Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: The two diagrams show a side view and a plan view of a tower made with light and dark coloured blocks. In the tower, only dark coloured blocks are placed on top of dark coloured blocks and only light coloured blocks are placed on top of light
<image1>
coloured blocks. How many blocks in the tower are light coloured?
Answer: 20
Problem: <image>Question: The diagram shows a triangle joined to a square to form an irregular pentagon. The triangle has the same perimeter as the square.
<image1>
What is the ratio of the perimeter of the pentagon to the perimeter of the square?
Choices:
A. 2: 1
B. 3: 2
C. 4: 3
D. 5: 4
E. 6: 5
Answer: B
Problem: <image>Question: My TV screen has sides in the ratio $16: 9$. My mother's TV screen has sides in the ratio $4: 3$. A picture which exactly fills the screen of my TV only fills the width of the screen of my mother's TV.
What fraction of the screen on my mother's TV is not covered?
<image1>
Choices:
A. $\frac{1}{6}$
B. $\frac{1}{5}$
C. $\frac{1}{4}$
D. $\frac{1}{3}$
E. It depends on the size of the screen.
Answer: C
Problem: <image>Question: In triangle $P Q R$, the point $S$ is on $P Q$ so that the ratio of the length of $P S$ to the length of $S Q$ is $2: 3$. The point $T$ lies on $S R$ so that the area of triangle $P T R$ is 20 and the area of triangle $S Q T$ is 18 , as shown in the diagram.
<image1>
What is the area of triangle $P Q R$ ?
Answer: 80
Problem: <image>Question: The diagram shows a plan of a town with various bus stops. There are four bus routes in the town.
Route 1 goes $\mathrm{C}-\mathrm{D}-\mathrm{E}-\mathrm{F}-\mathrm{G}-\mathrm{H}-\mathrm{C}$ and is $17 \mathrm{~km}$ long.
Route 2 goes $\mathrm{A}-\mathrm{B}-\mathrm{C}-\mathrm{F}-\mathrm{G}-\mathrm{H}-\mathrm{A}$ and is $12 \mathrm{~km}$ long.
Route 3 goes $\mathrm{A}-\mathrm{B}-\mathrm{C}-\mathrm{D}-\mathrm{E}-\mathrm{F}-\mathrm{G}-\mathrm{H}-\mathrm{A}$ and is $20 \mathrm{~km}$ long.
Route 4 goes $\mathrm{C}-\mathrm{F}-\mathrm{G}-\mathrm{H}-\mathrm{C}$.
<image1>
How long is route 4 ?
Choices:
A. $10 \mathrm{~km}$
B. $9 \mathrm{~km}$
C. $8 \mathrm{~km}$
D. $7 \mathrm{~km}$
E. $6 \mathrm{~km}$
Answer: B
Problem: <image>Question: In the sum each letter stands for a different digit.
What is the answer to the subtraction $ RN - KG $ ? <image1>
Answer: 11
Problem: <image>Question: The diagram shows squares of three different sizes arranged into a rectangle. The length of each side of the smallest squares is $20 \mathrm{~cm}$. Adam Ant walks along the path marked from $P$ to $Q$. How far does Adam walk? <image1>
Choices:
A. $380 \mathrm{~cm}$
B. $400 \mathrm{~cm}$
C. $420 \mathrm{~cm}$
D. $440 \mathrm{~cm}$
E. $460 \mathrm{~cm}$
Answer: C
Problem: <image>Question: Two rectangles of dimensions $8 \mathrm{~cm}$ by $10 \mathrm{~cm}$ and $9 \mathrm{~cm}$ by $12 \mathrm{~cm}$ overlap as shown in the diagram. The area of the black region is $37 \mathrm{~cm}^{2}$. What is the area of the grey region? <image1>
Choices:
A. $60 \mathrm{~cm}^{2}$
B. $62 \mathrm{~cm}^{2}$
C. $62.5 \mathrm{~cm}^{2}$
D. $64 \mathrm{~cm}^{2}$
E. $65 \mathrm{~cm}^{2}$
Answer: E
Problem: <image>Question: In the quadrilateral $P Q R S$, the length of $P Q$ is $11 \mathrm{~cm}$, the length of $Q R$ is $7 \mathrm{~cm}$, the length of $R S$ is $9 \mathrm{~cm}$ and the length of $S P$ is $3 \mathrm{~cm}$. Both $\angle Q R S$ and $\angle S P Q$ are $90^{\circ}$. What is the area of the quadrilateral $P Q R S$ ? <image1>
Choices:
A. $30 \mathrm{~cm}^{2}$
B. $48 \mathrm{~cm}^{2}$
C. $50 \mathrm{~cm}^{2}$
D. $52 \mathrm{~cm}^{2}$
E. $60 \mathrm{~cm}^{2}$
Answer: B
Problem: <image>Question: Two of the following four facts about a positive integer $N$ are true and two are false. <image1>
What is the value of $N$ ?
Answer: 5
Problem: <image>Question: The shape in the diagram is made up of a rectangle, a square and an equilateral triangle, all of which have the same perimeter. The length of the side of the square is $9 \mathrm{~cm}$. What is the length of the shorter sides of the rectangle? <image1>
Choices:
A. $4 \mathrm{~cm}$
B. $5 \mathrm{~cm}$
C. $6 \mathrm{~cm}$
D. $7 \mathrm{~cm}$
E. $8 \mathrm{~cm}$
Answer: C
Problem: <image>Question: The diagram shows a square, an equilateral triangle and a regular pentagon. What is the size of $\angle W U V$ ? <image1>
Choices:
A. $21^{\circ}$
B. $23^{\circ}$
C. $25^{\circ}$
D. $27^{\circ}$
E. $29^{\circ}$
Answer: A
Problem: <image>Question: In the diagram, each symbol represent a positive integer. The sums of the numbers in each row and in each column are as shown.
<image1>
What is the value of <image2>?
Answer: 23
Problem: <image>Question: In the diagram, $P Q R S$ is a square of side $10 \mathrm{~cm}$. The distance $M N$ is $6 \mathrm{~cm}$. The square is divided into four congruent isosceles triangles, four congruent squares and the shaded region.
<image1>
What is the area of the shaded region?
Choices:
A. $42 \mathrm{~cm}^{2}$
B. $46 \mathrm{~cm}^{2}$
C. $48 \mathrm{~cm}^{2}$
D. $52 \mathrm{~cm}^{2}$
E. $58 \mathrm{~cm}^{2}$
Answer: C
Problem: <image>Question: The diagram shows a $2 \times 4$ table in which the numbers in each column except the first column are the sum and the difference of the numbers in the previous column.
<image1>
Carl completes a $2 \times 7$ table in the same way and obtains the numbers 96 and 64 in the final column. What is the sum of the numbers in the first column of Carl's table?
Answer: 20
Problem: <image>Question: Ellis's Eel Emporium contains a large tank holding three different types of eel: electric eels, moray eels and freshwater eels. A notice on the tank reads as follows:
<image1>
How many eels are in the tank?
Answer: 21
Problem: <image>Question: Sid is colouring the cells in the grid using the four colours red, blue, yellow and green in such a way that any two cells that share a vertex are coloured differently. He has already coloured some of the cells as shown.
What colour will he use for the cell marked $X$ ?
<image1>
Choices:
A. Red
B. Blue
C. Yellow
D. Green
E. You can't be certain
Answer: A
Problem: <image>Question: Lily pours 296 litres of water into the top of the pipework shown in the diagram. Each time a pipe forks, half the water flows to one side and half to the other. How many litres of water will reach container $\mathrm{Y}$ ? <image1>
Answer: 222
Problem: <image>Question: Andrew wants to write the letters of the word KANGAROO in the cells of a $2 \times 4$ grid such that each cell contains exactly one letter. He can write the first letter in any cell he chooses but each subsequent letter can only be written in a cell with at least one common vertex with the cell in which the previous letter was written. Which of the following arrangements of letters could he not produce in this way?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: The diagram shows four congruent right-angled triangles inside a rectangle. What is the total area, in $\mathrm{cm}^{2}$, of the four triangles? <image1>
Answer: 56
Problem: <image>Question: The diagram shows three triangles which are formed by the five line segments $A C D F, B C G, G D E, A B$ and $E F$ so that $A C=B C=C D=G D=D F=E F$. Also $\angle C A B=\angle E F D$. What is the size, in degrees, of $\angle C A B$ ? <image1>
Answer: 60
Problem: <image>Question: Each square in the grid shown is $1 \mathrm{~cm}$ by $1 \mathrm{~cm}$. What is the area of the shaded figure, in $\mathrm{cm}^{2}$ ? <image1>
Answer: 17
Problem: <image>Question: The diagram shows the eight vertices of an octagon connected by line segments. Jodhvir wants to write one of the integers 1,2,3 or 4 at each of the vertices so that the two integers at the ends of every line segment are different. He has already written three integers as shown.
<image1>
How many times will the integer 4 appear in his completed diagram?
Answer: 4
Problem: <image>Question: In the diagram $P Q R S$ is a rhombus. Point $T$ is the mid-point of $P S$ and point $W$ is the mid-point of $S R$.
<image1>
What is the ratio of the unshaded area to the shaded area?
Choices:
A. $1: 1$
B. $2: 3$
C. $3: 5$
D. $4: 7$
E. $5: 9$
Answer: A
Problem: <image>Question: Using only pieces like the one shown in the diagram, Zara wants to make a complete square without gaps or overlaps.
<image1>
What is the smallest number of pieces she can use?
Answer: 20
Problem: <image>Question: Claudette has eight dice, each with one of the letters $P, Q, R$ and $S$ written on all six faces. She builds the block shown in the diagram so that dice with faces which touch have different letters written on them.
What letter is written on the faces of the one dice which is not shown on the picture? <image1>
Choices:
A. P
B. Q
C. R
D. S
E. It is impossible to say
Answer: B
Problem: <image>Question: A map of Wonderland shows five cities. Each city is joined to every other city by a road. Alice's map, as shown, is incomplete. How many roads are missing? <image1>
Answer: 3
Problem: <image>Question: Both of the shapes shown in the diagram are formed from the same five pieces, a $5 \mathrm{~cm}$ by $10 \mathrm{~cm}$ rectangle, two large quarter circles and two small quarter circles. What is the difference in $\mathrm{cm}$ between their perimeters? <image1>
Answer: 20
Problem: <image>Question: In the diagram, lines $Q T$ and $R S$ are parallel and $P Q$ and $Q T$ are equal. Angle $S T Q$ is $154^{\circ}$. What is the size of angle $S R Q$ ? <image1>
Choices:
A. $120^{\circ}$
B. $122^{\circ}$
C. $124^{\circ}$
D. $126^{\circ}$
E. $128^{\circ}$
Answer: E
Problem: <image>Question: A regular octagon is folded exactly in half three times until a triangle is obtained. The bottom corner of the triangle is then cut off with a cut perpendicular to one side of the triangle as shown.
<image1>
Which of the following will be seen when the triangle is unfolded?
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: Rectangle $W X Y Z$ is cut into four smaller rectangles as shown. The lengths of the perimeters of three of the smaller rectangles are 11, 16 and 19 . The length of the perimeter of the fourth smaller rectangle lies between 11 and 19. What is the length of the perimeter of $W X Y Z$ ? <image1>
Answer: 30
Problem: <image>Question: Joseph writes the numbers 1 to 12 in the circles so that the numbers in adjacent circles differ by either 1 or 2 . Which pair of numbers does he write in adjacent circles? <image1>
Choices:
A. 3 and 4
B. 5 and 6
C. 6 and 7
D. 8 and 9
E. 8 and 10
Answer: E
Problem: <image>Question: Patricia painted some of the cells of a $4 \times 4$ grid. Carl counted how many red cells there were in each row and in each column and created a table to show his answers.
Which of the following tables could Carl have created?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: Andrew wants to write the numbers $1,2,3,4,5,6$ and 7 in the circles in the diagram so that the sum of the three numbers joined by each straight line is the same. Which number should he write in the top circle? <image1>
Answer: 4
Problem: <image>Question: A square piece of paper of area $64 \mathrm{~cm}^{2}$ is folded twice, as shown in the diagram. What is the sum of the areas of the two shaded rectangles?
<image1>
Choices:
A. $10 \mathrm{~cm}^{2}$
B. $14 \mathrm{~cm}^{2}$
C. $15 \mathrm{~cm}^{2}$
D. $16 \mathrm{~cm}^{2}$
E. $24 \mathrm{~cm}^{2}$
Answer: D
Problem: <image>Question: Which single digit should be placed in all three of the boxes shown to give a correct calculation? <image1>
Answer: 4
Problem: <image>Question: Nico is learning to drive. He knows how to turn right but has not yet learned how to turn left. What is the smallest number of right turns he could make to travel from $\mathrm{P}$ to $\mathrm{Q}$, moving first in the direction shown? <image1>
Answer: 4
Problem: <image>Question: When she drew two intersecting circles, as shown, Tatiana divided the space inside the circles into three regions. When drawing two intersecting squares, what is the largest number of regions inside one or both of the squares that Tatiana could create? <image1>
Answer: 9
Problem: <image>Question: What is the largest number of " $\mathrm{T}$ " shaped pieces, as shown, that can be placed on the $4 \times 5$ grid in the diagram, without any overlap of the pieces? <image1>
Answer: 4
Problem: <image>Question: Maria has drawn some shapes on identical square pieces of paper, as shown. Each line she has drawn is parallel to an edge of her paper. How many of her shapes have the same perimeter as the sheet of paper itself? <image1>
Answer: 4
Problem: <image>Question: Christopher has made a building out of blocks. The grid on the right shows the number of blocks in each part of the building, when viewed from above. Which of the following gives the view you see when you look at Christopher's building from the front?
\begin{tabular}{|l|l|l|l|}
\hline 4 & 2 & 3 & 2 \\
\hline 3 & 3 & 1 & 2 \\
\hline 2 & 1 & 3 & 1 \\
\hline 1 & 2 & 1 & 2 \\
\hline \multicolumn{4}{|c|}{ front }
\end{tabular}
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: The diagram shows a wooden cube of side $3 \mathrm{~cm}$ with a smaller cube of side $1 \mathrm{~cm}$ cut out at one corner. A second cube of side $3 \mathrm{~cm}$ has a cube of side $1 \mathrm{~cm}$ cut out at each corner. How many faces does the shape formed from the second cube have? <image1>
Answer: 30
Problem: <image>Question: Alisha wrote an integer in each square of a $4 \times 4$ grid. Integers in squares with a common edge differed by 1 . She wrote a 3 in the top left corner, as shown. She also wrote a 9 somewhere in the grid. How many different integers did she write? <image1>
Answer: 7
Problem: <image>Question: The diagram shows a square $P Q R S$ with area $120 \mathrm{~cm}^{2}$. Point $T$ is the mid-point of $P Q$. The ratio $Q U: U R=2: 1$, the ratio $R V: V S=3: 1$ and the ratio $S W: W P=4: 1$. What is the area, in $\mathrm{cm}^{2}$, of quadrilateral $T U V W$? <image1>
Answer: 67
Problem: <image>Question: Emily makes four identical numbered cubes using the net shown. She then glues them together so that only faces with the same number on are glued together to form the $2 \times 2 \times 1$ block shown. What is the largest possible total of all the numbers on the faces of the block that Emily could achieve? <image1>
Answer: 68
Problem: <image>Question: One slice of a circular cake is $15 \%$ of the whole cake. What is the size of the angle marked with the question mark? <image1>
Choices:
A. $30^{\circ}$
B. $45^{\circ}$
C. $54^{\circ}$
D. $15^{\circ}$
E. $20^{\circ}$
Answer: C
Problem: <image>Question: In the picture, the three strips labelled 1,2,3 have the same horizontal width $a$. These three strips connect two parallel lines. Which of these statements is true? <image1>
Choices:
A. All three strips have the same area.
B. Strip 1 has the largest area.
C. Strip 2 has the largest area.
D. Strip 3 has the largest area.
E. It is impossible to say which has the largest area without knowing $a$.
Answer: A
Problem: <image>Question: In figure 1, alongside, the area of the square equals $a$. The area of each circle in both figures equals $b$. Three circles are lined up as shown in figure 2. An elastic band is placed around these three circles without moving them. What is the area inside the elastic band?

<image1>
Choices:
A. $3 b$
B. $2 a+b$
C. $a+2 b$
D. $3 a$
E. $a+b$
Answer: B
Problem: <image>Question: The cuboid shown has been built using four shapes, each made from four small cubes. Three of the shapes can be completely seen, but the dark one is only partly visible. Which of the following shapes could be the dark one? <image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: In the sum on the right, each of the letters $X, Y$ and $Z$ represents a different $X X$ non-zero digit. What does $X$ represent? <image1>
Answer: 8
Problem: <image>Question: What is the ratio of the areas of the triangles $A D E$ and $A B C$ in the picture? <image1>
Choices:
A. $9: 4$
B. $7: 3$
C. $4: 5$
D. $15: 10$
E. $26: 9$
Answer: A
Problem: <image>Question: A rectangular sheet of paper which measures $6 \mathrm{~cm} \times 12 \mathrm{~cm}$ is folded along its diagonal (Diagram A). The shaded areas in Diagram B are then cut off and the paper is unfolded leaving the rhombus shown in Diagram C. What is the length of the side of the rhombus? <image1>
Choices:
A. $\frac{7}{2} \sqrt{5} \mathrm{~cm}$
B. $7.35 \mathrm{~cm}$
C. $7.5 \mathrm{~cm}$
D. $7.85 \mathrm{~cm}$
E. $8.1 \mathrm{~cm}$
Answer: C
Problem: <image>Question: In the diagram, $Q R=P S$. What is the size of $\angle P S R$ ? <image1>
Choices:
A. $30^{\circ}$
B. $50^{\circ}$
C. $55^{\circ}$
D. $65^{\circ}$
E. $70^{\circ}$
Answer: D
Problem: <image>Question: Roo has a very unusual chessboard of side 7, in which only the squares which lie on the diagonals are shaded. Kanga then asks the question "What would be the total white area of your chessboard if each side was 2003 squares long?" What is the correct answer? <image1>
Choices:
A. $2002^{2}$
B. $2002 \times 2001$
C. $2003^{2}$
D. $2003 \times 2004$
E. $2004^{2}$
Answer: A
Problem: <image>Question: The target shown consists of an inner black circle with two rings, one black and one white, around it. The width of each ring is equal to the radius of the black circle. What is the ratio of the area of the black ring to the area of the inner black circle? <image1>
Choices:
A. $2: 1$
B. $3: 1$
C. $4: 1$
D. $5: 1$
E. $6: 1$
Answer: D
Problem: <image>Question: A chain is made from circular links with external radius $3 \mathrm{~cm}$ and internal radius $2 \mathrm{~cm}$. When the rings are linked together as shown in the diagram, the length of the chain is $1.7 \mathrm{~m}$. How many rings are used? <image1>
Answer: 42
Problem: <image>Question: Sol is having fun playing with water in two tanks. Tank $X$ has a base of area of $200 \mathrm{~cm}^{2}$.
Tank $Y$ has a base of area $100 \mathrm{~cm}^{2}$ and height $7 \mathrm{~cm}$. Sol has partly filled Tank X to a depth of $5 \mathrm{~cm}$. He then places Tank Y, which is empty, on the bottom of Tank X. The water in Tank X rises, of course, and spills over into in Tank Y. What level does the water reach in Tank Y ? <image1>
Choices:
A. $1 \mathrm{~cm}$
B. $2 \mathrm{~cm}$
C. $3 \mathrm{~cm}$
D. $4 \mathrm{~cm}$
E. $5 \mathrm{~cm}$
Answer: C
Problem: <image>Question: Zoli wants to make a bench for his garden from some tree trunks sawn in half, as shown in the picture. The diameters of the two bottom trunks are 20 centimetres, and the diameter of the top trunk is 40 centimetres. What is the height of the bench in centimetres? <image1>
Choices:
A. 25
B. $20 \sqrt{ } 2$
C. 28.5
D. 30
E. $10 \sqrt{ } 10$
Answer: B
Problem: <image>Question: Barney has 16 cards: 4 blue $(B), 4$ red $(R), 4$ green $(G)$ and 4 yellow (Y). He wants to place them in the square shown so that every row and every column has exactly one of each card. The diagram shows how he started. How many different ways can he finish?
<image1>
Answer: 4
Problem: <image>Question: Andrew, the absent-minded mountain climber, climbed a mountain range, with a profile as shown in Figure 1 from point $P$ to point $Q$. From time to time he had to turn back to find bits of his equipment that he had dropped. The graph of the height $H$ of his position at time $t$ is shown in Figure 2 to the same scale as Figure 1. How many times did he turn back?
<image1>
Answer: 3
Problem: <image>Question: The diagram shows a square and an equilateral right-angled crossshaped dodecagon. The length of the perimeter of the dodecagon is $36 \mathrm{~cm}$. What is the area of the square in $\mathrm{cm}^{2}$ ? <image1>
Answer: 72
Problem: <image>Question: The shaded area is equal to $2 \pi$. What is the length of $P Q$ ? <image1>
Answer: 4
Problem: <image>Question: In the grid on the right, there are eight kangaroos. A kangaroo may jump into any empty square. Find the least number of the kangaroos which have to jump into an empty square so that in each row and column there are exactly two kangaroos. <image1>
Answer: 1
Problem: <image>Question: In the diagram alongside, the five circles have the same radii and touch as shown. The square joins the centres of the four outer circles. <image1>
The ratio of the area of the shaded parts of all five circles to the area of the unshaded parts of all five circles is
Choices:
A. $5: 4$
B. $2: 3$
C. $2: 5$
D. $1: 4$
E. $1: 3$
Answer: B
Problem: <image>Question: In the diagram there are 7 squares. What is the difference between the number of triangles and the number of squares in the diagram? <image1>
Answer: 3
Problem: <image>Question: Kanga and Roo are hopping around a stadium with a perimeter of $330 \mathrm{~m}$. Each of them makes one jump every second.
Kanga's jumps are $5 \mathrm{~m}$ long, while Roo's jumps are $2 \mathrm{~m}$ long. They both start at the same point and move in the same direction. Roo gets tired and stops after 25 seconds whilst
<image1>
Kanga keeps jumping. How much more time passes before Kanga is next beside Roo?
Choices:
A. 15 seconds
B. 24 seconds
C. 51 seconds
D. 66 seconds
E. 76 seconds
Answer: C
Problem: <image>Question: What entry should replace $x$ in the table so that the numbers in each row, each column and each diagonal form an arithmetic sequence?
(In an arithmetic sequence, there is a constant difference between successive terms.) <image1>
Answer: 42
Problem: <image>Question: The diagram shows 3 semicircular arcs with the endpoints $A, B$ of one arc and the centres $E, F$ of the other two arcs at the vertices of a rectangle. What is the area of the shaded region when the radius of each semicircle is $2 \mathrm{~cm}$ ? <image1>
Choices:
A. $2 \pi+2 \mathrm{~cm}^{2}$
B. $8 \mathrm{~cm}^{2}$
C. $2 \pi+1 \mathrm{~cm}^{2}$
D. $7 \mathrm{~cm}^{2}$
E. $2 \pi \mathrm{cm}^{2}$
Answer: B
Problem: <image>Question: What is the sum of the 10 angles marked on the diagram on the right? <image1>
Choices:
A. $300^{\circ}$
B. $450^{\circ}$
C. $360^{\circ}$
D. $600^{\circ}$
E. $720^{\circ}$
Answer: E
Problem: <image>Question: The numbers on each pair of opposite faces on a die add up to 7 . A die is rolled without slipping around the circuit shown. At the start the top face is 3 . What number will be displayed on the top face at the end point? <image1>
Answer: 6
Problem: <image>Question: How many ways are there to choose a white square and a black square, such as those shown, from an $8 \times 8$ chess board so that these squares do not lie in either the same row or the same column? <image1>
Answer: 768
Problem: <image>Question: The flag shown in the diagram consists of three stripes, each of equal height, which are divided into two, three and four equal parts, respectively. What fraction of the area of the flag is shaded? <image1>
Choices:
A. $\frac{1}{2}$
B. $\frac{2}{3}$
C. $\frac{3}{5}$
D. $\frac{4}{7}$
E. $\frac{5}{9}$
Answer: D
Problem: <image>Question: The circle shown in the diagram is divided into four arcs of length 2, 5, 6 and $x$ units. The sector with arc length 2 has an angle of $30^{\circ}$ at the centre. Determine the value of $x$. <image1>
Answer: 11
Problem: <image>Question: The rectangle shown is divided into six squares. The length of the sides of the smallest square is 1 . What is the length of the sides of the largest square? <image1>
Answer: 7
Problem: <image>Question: Each letter in the sum shown represents a different digit and the digit for $\mathrm{A}$ is odd. What digit does $\mathrm{G}$ represent?
<image1>
Answer: 9
Problem: <image>Question: Two identical equilateral triangles overlap with their sides parallel, so that the overlapping region is the hexagon shown shaded in the diagram. The perimeter length of each triangle is 18 . What is the perimeter length of the shaded hexagon? <image1>
Answer: 12
Problem: <image>Question: Two squares with side 1 have a common vertex, and the edge of one of them lies along the diagonal of the other. What is the area of the overlap between the squares? <image1>
Choices:
A. $\sqrt{2}-1$
B. $\frac{\sqrt{2}}{2}$
C. $\frac{\sqrt{2}+1}{2}$
D. $\sqrt{2}+1$
E. $\sqrt{3}-\sqrt{2}$
Answer: A