Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
30 questions · official format, shuffled order
1 What is the value of 0.2 × 0.3 × 500?
Show worked solution
0.2 × 0.3 = 0.06, and 0.06 × 500 = 30. (300 comes from treating 0.2 × 0.3 as 0.6 — two decimal places, not one.)
2 Which of these is NOT a factor of 720?
Show worked solution
720 = 24 × 32 × 5 has only two 3s, but 27 = 33 needs three, so 27 is not a factor. The others fit: 15 = 3 × 5, 40 = 23 × 5, 45 = 32 × 5, 48 = 24 × 3.
3 How many whole numbers n satisfy 3 < √n < 7?
Show worked solution
Squaring gives 9 < n < 49, so n runs from 10 to 48: 48 − 10 + 1 = 39. (40 includes one of the ends; 41 includes both 9 and 49, where √n equals 3 or 7 exactly.)
4 In a chess club the ratio of boys to girls is 3 : 5. Then 6 more boys join, and now there are equal numbers of boys and girls. How many girls are in the club?
Show worked solution
3k + 6 = 5k, so k = 3: there are 9 boys at first and 15 girls. (9 is the number of boys before the new members joined.)
5 ABCD is a square, and ABE is an equilateral triangle drawn inside it. What is the size of ∠DEC?
Show worked solution
∠DAE = 90° − 60° = 30°, and AD = AE, so ∠AED = (180° − 30°) ÷ 2 = 75°. Likewise ∠BEC = 75°. The angles round E add to 360°: ∠DEC = 360° − 60° − 75° − 75° = 150°. (75° is ∠AED, one step short.)
6 What is the smaller angle between the hour hand and the minute hand of a clock at 4:20?
Show worked solution
The minute hand is at 120°. The hour hand moves 30° per hour, so at 4:20 it is at 4 × 30° + 10° = 130°. The gap is 10°. (0° forgets that the hour hand has moved past the 4.)
7 How many whole numbers from 1 to 1000 are perfect squares or perfect cubes (or both)?
Show worked solution
Squares 12 to 312: 31. Cubes 13 to 103: 10. Both (sixth powers): 1, 64, 729 — 3. Total 31 + 10 − 3 = 38. (41 counts the sixth powers twice; 31 counts the squares only.)
8 The interior angles of a convex polygon add up to 1980°. How many diagonals does the polygon have?
Show worked solution
1980 ÷ 180 = 11 = n − 2, so n = 13. Each corner starts 13 − 3 = 10 diagonals, and each diagonal is counted from both ends: 13 × 10 ÷ 2 = 65. (130 forgets to halve; 78 counts every pair of corners, sides included.)
9 If 2x = 3, what is the value of 4x + 1?
Show worked solution
4x = (2x)2 = 9, so 4x + 1 = 4 × 9 = 36. (12 uses 4x = 3; 18 doubles instead of squaring.)
10 The numbers 1, 2, 3, 4, 5 and 6 are placed in the six circles, one in each, so that the three numbers along each side of the triangle add to 12. What is the sum of the three numbers in the corner circles?
Show worked solution
The three sides total 3 × 12 = 36. That counts every number once (1 + … + 6 = 21) and each corner a second time, so the corners add to 36 − 21 = 15. It works: corners 4, 5, 6 with 3, 1, 2 between them. (21 is the sum of all six numbers.)
11 Four consecutive odd numbers add up to 136. What is the largest of them?
Show worked solution
The mean is 34, so the middle two are 33 and 35, and the four numbers are 31, 33, 35, 37. The largest is 37. (31 is the smallest; 34 is the mean, which is even and is not one of them.)
12 Three straight lines can cut a large sheet of paper into at most 7 regions, as shown. What is the largest number of regions that four straight lines can make?
Show worked solution
The 4th line can cross the 3 others at 3 different points, so it passes through 4 regions and splits each: 7 + 4 = 11. (9 is a 3 by 3 grid from two pairs of parallel lines; 8 adds only one region.)
13 A rectangle 8 cm by 6 cm has all four corners on a circle. What is the area of the circle?
Show worked solution
Diagonal = √(82 + 62) = 10 cm, so the radius is 5 cm and the area is 25π cm2. (100π uses the diameter as the radius; 16π and 36π use a side.)
14 How many whole numbers less than 100 have exactly three positive divisors?
Show worked solution
Exactly three divisors means p2 for a prime p: 4, 9, 25, 49 — 4 numbers. (8 counts every square from 4 to 81, but 16 has five divisors; 25 is the number of primes below 100.)
15 Two different cards are drawn at random from nine cards numbered 1 to 9. What is the probability that the sum of the two numbers is even?
Show worked solution
Pairs: 9 × 8 ÷ 2 = 36. Two odds: 10; two evens: 6. So 16/36 = 4/9. (1/2 assumes even and odd sums are equally likely, but there are more odd cards than even.)
16 Three faces of a cuboid have areas 8 cm2, 18 cm2 and 36 cm2. What is its volume?
Show worked solution
(ab)(bc)(ca) = (abc)2 = 8 × 18 × 36 = 5184 = 722, so abc = 72 cm3 (edges 2, 4 and 9 cm). (62 adds the areas; 36 cm3 takes the largest face for the volume.)
17 If 3x + 2y = 17 and 2x + 3y = 13, what is the value of x − y?
Show worked solution
Subtracting: x − y = 17 − 13 = 4 (x = 5, y = 1). (5 is x itself; 6 is x + y, from adding and dividing by 5; 30 adds 17 and 13.)
18 Two lines are drawn from the top corner of a triangle to its base, and one line is drawn parallel to the base, as shown. How many triangles of any size are in the figure?
Show worked solution
There are 4 lines from the top corner, giving 6 pairs, and 2 lines across (the base and the parallel line). Each pair with each cross line makes a triangle: 6 × 2 = 12. (6 counts with one cross line only.)
19 AB and BA are two-digit numbers, where A and B are digits and A > B. If AB + BA = 132, how many possible values of AB are there?
Show worked solution
11(A + B) = 132, so A + B = 12 with A > B: 93, 84, 75 — 3 values. (4 includes 66, where A = B; 7 drops the condition A > B.)
20 Two circles have the same centre. A chord of the larger circle is 12 cm long and just touches the smaller circle. What is the area of the shaded ring between the circles?
Show worked solution
R2 = r2 + 62, so R2 − r2 = 36, and the ring has area π(R2 − r2) = 36π cm2 — the radii themselves are not needed. (144π uses the whole chord, 12, instead of half of it.)
21 For how many whole numbers n (positive, negative or zero) is (n + 10) ÷ (n − 2) also a whole number?
Show worked solution
n − 2 must be one of ±1, ±2, ±3, ±4, ±6, ±12: 12 values of n (from −10 to 14). (6 forgets the negative divisors.)
22 A right-angled triangle has sides 20 cm, 21 cm and 29 cm. What is the radius of the circle that touches all three sides?
Show worked solution
Area = ½ × 20 × 21 = 210 = ½ × r × (20 + 21 + 29) = 35r, so r = 6 cm. (Check: in a right triangle r = (20 + 21 − 29) ÷ 2 = 6.) (10 is half the shortest side.)
23 Real numbers x and y satisfy x + y = 10 and x2 + y2 = 58. What is x3 + y3?
Show worked solution
100 = 58 + 2xy, so xy = 21. Then x3 + y3 = 10 × (58 − 21) = 370 (x and y are 3 and 7: 27 + 343). (37 forgets the factor x + y = 10; 210 is 10 × xy.)
24 How many rectangles in this 4 by 5 grid are NOT squares? (Count rectangles of every size whose sides lie along the grid lines.)
Show worked solution
Rectangles: 10 × 15 = 150. Squares: 1 by 1: 20, 2 by 2: 12, 3 by 3: 6, 4 by 4: 2 — total 40. Not squares: 150 − 40 = 110. (150 counts the squares too.)
25 How many different triangles have whole-number side lengths and a perimeter of 20? (Triangles with the same three side lengths count as the same.)
Show worked solution
c = 9: (2, 9, 9), (3, 8, 9), (4, 7, 9), (5, 6, 9); c = 8: (4, 8, 8), (5, 7, 8), (6, 6, 8); c = 7: (6, 7, 7). That is 8. (10 lets c = 10, giving flat "triangles" such as 5, 5, 10.)
26 Three fair six-sided dice are rolled. The probability that the largest of the three numbers shown is exactly 4 is m/n, where m/n is a fraction in lowest terms. What is m + n?
Show worked solution
All dice ≤ 4: 43 = 64 ways. All ≤ 3: 33 = 27 ways. Largest exactly 4: 64 − 27 = 37 out of 63 = 216. 37 is prime and does not divide 216, so m/n = 37/216 and m + n = 253.
27 The whole numbers from 1 to 60 are written one after another to make one long number N = 123456789101112…5960. What is the remainder when N is divided by 36?
Show worked solution
Last two digits 60, so N ≡ 0 (mod 4). The digit sum of N has the same remainder by 9 as 1 + 2 + … + 60 = 1830, whose digits add to 12, so N ≡ 3 (mod 9). From 0 to 35, the multiples of 4 are 0, 4, 8, 12, …; the first that leaves 3 by 9 is 12. Remainder 12.
28 How many points (x, y) with both coordinates whole numbers lie inside or on the triangle with corners (0, 0), (20, 0) and (0, 30)?
Show worked solution
For even x = 2k the top is y = 30 − 3k, giving 31 − 3k points; for odd x = 2k + 1 the top is 28 − 3k (rounded down), giving 29 − 3k. Even x (k = 0…10): 31 × 11 − 3 × 55 = 176. Odd x (k = 0…9): 29 × 10 − 3 × 45 = 155. Total 176 + 155 = 331. (Check with Pick: area 300, boundary points 20 + 30 + 10 = 60, interior 300 − 30 + 1 = 271, and 271 + 60 = 331.)
29 A function f on the positive whole numbers has f(1) = 1, and for every n ≥ 1, f(2n) = f(n) and f(2n + 1) = f(n) + 1. For how many whole numbers n from 1 to 255 is f(n) = 4?
Show worked solution
Doubling appends a 0 in binary and 2n + 1 appends a 1, so f(n) counts the 1s in n written in binary. The numbers 1 to 255 are exactly the 8-digit binary strings (with leading zeros) except 00000000. Choosing which 4 of the 8 digits are 1 gives 8 × 7 × 6 × 5 ÷ 24 = 70.
30 A 2 by 9 strip is to be covered exactly, with no overlaps, by 1 by 2 dominoes (placed either way round) and 2 by 2 squares. One way is shown. In how many ways can it be done?
Show worked solution
The left end is either an upright domino (leaving 2 by (n − 1)) or a 2 by 2 block filled by two flat dominoes or one square (leaving 2 by (n − 2)). So t(n) = t(n − 1) + 2 t(n − 2), with t(1) = 1 and t(2) = 3. Then 5, 11, 21, 43, 85, 171, 341: t(9) = 341.