a number that has a maximum of two different digits. Their phone number must start with the digit 3 and be six digits long.
How many such numbers are possible?
130. How many games?
Cleo played 40 games of chess and scored 25 points.
A win counts as one point, a draw counts as half a point, and a loss counts as zero points.
How many more games did she win than lose?
131. How many draughts?
Barbara wants to place draughts on a 4 × 4 board in such a way that the number of draughts in each row and in each column are all different. (She may place more than one draught in a square, and a square may be empty.)
What is the smallest number of draughts that she would need?
132. To and from Jena
In a certain region these are five towns: Freiburg, Göttingen, Hamburg, Ingolstadt and Jena.
One day, 40 trains each made a journey, leaving one of these towns and arriving at another.
10 trains travelled either to or from Freiburg.
10 trains travelled either to or from Göttingen.
10 trains travelled either to or from Hamburg.
10 trains travelled either to or from Ingolstadt.
How many trains travelled either to or from Jena?
133. Largest possible remainder
What is the largest possible remainder that is obtained when a two-digit number is divided by the sum of its digits?
134. nth term n
The first term of a sequence of positive integers is 6. The other terms in the sequence follow these rules:
if a term is even then divide it by 2 to obtain the next term;
if a term is odd then multiply it by 5 and subtract 1 to obtain the next term.
For which values of n is the nth term equal to n?
135. How many numbers?
Rafael writes down a five-digit number whose digits are all distinct, and whose first digit is equal to the sum of the other four digits.
How many five-digit numbers with this property are there?
136. A square area
A regular octagon is placed inside a square, as shown.
The shaded square connects the midpoints of four sides of the octagon.
What fraction of the outer square is shaded?
137. ODD plus ODD is EVEN
Find all possible solutions to the ‘word sum’ shown.
Each letter stands for one of the digits 0−9 and has the same meaning each time it occurs. Different letters stand for different digits. No number starts with a zero.
138. Only odd digits
How many three-digit multiples of 9 consist only of odd digits?
139. How many tests?
Before the last of a series of tests, Sam calculated that a mark of 17 would enable her to average 80 over the series, but that a mark of 92 would raise her average mark over the series to 85.
How many tests were in the series?
140. Three primes
Find all positive integers p such that p, p + 8 and p + 16 are all prime.
ACROSS
1. The cube of a square (5)
4. Eight less than 5 DOWN (3)
6. One less than a multiple of seven (3)
7. A prime factor of 20 902 (4)
10. A number whose digits successively decrease by one (3)
12. Sixty per cent of 20 DOWN (3)
14. A multiple of seven (3)
15. A multiple of three whose digits have an even sum (3)
16. The square of a square (3)
17. A prime that is
