 0 1 2 3 4 5 6 7 8 End of Day 0 9 18 27 36 45 54 63 72 Bournville Dreams A chocolate factory makes two different types of chocolate: dark and light. Each day, it receives 1200L of milk and 1200kg of cocoa. A batch of dark chocolate requires 20L milk and 50kg of cocoa. A batch of light chocolate requires 40L of milk and 30kg cocoa. The management insists that the workers produce at least 25 batches in total each day and that only full batches can be produced. The company makes a profit of £15 on a batch of dark chocolate and £10 on a batch of light chocolate. Work out how many batches of each type need to be produced to maximise profits. Bournville Dreams A chocolate factory makes two different types of chocolate: dark and light. Each day, it receives 1200L of milk and 1200kg of cocoa. A batch of dark chocolate requires 20L milk and 50kg of cocoa. A batch of light chocolate requires 40L of milk and 30kg cocoa. The management insists that the workers produce at least 25 batches in total each day and that only full batches can be produced. The company makes a profit of £15 on a batch of dark chocolate and £10 on a batch of light chocolate. Work out how many batches of each type need to be produced to maximise profits. First we identify constraints. Cocoa 50d + 30l ≤ 1200 Milk 20d + 40l ≤ 1200 Min Batches d + l ≥ 25 d ≥ 0, l ≥ 0 Bournville Dreams Bournville Dreams Number of Light Batches Maximise: profit = 15d + 10l The optimum solution will always be at a corner of the feasible region. These are: 26 • (0,30) = £300 25.9 • (0,25) = £250 25.8 25.7 • (22,3) = £360 25.6 25.5 • (9,25) = £385 25.4 • (8,26) = £380 25.3 25.2 • (8,25) = £370 25.1 25 Milk = 9 × 20 + 25 × 40 = 1180L Cocoa = 9 × 50 + 25 × 30 = 1200kg 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 Number of Dark Batches 9 Christmas Celebrations At Christmas, Ken likes to buy chocolates for family, friends and carol singers. He decides to spend £200 and has 100 people to buy for. He divides his spending between family-sized tins for £10, small selection packs for friends at £2, and fun-sized chocolate bars for carol singers at 20p each. Given that Ken has more friends than he anticipates carol singers, how many of each item should he buy? Christmas Celebrations T + S + B = 100 10T + 2S + 0.2B = 200 B must be a multiple of 5 and S > B 2T + 2S + 2B = 200 8T - 1.8B = 0 8T = 1.8B 40T = 9B Christmas Celebrations 40T = 9B B = 0, 40 or 80 S > B, so B ≠ 80 B = 0 => T = 0 => S = 100 B = 40 => T = 9 => S = 51 So 40 carol singers, 9 family and 51 friends. Box Clever The gift box for some Fairtrade hot chocolate powder is to be made from a single square sheet of card, with side length of 24cm, by cutting smaller squares (of integer side length) from the corners of the original square. It will then be folded up to make an open box (The lid is made separately). What is the maximum volume that can be obtained in this way? Show, using a graph, that this is the maximum volume. Box Clever cut out 0 1 2 3 4 5 6 7 8 9 10 11 12 base 24 22 20 18 16 14 12 10 8 6 4 2 0 volume 0 484 800 972 1024 980 864 700 512 324 160 44 0 Box Clever 1200 Volume of box (cm³) 1000 cut out 0 1 2 3 4 5 6 7 8 9 10 11 12 800 base 24 22 20 18 16 14 12 10 8 6 4 2 0 Maximum volume of 1024cm³ when cut out is 4cm 600 400 200 volume 0 484 800 972 1024 980 864 700 512 324 160 44 0 0 0 1 2 3 4 5 6 7 8 Size of cut out (cm) 9 10 11 12 Six After Eight A small selection bag of posh Fairtrade chocolates contains 6 chocolates. These are a random selection from 8 different chocolates. A quality-control system is put in place to ensure that no selection contains more than 2 of the same chocolate or fewer than 4 different chocolates. How many different bags of chocolates are possible? Six After Eight There are 3 possible scenarios: 6 unique chocolates 1 pair and 4 other unique chocolates 2 pairs and 2 other unique chocolates Six After Eight 6 unique chocolates 8! 6! 2! 8 7 2 28 Six After Eight 1 pair and 4 other unique chocolates 8 7! 4! 3! 8 7 65 3 2 8 7 5 280 Six After Eight 2 pairs and 2 other unique chocolates 8 7 2 6! 2! 4! 8 7 65 4 12 7 5 420 Six After Eight 28 + 280 + 420 = 728 Seg-sational! There are 20 segments in a chocolate orange. Modelling a chocolate orange as a sphere of diameter 5cm, with a cylindrical hollow of diameter 5mm running down the core, what is the volume of a segment? You may discount the small dome of chocolate at each of the ‘poles’ of the Chocolate Orange. Seg-sational! Volume of sphere of chocolate 4 3 r 3 4 3 2 . 5 65 . 45 cm 3 3 Seg-sational! Volume of hollow cylinder r h 0 . 25 5 0 . 98 cm 2 2 3 Seg-sational! Volume of segment 65 . 45 0 . 98 64 . 47 64 . 47 20 3 . 22 cm 3 Breaking the Mould Anna and Billy play a game with a rectangular chocolate bar that is 5 squares by 10 squares. Anna starts. They take turns breaking a piece of the bar (only one piece can be broken in one turn, always along the lines between the squares). The first player to break off a 1 by 1 square piece wins. Who wins the game?! Breaking the Mould Anna Billy A Year’s Supply of Chocolate Assume that all of the Cadbury’s Dairy Milk bars bought in England were the standard length of 11.4cm. When laid end-to-end, they would form a line around the Earth along the line of latitude passing through Liverpool (53½°N). Using the Earth’s circumference at 40,075km at the equator, how many bars are sold in England each year? A Year’s Supply of Chocolate Radius of Earth = 40075 ÷ 2π = 6378.134km Radius at Liverpool 6378.134 × cos (53.5) = 3793.860km Circumference at Liverpool = 3793.860 × 2π = 23837.523km A Year’s Supply of Chocolate Circumference at Liverpool = 3793.860 × 2π = 23837.523km = 2383752317.9cm Number of bars = 2383752317.9 ÷ 11.4 = 209101080.5118 ≈ 209,101,081 bars