Unizor – Creative Mind through Art of Mathematics: Logic+ 08: UNIZOR.COM – Math+ &Problems

Notes to a video lecture on http://www.unizor.com Logic+ 08Problem ATrains between points A and B are running regularly with the same speed and the same interval between trains in each direction.As you go in a train from A to B, you see trains going in an opposite direction every 10 minutes.How many trains come to point A in an hour?Answer A:Three trains come to point A during every hour.Problem BSome crime was committed, and there are only three suspects, only one of which has committed this crime.Suspect A said that suspect B committed this crime.Suspect B denied this accusation.Suspect C said that he is innocent as well.Who committed this crime, if it becomes known that only one of the suspects told the truth, while two others lied?Solution BIf A said the truth, and B indeed committed this crime, C said the truth as well, which contradicts the condition of only one of the suspects telling the truth.So, A has definitely lied, so the crime was committed by either himself or by C.If the crime was committed by A, then B and C both said the truth, so we again have a contradiction with a condition that there is only one truthful person.Therefore, the only conclusion we can come up with is that this crime was committed by C.Checking BIf C committed the crime, both A and C lied, while B said the truth, which satisfies the condition of a problem.Problem CSolve the equationa+2b+3c+4d = 22for a, b, c and d, if it’s given that these unknown numbers are all positive integers, are different and do not exceed 4.Hint CThere are four unknowns and four possible values:1, 2, 3 and 4.So, the question only is, which of them is a, which is b, which is c and which is d.From Combinatorics we know that there are 4!=24 ways to assign 4 values to 4 unknowns, so we can try all of them until we get the proper equation. But this is not a nice approach. Try to reduce the number of calculations by applying some logic.The easy way to reduce the number of calculations is to start assigning the highest possible value (4) to variable d, which participates in the equation with a multiplier 4. This immediately renders the impossibility to assign the other variables values because the left side of an equation will obviously exceed 22.Answer Ca=3, b=4, c=1, d=2Problem DThere are red and blue balls in a box.Only one ball is not red.All balls except one are blue.How many balls are in a box and what are they?AnswerThere are two balls in a box – one red and one blue.Problem EGiven three objects: a table, a bottle and a book.When a bottle is on a table and a book is on the floor, the distance from the top of a bottle to a top of a book is 100 cm.When a book is on a table and a bottle is on the floor, the distance from the top of a bottle to a top of a book is 50 cm.What is the height of a table?AnswerThe height of a table is 75 cm.