Express the number 111 as a sum of 51 natural numbers so that each of the terms has the same sum of digits.
There are 36 parcels weighing 1 kg, 2 kg, 3 kg, ..., 36 kg. Today only three cars are in service. Each car has a capacity of 12 parcels. Can one distribute all packages between the cars in such a way that each vehicle has the same total weight of parcels?
Can you cover the surface of a cube with 16 identical colourful rectangles? No overlappings are allowed and the cube has to be fully covered.
a) Express the number 221 as a sum of 52 natural numbers so that each of the terms has the same sum of digits.
(b) Express the number 226 as a sum of 52 natural numbers so that all terms have the same sum of digits.
a) In the context of Example 2 assume we have some number of parcels each weighing different amount of kilograms. We still have 3 identical cars of equal capacities (in numbers of packages) and we still want to distribute parcels in such a way that each car has the same total weight of parcels. Knowing that the number of parcels is not greater than 100 find the maximum and the minimum amounts of packages for which it is possible.
(b) Now we have 3 trucks so we do not really care about the sizes of parcels and their number. But yet we need to satisfy the condition of equal total weights of parcels in each vehicle. Can we do so if there are 27 packages weighing 1 kg, 2 kg, ..., 27 kg?
Sometimes life can make us do the craziest of things. In this problem you just need to find out how one can cut an \(8\times8\) chessboard into 20 pieces each having the same perimeter and consisting of a whole number of cells.
A battalion of soldiers was marching towards a captured city. Their progress was stopped by a wide river. Fortunately, close to the shore there were two boys sailing in a small boat. They escaped from the city and were eager to help the soldiers to cross the river. The only obstacle was that their boat could fit either two boys or one soldier. Taking into account one person was enough to handle that kind of boat (i.e. to sail from one shore to another) and the fact that on the next day the city was liberated so the boys could reunite with their families describe how the battalion was capable of crossing the river.
The cube from Example 3 is a present and one layer of a gift-wrap is totally not enough. Can you cover it with another 15 identical rectangles? You can assume the covering from Example 3 was thin and it did not affect the shape of a cube. As before no overlappings are allowed and the surface has to be fully covered by rectangles.
(a) Jimmy is working on a metal model of a grasshopper. He named it Kimmy. The boy keeps on adding new features to his robot. Besides being an accurate alarm clock, Kimmy can jump in one or two cells depending on how many times Jimmy claps his hands. Do you think the boy can choose a sequence of claps in such a way that the robot will visit all cells of a \(1\times101\) strip?
(b) What if the task is to visit all cells of a \(1\times99\) strip?
Two clowns A and B are playing the following game. They have 33 tomatoes on a plate. One of the tomatoes is rotten and both clowns know which one. Each move they can choose one, two, or three of the remaining tomatoes from the plate and smash them into their own faces. They take turns and the clown who chooses the rotten tomato looses the game. They cannot skip the moves. Clown A starts the game. Does A or B have a winning strategy? (A winning strategy is a strategy following which you win no matter how your opponent plays.)