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Week 10 1. Three logicians are placed in a room and hats, either black or white, are placed on their heads. Each, secretly, either guesses the colour of their hat or chooses not to guess at all. If any of them are wrong, or they all choose not to guess, they die painful deaths. Find a strategy such that they have a 75% chance of surviving if they follow it. 2. Can they do any better? Is 75% the best they can hope for? 3. Show that, if we partition 1,2,. . . ,1978 into 6 disjoint sets, that in each set there is a number that is either the sum of two other numbers in the set, or is twice a number in the set. 4. Let m, n ∈ N be such that m 1 1 1 1 1 = 1 − + − + ··· − + n 2 3 4 1318 1319 Show that m is divisble by 1979. P n2 5. Compute ∞ n=1 2n 6. A stick is broken in random in 2 places (the 2 break-points being chosen independently). What is the probability that the 3 pieces form a triangle? 7. Can you find an n such that when you move the last digit, of the decimal expansion, to the start of the number, you obtain 3n? How many n are there? 8. Consider a 3 × 3 magical square: a square of numbers such that the sums across each row, the sums down each column and the sums along each diagonal are all equal. Show that they form a vector space. Find a basis for this space. 9. There is an unlimited supply of fuel at one edge of a desert that is 800km wide, but no source of fuel in the desert itself. A truck can carry enough fuel to travel 500km and can also build refueling stations at any spot along its travels, which it can leave some of its fuel at. How much fuel, in km, will the truck need to use to cross the desert? Is there a limit to how far it can go? 1