The figure below is made of a number of blocks, some of them are invisible. Can you count how many invisible blocks are there?
Modelling convention: count the blocks needed to support the shown staircase in solid vertical columns; do not add extra blocks behind the depicted structure.
The digital clock displays hours, minutes, and seconds as shown. In a day from 00:00:00 to 23:59:59, how many times do all six digits change at the same time?
Boundary convention: count transitions whose before and after readings both lie in the stated interval.
Given 10 numbers 0,1,2,3,4,...,9. What is the least number of numbers we must choose to be sure that there are 2 numbers whose sum is a perfect square? (A perfect square is the product of two equal natural numbers.)
In the 3 × 3 table shown, the numbers 1, 2 and 3 are placed so that each number occurs only once in each row and only once in each column. The value of X + Y is…
A knockout boxing tournament was held. There were 114 participants and so there were 57 fights in the first round of the tournament, in this round 57 boxers lost and were eliminated. In the second round, the remaining 57 fighters were paired, resulting in 28 matches; a boxer who is not paired is given a chance to advance to the next round without having to compete. The remaining 29 fighters after that round were paired again, and so on. How many matches in all tournaments are held to determine the champion of the tournament?
On the table there are 2 cloth bags including a large bag and a small bag. In the big bag there are 10 marbles, in the small bag there are 5 marbles. How many marbles are there at least on the table?
Interpretation for the minimum: a bag may be inside the other, and “on the table” includes contents of bags supported by the table.
Tom wrote down two positive integers consisting of the following digits: 1, 2, 3, 4, 5 and 6. Each of the digit appeared in only one of the two numbers, and only once. When Tom added up these numbers, he got 750. What is the smallest positive integer that Tom could write?
Out of the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9, a nine-digit number was formed such that each of the digit occurred only once, and each digit was either greater by 5 or smaller by 4 than the preceding one. How many such numbers can be written?
On an island 26,000 people live and there is a custom that they only marry the inhabitants of the island, known that 2/3 of all men are married and 4/7 of all women are married. How many people on this island are married?
Mathematical model: the population is partitioned into the men and women counted in the question, with each married couple consisting of one man and one woman, each person in at most one such couple.
There are six blue cards numbered 1 to 6, five red cards numbered 1 to 5 and four yellow cards numbered 1 to 4. How many ways are there to draw three cards such as that all three have different color and different number?
