Section A
Multiple Choice — 15 Questions
(+2 / -1)
1
arithmetic
easy
Calculate the value of 7 + 16 + 34 + 45 + 50 − 6 − 15 − 4 − 7.
Answer: B
Group matching terms: (7 − 7) + (16 − 6) + (34 − 4) + (45 − 15) + 50 = 0 + 10 + 30 + 30 + 50 = 120.
2
patterns
medium
Study the pattern below. How many black circles are in the circle with the question mark?
Answer: D
The 10 circles follow 3 interleaved cycles of 3: the 1st, 4th, 7th circles have (1,2,3) black & white pairs increasing; the 2nd, 5th, 8th circles have 10, 8, 6 black circles (decreasing by 2); the 3rd, 6th, 9th circles have 2, 3, 4 white circles (increasing by 1). The 10th circle (with '?') follows the first cycle's pattern one step further, giving 4 black & 4 white circles.
3
number-theory
easy
Kevin has 3 regular dice. Each dice has numbers from 1 to 6. Which of the following could not be the sum of the numbers on top of the 3 dice?
Answer: D
The smallest possible sum of 3 dice is 1 + 1 + 1 = 3, and the largest is 6 + 6 + 6 = 18. Since 22 is greater than 18, it is impossible to obtain a sum of 22.
4
spatial-reasoning
medium
Find the top view of the figure on the right.
Options A–E are the top-view diagrams shown in the image.
Answer: C
All the options except C show 4 vertical dividing lines, but the top view of the actual figure has only 3 vertical lines. So the answer is option C.
5
logic
medium
Mitchell, Jason, Tom and Mark race across a field. Mark finishes one place ahead of Jason. Mitchell finishes one place behind the winner. Who is the winner of the race?
Answer: C
Mitchell finishes second, one place behind the winner, so the winner finishes one place ahead of Mitchell — meaning Mitchell is not the winner. Mark is not the winner either, since he finishes one place ahead of Jason. Hence the winner is Tom.
6
combinatorics
medium
In a toy store, cars are available in 5 different colours: blue, white, yellow, black and red. A car has either 2 or 4 doors. How many different versions of the car are available?
Answer: A
Each of the 5 colours can be combined with either of the 2 door-count options, giving 5 × 2 = 10 different versions of the car.
7
spatial-reasoning
medium
Find the mirror image of the picture on the right.
Options A–E are the rocket pictures shown in the image.
Answer: A
Comparing each option's details (the dotted spots on the rocket body) against the true mirror image of the reference picture, only option A has every detail correctly mirrored.
8
divisibility
medium
The four-digit number 32B9 is divisible by 3. If B is even, find the digit B.
Answer: C
The digit sum 3 + 2 + 9 + B = 14 + B must be divisible by 3. This gives B = 1, 4 or 7 among single digits. Since B must be even, B = 4.
9
logic-arithmetic
hard
Study the picture below. How many balloons are needed to carry the sheep, cat and dog? (All balloons are visible in the picture and the same size. None of the balloons are hidden.)
Answer: A
From the 2nd basket: 1 cat + 1 sheep needs 5 balloons. From the 1st basket: 1 cat + 1 sheep + 2 dogs needs 11 balloons, so 2 dogs need 11 − 5 = 6 balloons, meaning 1 dog needs 3 balloons. So 1 cat + 1 sheep + 1 dog needs 5 + 3 = 8 balloons.
10
scheduling
hard
A group of friends arrive at SASMO Cinema at 9:30. They watch any movie together. What is the earliest time they could finish watching both movies?
SASMO Cinema
Four Princesses — 115 minutes — Show times: 10:00, 12:15, 13:55, 17:05, 19:35
Dragon Fighter — 105 minutes — Show times: 9:00, 10:55, 14:10, 16:50, 19:35
Answer: C
Case 1 (Four Princesses first): earliest showing is 10:00, ending 11:55; next Dragon Fighter showing is 14:10, ending 15:55. Case 2 (Dragon Fighter first): earliest showing they can catch is 10:55 (they missed the 9:00 show), ending 12:40; next Four Princesses showing is 13:55, ending 15:50. Case 2 is earlier, so the earliest finish time is 15:50.
11
patterns
hard
Find the missing shape in the diagram below.
Options A–E are the shapes shown in the image.
Answer: B
In each row, the third image is formed by combining the second image with the first and removing its smallest circle. Applying the same rule to the third row (combining its first two images and removing the smallest circle) gives option B as the missing shape.
12
geometry-perimeter
hard
All the figures below are made up of identical squares. The area of the figure with the largest area is 48 cm². What is the perimeter of the figure with the largest perimeter?
Answer: D
Counting squares and outline sides for each figure: Figure 1 has 10 squares (16 sides), Figure 2 has 12 squares (16 sides) — the largest area, Figure 3 has 8 squares but 24 sides — the largest perimeter, Figure 4 has 11 squares (16 sides). Each square has area 48 ÷ 12 = 4 cm², so each side is 2 cm. Figure 3's perimeter = 24 × 2 = 48 cm.
13
patterns
hard
On Saturday, Diana read 102 pages of a book. On Sunday, she read 100 pages of the same book. Every day, she read 2 pages less than the day before. On which day of the week did she read exactly 4 pages of the book?
Answer: E
Continuing the pattern (2 pages fewer each day) from Saturday = 102: the following Saturday she reads 102 − 14 = 88, and the Saturday after that 88 − 14 = 74. Continuing 5 more days from that Saturday (74 − 14 − 14 − 14 − 14 − 14) gives 4 pages read on a Saturday — a day not listed among options A–D, so the answer is 'None of the above'.
14
number-theory
medium
How many two-digit numbers can be divided by both 2 and 3 at the same time?
Answer: C
A number divisible by both 2 and 3 is divisible by 6. The largest two-digit multiple of 6 is 96, the 16th multiple of 6, so there are 16 multiples of 6 up to 96 counting from 6. Excluding the one-digit multiple '6' itself, there are 16 − 1 = 15 two-digit numbers divisible by both 2 and 3.
15
logic-arithmetic
medium
Study the picture below (three balance scales comparing toys). Which toy is the heaviest?
Answer: B
From the scales: 2 trains balance 3 cars, so a train is heavier than a car. 4 cars balance 5 ships, so a car is heavier than a ship. 4 trains balance 3 submarines, so a submarine is heavier than a train. Ordering from lightest to heaviest: Ship – Car – Train – Submarine. The submarine is the heaviest.
Section B
Integer-type — 10 Questions
(+4)
16
counting-figures
hard
How many rectangles are there in the picture below (a house made of rectangular panels)?
Answer: 30
Counting rectangles formed by 1, 2, 3 and 4 adjoining parts: 16 (1-part) + 10 (2-part) + 1 (3-part) + 3 (4-part) = 30 rectangles.
17
patterns
hard
The operator ⊛ acts on two numbers to give the following outcomes:
2 ⊛ 4 = 44
5 ⊛ 7 = 104
1 ⊛ 5 = 28
7 ⊛ 10 = 146
What is 6 ⊛ 12 equal to?
Answer: 1212
The rule is: multiply the left number by 2 to form the first part of the result, and multiply the difference between the two numbers by 2 to form the second part, then place the two parts side by side (e.g. 2 ⊛ 4 = (2×2)(2×2) = 44, since 4−2=2; 7 ⊛ 10 = (2×7)(2×3) = 14 6, written as 146). So 6 ⊛ 12 = (2×6)(2×(12−6)) = 12 and 12, giving 1212.
18
data-interpretation
hard
The picture graph below shows the number of muffins 5 siblings have at first (each muffin icon = 6 muffins). Their mom takes all their muffins and then distributes them equally among Anastacia, Braiden, Darla and Eddie. How many more muffins does Eddie have now than before?
Answer: 12
From the picture graph: Anastacia = 1, Braiden = 7, Clarke = 8, Darla = 4, Eddie = 4 muffin icons, so altogether there are 24 icons × 6 = 144 muffins. Clarke is not one of the 4 who receive the redistribution, so the 144 muffins are split equally among Anastacia, Braiden, Darla and Eddie: 144 ÷ 4 = 36 muffins each. Eddie originally had 4 × 6 = 24 muffins, so Eddie now has 36 − 24 = 12 more muffins than before.
19
word-problem
medium
Andrew, Kim and Shaun have 256 stamps. Shaun has twice as many stamps as Kim. Andrew has 24 less stamps than Shaun. How many stamps does Andrew have?
Answer: 88
Let Kim = 1 unit, so Shaun = 2 units and Andrew = 2 units − 24. Together: 1 + 2 + (2 units − 24) = 5 units − 24 = 256, so 5 units = 280 and 1 unit = 56. Andrew = 2 × 56 − 24 = 112 − 24 = 88 stamps.
20
algebra
medium
In the number sentence below, what is the value of ⊟?
⊟ × 6 + 13 − ⊟ = 17 + 46
Answer: 10
⊟ × 6 − ⊟ + 13 = 63, so ⊟ × 5 + 13 = 63, giving ⊟ × 5 = 50, so ⊟ = 10.
21
logic-arithmetic
hard
Use the clues below to find the number being described:
• This number is a 3-digit number.
• The sum of its digits is 15.
• The digit in the tens place is twice the digit in the hundreds place.
• The digit in the hundreds place is 1 more than the digit in the ones place.
Answer: 483
Let the hundreds digit = 1 unit, so the tens digit = 2 units and the ones digit = 1 unit − 1. Sum: 1 unit + 2 units + (1 unit − 1) = 4 units − 1 = 15, so 4 units = 16 and 1 unit = 4. Hundreds = 4, tens = 8, ones = 3. The number is 483.
22
word-problem
hard
A group of children and adults went to a zoo. The price of an adult's ticket is $8 and a child's ticket is $7. The group paid $90 for their tickets. Given that there were less than 14 people in the group, how many adults were there?
Answer: 6
Testing possible numbers of adults (with remaining money spent exactly on $7 child tickets): 10 adults leaves $10 (not divisible by 7), 9 leaves $18 (no), 8 leaves $26 (no), 7 leaves $34 (no), 6 adults leaves $90 − $48 = $42, and $42 ÷ $7 = 6 children exactly. So there were 6 adults.
23
number-theory
hard
A book has 130 pages numbered from 1 to 130. How many digit '1's are there in all the page numbers of the book?
Answer: 64
Count digit '1' by place value. Ones place: 1,11,21,...,121 — 13 numbers. Tens place: 10–19 and 110–119 — 20 numbers. Hundreds place: 100–130 — 31 numbers. Total: 13 + 20 + 31 = 64.
24
logic-arithmetic
hard
Picture 2 shows the completed version of the puzzle in Picture 1 (a grid where each cell relates to its neighbours by the shown ×, +, − and = operations). Complete the puzzle in Picture 3 and find the missing number '?'.
Answer: 10
Let the top-left value be A and the middle-middle value be C. Following Picture 1/2's pattern: top-right = A × 2, mid-right = 7 × C, bottom-left = A × 7, bottom-mid ('?') = 2 × C, and bottom-right = top-right + mid-right = 53 while also bottom-left − bottom-mid = 53. This gives two equations: 2A + 7C = 53 and 7A − 2C = 53. Solving them together gives A = 9 and C = 5, so bottom-mid = 2 × C = 10.
25
cryptarithm
hard
In the following, all the different letters stand for different digits. What is the value of A + B + C?
A B B B
− C C C
-------
A
Answer: 10
A must be 1 (otherwise the subtraction result would exceed 3 digits). Rewriting as an addition: CCC + A = ABBB. Since A = 1, this forces C = 9 and B = 0 (999 + 1 = 1000). So A + B + C = 1 + 0 + 9 = 10.