Girls who skipped leg days

www.mathe-treff.de Maths-Treff: puzzles for grades 5 and 6 June - August 2016 Submission deadline: August 31, 2016 Task 1 time back yard service In class 6c there are again discussions about who should do the yard service. The class consists of 15 girls and 15 boys. Both the girls and the boys claim that it was the last time they took over the entire farm service. Since they cannot agree, the lot should now decide who is to act. a) Jupiter brings the boys together, there is a lot of whispering. Then they propose the following deal to the girls: “We all stand up in a circle, then we count clockwise. Every ninth of us will then be assigned to court service. Those that have already been counted will be skipped in the next round. ”This is how it is done. In the end, only girls have to do court service. How can that be? Make a drawing that shows how the boys must have positioned themselves. b) The girls do not want to let this defeat sit on them. They propose a new deal for the next but one court service (the boys have to take over the next one anyway for reasons of justice): “The boys and girls each choose 6 representatives who line up in a circle in the form of a clock face. The class representative thinks of any number between 1 and 12 and counts clockwise from 12 as many hours as this number of letters has in the English translation. (Example: The number three in the English translation has five letters). Then it lands on a new number. (In our example you would land on the five.) Now he again counts as many hours as this new number on which he stands has letters in the English translation. He repeats this procedure a third time. The one of all of us who is on the clock, who was the last to be “hit”, can decide who should do court service. ”Since the class representative is a boy and he thinks the game depends on his decision, which number he should at the beginning chooses to have ??????? \$ ?????????? in hand, the boys respond to the suggestion. This time the girls win. They say they didn't take any chances. How can that be? Exercise 2 Dekominos from Pentominos The so-called "Pentominos" are composed of five equal squares, which are connected to each other at the edges. There are a total of 12 of them, with rotated and mirrored pentominos to be considered the same. The same principle applies to the so-called Dekominos, only that they are made up of ten squares each. There are many more of them, namely 4655 pieces. a) Put the Dekomino in the picture together from two pentominos. Remember that the pentominoes could also be mirrored. How many different solutions are there? b) Which of the pentominos cannot be used? Explain why certain of them can be excluded from the start. Exercise 3 Can you copy it from the neighbor? The five friends Luca, Tim, Julia, Paula and Aisha sit next to each other in the first row in front of the desk. There are three tables next to each other, each of which can seat two people. Luca and Tim only have one table neighbor. Julia can look directly into the notebook with both Paula and Aisha. That doesn't work with Aisha and Luca. Paula is sitting further to the left than Tim, seen from the desk. All five have left no space between them. Who sits where?