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Can you help find the treasure?

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It was the summer holidays. Archibald, Beatrice, and Cassandra were outside enjoying their picnic of potted meat sandwiches and lashings of ginger beer, when they saw Dave come rushing down the lane with a battered old book in his hand. Dave was their younger, less pretty, and more dopey sibling. Archibald, Beatrice and Cassandra didn’t like to tell him this to his face – mainly because they didn’t like looking at his face. But he was their brother, and they loved him all the same.

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“I’ve just found this in Aunt Mathelina’s library,” panted Dave, out of breath from running through the village. “It’s a book.” “No way. A book? In a library? Whatever next?” piped up Beatrice sarcastically. “Shut your mouth, Beatrice. It looks really old. I’ve flicked through it and it seems like a history of the island. There’s a page that says there has been treasure hidden around here somewhere for hundreds of years, but it’s yet to be found.”

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“Golly, that sounds frightfully exciting!” exclaimed Archibald. Archibald loved a good adventure, almost more than he liked potted meat sandwiches and ginger beer. “Where do we start?” “Well,” said Dave, “there’s a map here. It says something about victors. What are victors? Wasn’t our great-grandad called Victor? Maybe he has something to do with it.” “VECTORS, you muppet. It says VECTORS. Give it here,” muttered Cassandra as she grabbed the dusty book and read the first of what was to be many puzzles in their adventure….

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‘There is more to be found at Ye Olde Mill, but watch out for the mines,’ read the note. ‘Start at the Farmhouse, and ye must only use these vectors’ What route must the children take?

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When Archibald, Beatrice, Cassandra and Dave reach the Old Mill, they are faced with a 3 metre high wall. There is a 3.5m long ladder lying on the floor. The ladder must be placed at an incline of at least 60 o to climb it safely. Will the ladder reach the top of the wall?

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The ladder just reaches the top of the wall! With centimetres to spare! Over the wall, the terrific trio (and Dave) are then faced with 4 cylindrical tunnels. “Sheesh, they’re not making it easy to find this treasure!” moaned Dave. “Well, if it was easy, it would have been found a long time ago, you dope,” replied Archibald rolling his eyes.

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The first tunnel’s face has a diameter of 50cm. The second tunnel’s face has a circumference of 2m. The perimeter of half of the third tunnel’s face is 145cm. The radius of the final tunnel’s face is 30cm. The widest points of the terrific trio (and Dave) are: Archibald – 48cm Beatrice – 55cm Cassandra – 58cm Dave – 62cm. They must enter the tunnels at the same time, so only one person can go in each tunnel. Who should go in which tunnel?

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Archibald, Dave, Beatrice and Cassandra made their way through their respective tunnels and came out to see a message scrawled on to the wall in front of them. “It’s those victor….I mean vector things again!” shouted Dave. “I think I understand them now after we used them to dodge those mines.” “But how are we meant to follow the vectors when they haven’t said what a or b is?” whined Cassandra. “The wall must have crumbled away where the vectors were,” said Archibald. “Look, the ground is covered in bits of rock and dust where it looks like it’s worn away.” “Oh great, well that’s our adventure over then,” sighed Beatrice. “I suppose we might as well head back. At least we’ll be back in time for tea.”

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The terrific trio turned to head back into the tunnels, but Dave was still stood scratching his head and staring at the message. “I think….yes….I think if you followed those vectors, you’d just end up back where you started.” “Don’t be ridiculous, Dave. It’s impossible to solve the puzzle without knowing what a and b are,” said Cassandra. “Don’t be a dingbat.” “Yeah, Dave, don’t be a dingbat,” said Archibald and Beatrice in unison. But Dave wasn’t being a dingbat. He was right. How did he know?

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“Bloomin’ ‘eck! He’s right!” exclaimed a shocked Cassandra. Beatrice still looked confused. “But, if we don’t have to move anywhere, what do we do now? I don’t get it. Dave? Do you have any bright ideas?” “Nah, I’m done for the year now. I’ve had my one bright idea,“ said Dave who was sat down on the floor from exhaustion, making patterns in the dust with his finger. “Woah, woah, woah!” shouted Archibald, pointing at the ground where Dave was unintentionally revealing what looked like a grid scratched into the floor. “Look!” After much scrabbling around, the terrific trio (and Dave) uncovered a set of axes with five shapes, along with five instructions:

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“Oh I say!” giggled Cassandra giddily. “Now we know what the key looks like! We just need to get the key cut! And I bet old Mr Kobbler in the village will be able to help us with that tomorrow.” “Well yes, but what happens once we’ve got the key?” asked Beatrice. “A key is no good without a keyhole!” “Well we’ll just have to see what tomorrow brings,” announced Archibald. And so the children headed back to the farmhouse, and rested for the exciting day ahead.

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The next day, Cassandra headed to Mr Kobbler’s hut in the village to see if he would cut the key that matched the shape. “Now then, now then,” mumbled Mr Kobbler. “What have you got there, Missy?” “I was wondering if you could cut a key to match this,” Cassandra enquired, holding out the diagram. “Hmmm, well I’ll have a go, lass, that’s for sure. What on earth is it for, may I ask?” “Well, it’s….erm….it’s the key to some treasure,” Cassandra replied sheepishly, thinking Mr Kobbler would think she was a little mad.

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“Ah, a little game you’re playing with the others is it? Shame you haven’t found the key to the treasure at Smayth Castle!” chuckled Mr Kobbler. “What do you mean?” asked Cassandra, taken aback. Maybe Mr Kobbler knew something that would help in their adventure. “There’s a great iron door at the very top of the castle. It’s been shut tight for as long as I can remember, and come to think of it, as long as any of my relatives can remember. Rumour has it that there’s some kind of treasure behind it, but you need a special key to get in, you see. Maybe you should give this key a go, haha!"

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Cassandra couldn’t contain her excitement, and without thinking of the consequences, spilled the story of how Dave had found the book in Aunt Mathalina’s library, and how they had solved the puzzle at the Old Mill to make the shape of the key. “My word,” gasped Mr Kobbler, “that’s incredible! You could be into some serious money there! In fact, I should really charge you a lot more for this key cutting. I tell you what, in return for telling you the location of the castle and for cutting the key for you, I’ll give you two options. You can either pay me £7 per cubic centimetre of metal, or a flat fee of £100.” Which option should Cassandra have gone for?

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“Hmm, I reckon the price for 15.42cm 3 would be £107.96. I suppose I’d best go for the set rate of £100,” calculated Cassandra. “Thank you very much,” she said to Mr Kobbler, shaking his hand and passing him over five £20 notes. “No my dear, thank you,” Mr Kobbler replied with a grin that Cassandra thought seemed a little more sinister than the old man’s usual gentle smile. Thinking no more of it, she picked up the key, and set off to meet her three siblings at their agreed meeting place at the Black Swan pub.

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In order to get the pub, the children took two different routes; Archibald and Beatrice walked along the line 3x + 4y = 18 Dave walked along the line 2x – 5y = –11 Where is the Black Swan? Cassandra is coming from Mr Kobbler’s hut at (4,6). The equation of what line will take her to the Black Swan?

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Whilst the four children enjoyed some homemade lemonade at the Black Swan before they set off to the castle, Cassandra told the others about what had happened with Mr Kobbler and the price he had set. “What a money-grabber!” said Dave. “He obviously knew this treasure might be worth something and charged you a fortune! Wouldn’t surprise me if he’s making a copy of the key right now, the scoundrel! And I thought he was a nice old m….” Dave stopped mid-sentence when he realised that Archibald, Beatrice and Cassandra were looking at each other in horror.

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“Oh my goodness! The treasure! He’s after it too!” gasped Beatrice. “We need to get to the castle before he does!” And so, the children abandoned their lemonades and ran as quickly as they could to the castle.

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The children and Mr Kobbler set off running to the castle at (6,5) at the same time. The children run from the Black Swan at (2,3) at 9km/hr. Mr Kobbler runs from his hut at (4,6) at 5km/hr. They both take the shortest route possible. You may assume that one square on the grid represents 1km. How long does it take the children and Mr Kobbler to reach the castle?

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The children got to the castle in 30 minutes, but Mr Kobbler arrived sooner in 27 minutes. The children raced up the spiral staircase to see Mr Kobbler about to put the key in the lock. “STOP!” yelled Dave. Mr Kobbler spun round and gave a cackle. “Hahaaaa, I seem to have got here before you, you silly children! And now the treasure that lies behind this door is mine, all mine! In fact, you can keep your measly hundred pounds – after all, it will be pennies compared to the riches that await me!” Mr Kobbler threw the money that Cassandra had given him for the key in the faces of the four children with another evil laugh, and turned to finish turning the heavy key in the lock.

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The lock was opened, but the rusty iron door was stiff and Mr Kobbler needed to push with his whole body to open it. After a big heave, the door burst open, and Archibald, Beatrice, Cassandra and Dave watched with mouths wide open as Mr Kobbler fell into the room and plunged down a great big trapdoor in the floor. There followed an almighty splash, and the four children peered over the drop to see a very dirty Mr Kobbler spluttering around in what looked (and smelled) like a filthy sewer.

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“Why you little pesky kids!” he spluttered from the pit below. “I’ll see you get what you deserve for this!” And Archibald, Beatrice, Cassandra and Dave had got what they deserved – a wonderful weekend of puzzling adventure, and the chance to see an evil man get his come-uppance.

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And so, the four children made their way down the spiral staircase of the castle tower, and skipped merrily through the fields back to the farmhouse. Just in time for tea.

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‘There is more to be found at Ye Olde Mill, but watch out for the mines,’ read the note. ‘Start at the Farmhouse, and ye must only use these vectors What route must the children take?

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When Archibald, Beatrice, Cassandra and Dave reach the Old Mill, they are faced with a 3 metre high wall. There is a 3.5m long ladder lying on the floor. The ladder must be placed at an incline of at least 60 o to climb it safely. Will the ladder reach the top of the wall?

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They must enter the tunnels at the same time, so only one person can go in each tunnel. Who should go in which tunnel? The first tunnel’s face has a diameter of 50cm. The second tunnel’s face has a circumference of 2m. The perimeter of half of the third tunnel’s face is 145cm. The radius of the final tunnel’s face is 30cm. The widest points of the terrific trio (and Dave) are: Archibald – 48cm Beatrice – 55cm Cassandra – 58cm Dave – 62cm.

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‘To find how to get the key to the treasure, you must follow the vectors below. 2a + b + 2(a – b) – 3a + (b – a).’

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Complete the transformations to.... construct the shape of the key to the treasure Reflect shape A in the line x = 1. Rotate shape B 90 o anticlockwise around the point (0,0). Translate shape C by the vector Reflect shape D in the line x = –1. Rotate shape E 180 o around the point (3,0).

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You can either pay me £7 per cubic centimetre of metal, or a flat fee of £100.” Which option should Cassandra choose?

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Where is the Black Swan? Archibald and Beatrice walked along the line 3x + 4y = 18 to the pub. Dave walked along the line 2x – 5y = –11 to the pub. Cassandra is coming from Mr Kobbler’s hut at (4,6). The equation of what line will take her to the Black Swan?

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How long does it take the children and Mr Kobbler to reach the castle? Who gets there first? The children and Mr Kobbler set off running to the castle at (6,5) at the same time. The children run from the Black Swan at (2,3) at 9km/hr. Mr Kobbler runs from his hut at (4,6) at 5 km/hr. They both take the shortest route possible. You may assume that one square on the grid represents 1km.

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