(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?
There are 30 people standing in a queue in front of a candy shop. We know that among every ten people standing one after another there are more adults than kids. Is it possible that among all 30 people there are more kids than adults?
Half an hour later (see example 3) there are only 15 people standing in the queue. The condition about every ten consecutive members of the queue is still satisfied. Can we guarantee that there are more adults than kids?
Three liberals and three democrats are about to cross the river. The only available boat has two seats. The difficult part is that the democrats do not want to be outnumbered on any of the shores while the boat is on water (they are afraid in such case they will get pushed into the river by the liberals). To avoid another civil war you need to organise their transit properly.
Ten ladies and ten gentlemen regularly attend a dancing club. Last week the participants gave a short performance for their relatives and friends. They showed ten different dances. Every gentleman had a chance to dance once with every lady. It turns out that every lady danced her next dance with either blonder or taller partner than the previous dance. Explain how that could be possible.
(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 the distance of 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 exactly once? (The robot is not allowed to leave the strip.)
(b) What if the task is to visit exactly once all cells of a \(1\times99\) strip? (The robot is not allowed to leave the strip.)
Ten ladies and ten gentlemen regularly attend a dancing club. Last week the participants gave a short performance for their relatives and friends. They showed ten different dances. Every gentleman had a chance to dance once with every lady. It turns out that every lady danced her next dance with either blonder or taller partner than the previous dance. Explain how that could be possible.
After having a very “spiritual” conversation with her personal yoga instructor Mrs. Robinson decided to learn more about Feng shui. In her guest room there are 16 chairs and she never liked the way they are standing. She has read through some books on Feng shui and searched through the internet looking for a solution to her problem. Finally, Mrs. Robinson found a short paragraph in one of the old issues of a national magazine about health and home. In this paragraph it was explained that one has to rearrange chairs in such a way that there are exactly 5 chairs standing next to each wall. Taking into account Mrs. Robinson’s guest room has a shape of a square describe how she needs to rearrange the chairs in order to satisfy the condition. (Feng shui is a Chinese philosophical system of harmonizing everyone with the surrounding environment.
Six glasses are placed in a row. The leftmost three are full, and the other three are empty. By doing manipulations with just one glass you need to make empty glasses alternate with full glasses.
In the following puzzle an example on multiplication is encrypted with the letters of Latin alphabet: \[{BAN}\times {G}= {BOOO}.\] Different letters correspond to different digits, identical letters correspond to identical digits. The task is to solve the puzzle.