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The Answers

The Answers. Senior. Challenge ‘ 14. Game for Olaf.

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The Answers

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  1. The Answers Senior Challenge ‘14

  2. Game for Olaf Along a shoreline, running North-South, there are seaward-facing steep cliffs. There is a harbour in a narrow river mouth which forms a gap in the cliffs. Olaf, a Viking re-enactor, is on the open sea, somewhere to the East of the harbour. A thick fog descends, reducing visibility to just a few metres. Olaf has to sail towards the harbour using his compass. He knows that his compass is 5 degrees out from true, but can’t remember in which direction. How should he proceed in order to safely get to the harbour?

  3. Game for Olaf • Throw the compass overboard to test the depth • Watch for the currents around the river’s mouth • Wait for the fog to clear

  4. Game for Olaf

  5. Fenrir’s Chain Fenrir the wolf is bound by a magical chain. The chain is an endless piece made up of 30 links. It was originally forged as six pieces, each made up of 5 links. It costs 2 silver coins to cut a link and 4 to weld it together again. Find the cheapest way to create the endless chain.

  6. Fenrir’s Chain 6 × 6 = £36

  7. Fenrir’s Chain 5 × 6 = £30

  8. Odin, Norse god of war, has only one eye according to their myths. An expedition of one-eyed Viking warriors sets out in a longboat from Norway, with some chickens and some cows. At this point, there are 70 pairs of legs aboard. They raid a settlement on the Scottish coast. 2 warriors are imprisoned by the Scots, but they manage to seize a cow and 5 chickens for the onward voyage. There are now 90 eyes on the longboat. By their arrival in Greenland, one more warrior has been buried at sea, 2 cows have been slaughtered for beef, but 2 eggs have hatched into chickens! 68 mouths dock at Greenland. How many Vikings, cows and chickens began the trip? Eye, Eye

  9. Eye, Eye v = Vikings, 1 eye, 2 legs, 1 mouth c = cows, 2 eyes, 4 legs, 1 mouth h = chickens (hens), 2 eyes, 2 legs, 1 mouth

  10. Eye, Eye Legs: v + 2c + h = 70 Eyes: (v - 2) + 2(c + 1) + 2(h + 5) = 90 => v + 2c + 2h - 2 + 2 + 10 = 90 => v + 2c + 2h = 80 Mouths: (v - 3) + (c - 1) + (h + 7) = 68 => v + c + h - 3 - 1 + 7 = 68 => v + c + h = 65

  11. Eye, Eye Legs: v + 2c + h = 70 Eyes: v + 2c + 2h = 80 Mouths: v + c + h = 65 Comparing legs and eyes gives us 10 chickens Comparing legs and mouths gives us 5 cows

  12. Eye, Eye Mouths: v + c + h = 65 Substituting in 10 chickens and 5 cows gives us: v + 5 + 10 = 65 So v = 50 Vikings

  13. Viking Babies A Viking village is founded by a population of 100 couples. Each couple wishes to "continue the male line" but, as resources are scarce, they do not wish to overpopulate.  The agreed rule is that each couple has 1 baby per year, until the arrival of their first boy. For example, if, at some future date, a family has 5 children, then it must be 4 girls & a boy (the boy being the youngest and no more children planned), or 5 girls with another child planned.  Assume that children are equally likely to be born male or female.  Let p(n) denote the expected fraction of male children born in the nth year from the founding of the village. Give a formula for p(n) for arbitrary positive n.

  14. Viking Babies Let p(n) denote the expected fraction of male children born in the nth year from the founding of the village. So p(n) = ½

  15. Viking Babies “Alternatively, the villagers could stop being so misogynistic and stop subscribing to the ideas of the patriarchy, thus – hopefully – realising that ‘continuing the male line’ is an outdated, stupid idea.” SS

  16. Escape the Monastery Mervin the Monk is running through the monastery to collect all of the treasure to take it to safety before the Viking raiders arrive (see the map on the left). He only has time to visit every cell once, and must start through the entrance door on the South side of the monastery and leave through the exit door on the North side. How many different routes are there where he visits all of the cells only once? N

  17. Escape the Monastery N N N N

  18. Land Ahoy! Sigmund’s long boat is sailing towards the shore. His wife, Freya, is standing on the beach (at sea level) waiting for her husband to arrive. As the sea is calm, Sigmund is standing in the crow’s nest at the top of the ship’s 15m mast. Given that the radius of the Earth is 6.4 x 106 m, and the long boat is sailing at a constant speed of 2m/s, for how long will Sigmund be able to see Freya before she can see the long boat’s waterline? Assume they both are 1.5m tall and have perfect eyesight. Give your answer to the nearest minute.

  19. Land Ahoy! So Distance along curve = =14532m R+16.5 R R+1.5 A

  20. Land Ahoy! Time = 14532 / 2 = 7266 seconds Sigmund can see Freya for 2 hours and 1 minute before she can see the waterline of the boat. R+16.5 R R+1.5 A

  21. Mead and Drink Sven, a young Viking, has come of age and has outgrown his old helmet. A new one is made for him. You may assume that the basic helmet is a hemisphere, but with a cylindrical band attached underneath, which measures 60cm around the outside and is 5cm deep. What is the area of sheepskin needed to cover Sven’s helmet? How many litres of mead could Sven drink if he were to fill his helmet to the brim (whilst holding it upside down)?

  22. Mead and Drink - Volume Radius = Volume of cylinder =

  23. Mead and Drink - Volume Volume of hemisphere = So total volume = 1432.394 + 1823.781 = 3256.175cm3= 3.26 litres

  24. Mead and Drink – Surface Area Surface area of cylinder = Surface area of hemisphere = Total surface area = 300 + 572.96 = 872.96cm2

  25. Eating Seating Erik is holding a feast for himself and 9 other Vikings. As host, he will sit at the head of the long rectangular table; one guest will be seated at the foot of the table, with four on each side. Hagar doesn’t want to sit at the foot. Olaf and Leif insist on being adjacent to one another and sitting on the side to Erik’s left; Arvid, Balder & Colby are superstitious and refuse to be on the side to Erik’s left. Delling, Frode and Gilby have no preference. How many arrangements are possible?

  26. Eating Seating E • Erik is at the head of the table • Olaf and Leif must be together on Eric’s left, this gives 2×3=6 options. • There are now 2 possibilities. Either Hagar is on the left as well or he is on the right

  27. Eating Seating E • If Hagar is on the left, he can sit in one of 2 seats • The remaining seat can only be filled by Delling, Frode or Gilby • This gives 6 options. • The remaining 5 can be filled by anyone, so there are 5! = 120 options. • Altogether, this gives 6 × 6 × 120 = 4320 options. D, F, or G

  28. Eating Seating E • If Hagar is not on the left, these two seats must be filled by 2 of Delling, Frode or Gilby • This gives 6 options. • Hagar now has 4 options. • The remaining 4 can be filled by anyone, so there are 4! = 24 options. • Altogether, this gives 6 × 6 × 4 × 24 = 3456 options. D, F, or G D, F, or G

  29. Eating Seating E • Overall then, there are: • 4320 + 3456 = 7776 options.

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