a multiple of six (3)
19. Eleven more than a cube (4)
22. A number all of whose digits are the same (3)
24. A number that leaves a remainder of eleven when divided by thirteen (3)
25. The square of a prime; the sum of the digits of this square is ten (5)
DOWN
1. A number with an odd number of factors (3)
2. Four less than a triangular number (3)
3. The square root of 9 DOWN (2)
4. A factor of 12 ACROSS (3)
5. The longest side of a right-angled triangle whose shorter sides are 3 DOWN and 4 ACROSS (3)
8. A Fibonacci number (5)
9. The square of 3 DOWN (4)
11. Three more than an even cube (5)
13. A prime factor of 34567 (4)
17. The mean of 10 ACROSS, 16 ACROSS, 18 DOWN, 20 DOWN and 21 DOWN (3)
18. A power of eighteen (3)
20. Two less than 22 ACROSS (3)
21. A number whose digits are those of 12 ACROSS reversed (3)
23. A multiple of twenty-three (2)
141. Joey’s and Zoë’s sums
Joey calculated the sum of the largest and smallest two-digit numbers that are multiples of three. Zoë calculated the sum of the largest and smallest two-digit numbers that are not multiples of three.
What is the difference between their answers?
142. When is the party?
Six friends are having dinner together in their local restaurant. The first eats there every day, the second eats there every other day, the third eats there every third day, the fourth eats there every fourth day, the fifth every fifth day and the sixth eats there every sixth day. They agree to have a party the next time they all eat together there. In how many days’ time is the party?
143. A multiple of 11
The eight-digit number ‘1234d678’ is a multiple of 11.
Which digit is d?
144. Two squares
ABCD is a square. P and Q are squares drawn in the triangles ADC and ABC, as shown.
What is the ratio of the area of the square P to the area of the square Q?
145. Proper divisors
Excluding 1 and 24 itself, the positive whole numbers that divide into 24 are 2, 3, 4, 6, 8 and 12. These six numbers are called the proper divisors of 24.
Suppose that you wanted to list in increasing order all those positive integers greater than 1 that are equal to the product of their proper divisors. Which would be the first six numbers in your list?
146. Kangaroo game
In the expression
the same letter stands for the same non-zero digit and different letters stand for different digits.
What is the smallest possible positive integer value of the expression?
147. A game with sweets
There are 20 sweets on the table. Two players take turns to eat as many sweets as they choose, but they must eat at least one, and never more than half of what remains. The loser is the player who has no valid move.
Is it possible for one of the two players to force the other to lose? If so, how?
148. A 1000-digit number
What is the largest number of digits that can be erased from the 1000-digit number 201820182018 … 2018 so that the sum of the remaining digits is 2018?
149. Gardeners at work
It takes four gardeners four hours to dig four circular flower beds, each of diameter four metres.
How long will it take six gardeners to dig six circular flower beds each of diameter six metres?
150. Overlapping squares
The diagram shows four overlapping squares that have sides of lengths 5 cm, 7 cm, 9 cm and 11 cm.
What is the difference between the total area shaded grey and the total hatched area?
151. What can T be?
Each of the numbers from 1 to 10 is to be placed in the circles so that the sum of each line of three numbers is equal to T. Four
