A kindergarten used cards for teaching children how to read: on some, the letter “MA” are written, on the rest – “DA”. Each child took three cards and began to compose words from them. It turned out that the word “MAMA” was created from the cards by 20 children, the word “DADA” by 30 children, and the word “MADA” by 40 children. How many children all had 3 of the same cards?
Write a number instead of the space (in letters, not numbers!) to get a true sentence:
THIS SENTENCE HAS ... LETTERS
(Note that the dash does not count as a letter, i.e. the word twenty-two is made up of 9 letters).
Find the largest six-digit number, for which each digit, starting with the third, is equal to the sum of the two previous digits.
Find the largest number of which each digit, starting with the third, is equal to the sum of the two previous digits.
James spent the first Tuesday of some month in Liverpool and the first Tuesday after the first Monday he spent in Newcastle. In the next month, James spent the first Tuesday in Dover and the first Tuesday after the first Monday he spent in Bristol. Could you determine the dates (day and month) spent by James in each of the cities?
Dave spent the first Tuesday of the month in Bath, and the first Tuesday after the first Monday in Cardiff. The following month Dave spent the first Tuesday in London, and the first Tuesday after the first Monday in Cambridge. Can you determine what month and date Dave was in each of the cities?
A schoolboy told his friend Bob:
“We have thirty-five people in the class. And imagine, each of them is friends with exactly eleven classmates...”
“It cannot be,” Bob, the winner of the mathematical Olympiad, answered immediately. Why did he decide this?
Alice the fox and Basilio the cat are counterfeiters. Basilio makes coins heavier than real ones, and Alice makes lighter ones. Pinocchio has 15 identical in appearance coins, but one coin is fake. How can Pinocchio determine who made the false coin – Basilio the cat or Alice the fox – with only 2 weighings?
Find the missing numbers:
a) 4, 7, 12, 21, 38 ...;
b) 2, 3, 5, 9, ..., 33;
c) 10, 8, 11, 9, 12, 10, 13, ...;
d) 1, 5, 6, 11, 28, ....
Once Alice was in one of two countries – A or Z. She knows that all of the residents of country A always tell the truth, and all the inhabitants of country Z always lie. Moreover, they often go to visit each other. Can Alice, after asking a single question to the first person that she meets, find out which country she is in?