C.1265. On a castle-building competition everybody has 10 cubes to build a building. You can place a cube on top of or next to another cube so that every cube matches up with a whole side with the other cubes it touches. The building cannot have a hole in it, and you should be able to climb to its top from the ground level by going one cube higher every time you make a step. The value of a building in the competition is calculated by multiplying the height by the area of the base of the building. Let’s suppose that the participating contestants built all possible buildings. Is it true that the winner built the tallest building?
C.1266. In the showroom of a lighting store you can see 2-, 3- or 4-armed chandeliers. They put out 8, 9, and 10 chandeliers from these kinds, but we do not know which number belongs to which kind of chandelier. There is one light bulb on each arm of each chandelier. What could the total number of light bulbs in the showroom be? (List all possible numbers.)
C.1267. Arshan and Ben are playing with a chair. First Arshan stands on the chair, while Ben is standing on the floor. This time the top of Arshan’s head is 30 cm higher than the top of Ben’s head. When they switch, so that Ben is standing on the chair and Arshan is standing on the floor, the top of Ben’s head is at a half a meter higher level than the top of Arshan’s head. How tall is the part of the chair they stood on?
C.1268. We glued together several unit cubes to make a solid (no hole in the middle) big cube. There are a total of 80 unit cubes that are either at the vertices or on the edges of the big cube (a cube at the vertex is not counted on the edge). How many unit cubes did we use to build the big cube?
C.1269. An isosceles Pythagorean tree grows the following way: In the first year it grows its trunk, which is a square. In the second year it grows an isosceles right triangle so that its diagonal is the top side of the square, and it grows two branches from the two legs of the triangle, which are also squares. Then it continues every year: each new square grows an isosceles right triangle so that its diagonal is the top side of the square, and each new triangle grows two branches from the two legs of those triangles, which are squares. Let’s suppose that the trunk of the tree, which is the first square, is 8 meters wide. How tall and how wide is the tree at the end of the fifth year? (Notice that the diagram shows the tree after the third year.)
C.1270. You have a 7-liter and an 11-liter container with no markings on them. Both of them are empty. You may fill up either of them from the water faucet any number of times, you may empty either of them at any time, and you may pour water from one container into the other any number of times. Using only these steps, can you get 2 liters of water in one of the containers?
C.1271. You can buy an energy-saving light bulb for $18.07, which can work for 10,000 hours. It costs you 0.8 cents to operate it for an hour. ($1 = 100 cents). A regular light bulb with the same intensity costs 39 cents. This light bulb can work for 1000 hours. It costs you 3.4 cents to operate it for an hour. After how many hours of operation is the use of the energy-saving light bulb more economical?
C.1272. Steve showed an interesting multiplication to his classmates. To multiply 37 and 64, he wrote the two numbers one under the other, and then he wrote the following numbers on the board while saying: 7 times 4 is 28, I write down the 8 and remainder 2. 3 times 4 is 12, I add the remainder and I write down the 14. Then 7 times 6 is 42, and finally, 3 times 6 is 18. Add them up and the product is 2368. In the next example, Steve multiplied 59 by 78. What did he write on the board?
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