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.
a) You have a \(10\times20\) chocolate bar and 19 friends. Since you are good at maths they ask you to split this bar into 19 pieces (always breaking along the lines between squares). All the pieces have to be of a rectangular shape. Your friends don’t really care how much they will get, they just want to be special, so you need to split the bar in such way that no two pieces are the same.
(b) The friends are quite impressed by your problem solving skills. But one of them is not that happy with the fact you didn’t get a single piece of the chocolate bar. He thinks you might feel that you are too special, therefore he convinces the others that you should get another \(10\times20\) chocolate bar and now split it into 20 different pieces, all of rectangular shapes (and still you need to break along the lines between squares). Can you do it now?
a) Joker prepares 13 blank cards. He writes a natural number on each of them. (Natural numbers are whole positive numbers.) Then for all 13 numbers he calculates their product and sum. Joker gets the same result for both. Is this some kind of trick or is it really possible? Why?
(b) What is the answer if we don’t know how many cards he uses but we know that both results are equal to 13?
The edges of a cube are assigned with integer values. For each vertex we look at the numbers corresponding to the three edges coming from this vertex and add them up. In case we get 8 equal results we call such cube “cute”. Are there any “cute” cubes with the following numbers corresponding to the edges:
(a) \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\);
(b) \(-6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6\)?
A candle that burns out completely in one hour costs \(60\) pence and a candle that lasts for \(11\) minutes costs \(11\) pence. For some reasons which he doesn’t want to explain to the seller, Mr. Fawkes wants to measure exactly \(1\) minute with some of these candles. Can he manage to do that if he has only \(1\) pound and \(50\) pence to spend on the candles?
Mr. Fawkes can only measure the whole interval with each of the candles. He is not guaranteed that any of the candles burns uniformly, so he cannot divide it into two parts. He cannot make it burn faster by igniting them from both ends, but he can extinguish a candle and light it up again later.
There are 36 checkers on a \(6\times6\) board (one checker per one cell). They are numbered from 1 to 36. You need to rearrange the checkers so that the ones with numbers which differ by 1 are neighbours (their cells share an edge) and additionally all square numbers have to correspond to the checkers of one (a) horizontal line; (b) of the diagonals. Is it possible or not?
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Could you formulate a question which both a liar and a knight would answer identically (either “Yes” or “No”)?
Inhabitant A tells Inhabitant B, “At least one of us is a liar”. Who is A and who is B?
Three inhabitants are passing by. You ask the three inhabitants, “How many among you are knights?” The first one replies, “There is none”. The second inhabitant argues, “There is only one”. What should the third inhabitant say?