Читать книгу The Ultimate Mathematical Challenge: Over 365 puzzles to test your wits and excite your mind - Литагент HarperCollins USD, F. M. L. Thompson - Страница 42

Week 24

162. An angle in a square

The diagram shows a square ABCD and an equilateral triangle ABE.

The point F lies on BC so that EC = EF.

Calculate the angle BEF.

[SOLUTION]

163. Areas in a quarter circle

The diagram shows a quarter circle with centre O and two semicircular arcs with diameters OA and OB.

Calculate the ratio of the area of the region shaded grey to the area of the region shaded black.

[SOLUTION]

164. How many extensions?

In a large office, each person has their own telephone extension consisting of three digits, but not all possible extensions are in use. To try to prevent wrong numbers, no used number can be converted to another just by swapping two of its digits.

What is the largest possible number of extensions in use in the office?

[SOLUTION]

165. The top ball

Six pool balls numbered 1 to 6 are to be arranged in a triangle, as shown.

After three balls are placed in the bottom row, each of the remaining balls is placed so that its number is the difference of the two below it.

Which balls can land up at the top of the triangle?

[SOLUTION]

166. Two squares

A square has four digits. When each digit is increased by 1, another square is formed.

What are the two squares?

[SOLUTION]

167. Four vehicles

Four vehicles travelled along a road with constant speeds. The car overtook the scooter at 12:00 noon, then met the bike at 14:00 and the motorcycle at 16:00. The motorcycle met the scooter at 17:00 and overtook the bike at 18:00.

At what time did the bike and the scooter meet?

[SOLUTION]

168. A marching band

A marching band is having difficulty lining up for a parade. When they line up in rows of 3, one person is left over. When they line up in rows of 4, two people are left over. When they line up in rows of 5, three people are left over. When they line up in rows of 6, four people are left over.

However, the band is able to line up in rows of 7 with nobody left over. What is the smallest possible number of marchers in the band?

[SOLUTION]