diagram shows three semicircles, each of radius 1.
What is the size of the total shaded area?
102. How many weeks?
How many weeks are there in 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 minutes?
103. A platinum question
Platinum is a very rare metal, even rarer than gold. Its density is 21.45 g/cm3. Assuming that the world production has been about 110 tonnes for each of the past 50 years, and negligible before that, which of the following has a comparable volume to that of the total amount of platinum ever produced?
(a) a shoe box;
(b) a cupboard;
(c) a house;
(d) Buckingham Palace;
(e) the Grand Canyon
104. Underlining numbers
Ten different numbers (not necessarily integers) are written down. Any number that is equal to the product of the other nine numbers is then underlined.
At most, how many numbers can be underlined?
105. Placing draughts
Barbara wants to place draughts on a board in such a way that the number of draughts in each row is equal to the number shown at the end of the row, and the number of draughts in each column is equal to the number shown at the bottom of the column. No more than one draught is to be placed in any cell.
In how many ways can this be done?
106. Square roots
How many of the numbers
are greater than 10?
107. How many lines?
The picture shows seven points and the connections between them.
What is the least number of connecting lines that could be added to the picture so that each of the seven points has the same number of connections with other points?
(Connecting lines are allowed to cross each other.)
108. Where in the list?
There are 120 different ways of arranging the letters U, K, M, I and C. All of these arrangements are listed in dictionary order, starting with CIKMU.
Which position in the list does UKIMC occupy?
109. Eva’s sport
Two sportsmen (Ben and Filip) and two sportswomen (Eva and Andrea) − a speed skater, a skier, a hockey player and a snowboarder − had dinner at a square table, with one person on each edge of the square.
The skier sat at Andrea’s left hand.
The speed skater sat opposite Ben.
Eva and Filip sat next to each other.
A woman sat at the hockey player’s left hand.
Which sport did Eva do?
110. Pedro’s numbers
Pedro writes down a list of six different positive integers, the largest of which is N. There is exactly one pair of these numbers for which the smaller number does not divide the larger.
What is the smallest possible value of N?
111. The speed of the train
A train travelling at constant speed takes five seconds to pass completely through a tunnel that is 85 m long, and eight seconds to pass completely through a second tunnel that is 160 m long.
What is the speed of the train?
112. What is ‘pqrst’?
The digits p, q, r, s and t are all different.
What is the smallest five-digit integer ‘pqrst’ that is divisible by 1, 2, 3, 4 and 5?
ACROSS
2. A power of two (4)
5. A prime factor of 12345 (3)
6. Six more than a multiple of 13 ACROSS (3)
8. A cube (2)
10. The product of the digits of 25 ACROSS and also less than half of 23 ACROSS (2)
11. The mean of
