Maths Circles Week 3.pdf

Maths Circles Week 31. You are given 10 bags of coins, 9 of them contain fake gold coins and 1 bag contains realgold coins. The real gold coins weigh 150g and the fake gold coins weigh 100g. You areallowed use the weighing scale just once, which bag has the real gold coins?2. There are several chickens and rabbits in a cage (with no other types of animals). Thereare 72 heads and 200 feet inside the cage. How many chickens are there and how manyrabbits?3. You are on an island and there are three crates of fruit that have washed up in front of you.One crate contains only apples. One crate contains only oranges. The other crate containsboth apples and oranges.Each crate is labeled. One reads "apples", one reads "oranges", and one reads "apples andoranges". You know that NONE of the crates have been labeled correctly - they are allwrong.If you can only take out and look at just one of the pieces of fruit from just one of the crates,how can you label ALL of the crates correctly?4. Try to make up the number 1000 using exactly 8 8s. (You can use addition, subtractionmultiplication, division etc).5. Move 4 Matches to make 3 squares, try moving just 3 matches to make 3 squares!

6. Shown below is the map of the roads connecting a number of towns. Create a path thattravels over each road once and only once. (The blue lines are the roads and the red dotsare the towns).

Solutions:1. Take 1 coin from bag 1, 2 coins from bag 2, etc. Weigh all of these and the total weight willbe 55*100g + n*50g where n is the number of the bag.2. 28 rabbits and 44 chickens.3. Take a piece of fruit out of the crate that says apples and oranges and whatever fruit youtake out, that’s the fruit that’s inside the crate. Then you can use this to switch around theother 2 since all of the labels are wrong.4. 888+88+8+8+8 or (888-8)+8*(8+8)-8 or ((8*(8+8))-((8+8+8)/8))*8 or(8+((8+8)/8))^((8+8+8)/8) or (8+((8+8)/8))^((88/8)-8) or ((8*(8+8))-((88/8)-8))*8 or(8888-888)/8.