56. A word product
What is the value of P + Q + R in the multiplication shown?
ACROSS
1. The highest common factor of 5 DOWN and 8 DOWN (2)
3. A prime factor of 2007 (3)
5. 3 DOWN plus the square root of 4 DOWN (3)
6. The product of three consecutive integers, two of which are prime (3)
7. One less than a multiple of 2 DOWN (3)
9. Five less than 14 ACROSS (4)
11. Seven more than the product of the digits of 22 ACROSS (2)
13. Three more than a triangular number (2)
14. 9 ACROSS plus five (4)
16. A square whose digit sum is three more than its square root (3)
18. Three times the product of two consecutive prime numbers (3)
21. The mean of 11 ACROSS and 21 ACROSS is 16 ACROSS (3)
22. Twice a prime number (3)
23. Two less than a square (2)
DOWN
1. Eight less than a multiple of nine (3)
2. A prime factor of 12 DOWN (2)
3. A Fibonacci number that is also a prime (3)
4. A square (2)
5. One less than twice a triangular number (3)
7. 10 DOWN minus three (4)
8. One third the product of three consecutive numbers, two of which are prime (3)
10. 7 DOWN plus three (4)
12. A number whose digit sum is equal to one of its factors (3)
15. The product of two consecutive prime numbers (3)
17. p4 + 1, where p is prime (3)
19. The sum of 16 ACROSS and 3 ACROSS (3)
20. Six less than twice 13 ACROSS (2)
21. Fifteen plus the mean of 1 ACROSS and 11 ACROSS (2)
57. Three Tuesdays
Three Tuesdays of a month fall on even-numbered dates.
Which day of the week was the twenty-first day of the month?
58. Crack the code
In a seven-digit numerical code, each group of four adjacent digits adds to 16 and each group of five adjacent digits adds to 19.
What is the code?
59. Mr Bean’s fruit
Despite his name, Mr Bean likes to eat lots of fruit. He finds that four apples and two oranges cost £1.54 and that two oranges and four bananas cost £1.70.
How much would he have to pay if he bought one apple, one orange and one banana?
60. Ali’s bookshelves
Ali is arranging the books on his bookshelves. He puts half his books on the bottom shelf and two-thirds of what remains on the second shelf. Finally, he splits the rest of his books over the other two shelves so that the third shelf contains four more books than the top shelf. There are three books on the top shelf.
How many books are on the bottom shelf?
61. An unfair dice
I have an unfair dice that has probability
62. A room in Ginkrail
The town of Ginkrail is inhabited entirely by knights and liars. Every sentence spoken by a knight is true, and every sentence spoken by a liar is false. One day some inhabitants of Ginkrail were alone in a room and three of them spoke.
The first one said: ‘There are no more than three of us in the room. All of us are liars.’
The second said: ‘There are no more than four of us in the room. Not all of us are liars.’
The third said: ‘There are five of us in the room. Three of us are liars.’
How many people were in the room and how many liars were among them?
63. Curious integers
In the following puzzle, each different capital letter represents a different digit. Thus ‘SEVEN’ represents a five-digit decimal number.
‘SEVEN’ is prime and, as one would expect, ‘SEVEN’ minus ‘THREE’ equals ‘FOUR’.
Curiously, ‘FOUR’ is prime (as is ‘RUOF’) but ‘THREE’ is not prime. Another oddity is that ‘TEN’ is a square.
Find the values of ‘FOUR’ and ‘TEN’.
64. Eight factors
A certain number has exactly eight factors including 1 and itself. Two of its factors are 21 and 35.
What is the number?
65. A nonagon problem
The diagram shows a regular nine-sided polygon (a nonagon or an enneagon) with two of the sides extended to meet at the point X.
What is the size