Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: Meike paddles around five buoys with her boat (see diagram). Which of the buoys does she paddle around in a clockwise direction?
<image1>
Choices:
A. 2, 3 and 4
B. 1, 2 and 3
C. 1, 3 and 5
D. 2, 4 and 5
E. 2, 3 and 5
Answer: E
Problem: <image>Question: The numbers 3, 4, 5, 6, 7 are written inside the five circles of the shape. The product of the numbers in the four outer circles is 360. Which number is in the inner circle?
<image1>
Answer: 7
Problem: <image>Question: Kengu likes to jump on the number line. He starts at 0 , then always starts with two big jumps and then three small jumps (see diagram). He keeps repeating this in the same way, over and over again. On which of the following numbers will he land in the course of his jumps?
<image1>
Choices:
A. 82
B. 83
C. 84
D. 85
E. 86
Answer: C
Problem: <image>Question: Sonja builds the cube shown, out of equally sized bricks. The shortest side of one brick is $4 \mathrm{~cm}$ long. What dimensions in $\mathrm{cm}$ does one brick have?
<image1>
Choices:
A. $4 \times 6 \times 12$
B. $4 \times 6 \times 16$
C. $4 \times 8 \times 12$
D. $4 \times 8 \times 16$
E. $4 \times 12 \times 16$
Answer: C
Problem: <image>Question: The black-white caterpillar shown, rolls up to go to sleep. Which diagram could show the rolled-up caterpillar?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: Gerhard writes down the sum of the squares of two numbers. Unfortunately, some ink has run out (see diagram) and therefore we cannot read all the digits. What is the last digit of the first number?
<image1>
Answer: 5
Problem: <image>Question: There are five gaps in the following calculation. Adriana wants to write a "+" into four of the gaps and a "−" into one of the gaps so that the equation is correct. Where does she have to insert the "-"?
<image1>
Choices:
A. between 6 and 9
B. between 9 and 12
C. between 12 and 15
D. between 15 and 18
E. between 18 and 21
Answer: D
Problem: <image>Question: There are 5 trees and 3 paths in a park as shown on the map. Another tree is planted so that there is an equal number of trees on both sides of each path. In
which section of the park will the new tree be planted?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: The distance between two shelves in Monika's kitchen is $36 \mathrm{~cm}$. She knows that a stack of 8 identical glasses is $42 \mathrm{~cm}$ high and a stack of 2 such glasses is $18 \mathrm{~cm}$ high. How many glasses has the biggest stack that will fit between two shelves?
<image1>
Answer: 6
Problem: <image>Question: On an ordinary die the numbers on opposite sides always add up to 7. Four such dice are glued together as shown. All numbers that can still be seen on the outside of the solid are added together. What is the minimum of that total?
<image1>
Answer: 58
Problem: <image>Question: Gardener Toni plants tulips <image1> and sunflowers <image2> in a square flowerbed with side length $12 \mathrm{~m}$, as shown in the diagram. How big is the entire area where sunflowers are planted?
<image3>
Choices:
A. $36 \mathrm{~m}^{2}$
B. $40 \mathrm{~m}^{2}$
C. $44 \mathrm{~m}^{2}$
D. $46 \mathrm{~m}^{2}$
E. $48 \mathrm{~m}^{2}$
Answer: E
Problem: <image>Question: The big rectangle $A B C D$ is made up of 7 congruent smaller rectangles (see diagram). What is the ratio $\frac{A B}{B C}$?
<image1>
Choices:
A. $\frac{1}{2}$
B. $\frac{4}{3}$
C. $\frac{8}{5}$
D. $\frac{12}{7}$
E. $\frac{7}{3}$
Answer: D
Problem: <image>Question: Two identical bricks can be placed side by side in three different ways as shown in the diagrams. The surface areas of the resulting cuboids are 72, 96 and $102 \mathrm{~cm}^{2}$. What is the surface area (in $\mathrm{cm}^{2}$ ) of one brick?
<image1>
Answer: 54
Problem: <image>Question: Jenny writes numbers into a $3 \times 3$ table so that the sums of the four numbers in each $2 \times 2$ area of the table are the same. The numbers in three of the cells in the corner can already be seen in the diagram. Which number does she write into the cell in the fourth corner?
<image1>
Answer: 1
Problem: <image>Question: A shape is made up of a triangle and a circle that partially overlap. The grey area is $45 \%$ of the entire area of the shape. The white part of the triangle is $40 \%$ of the total area of the shape. What percent of the area of the circle is the white part, outside the triangle?
<image1>
Choices:
A. $20 \%$
B. $25 \%$
C. $30 \%$
D. $35 \%$
E. $50 \%$
Answer: B
Problem: <image>Question: The numbers 1 to 8 are written into the circles shown so that there is one number in each circle. Along each of the five straight arrows the three numbers in the circles are multiplied. Their product is written next to the tip of the arrow. How big is the sum of the numbers in the three circles on the lowest row of the diagram?
<image1>
Answer: 17
Problem: <image>Question: What is the minimum number of cells of a $5 \times 5$ grid that have to be coloured in so that every possible $1 \times 4$ rectangle and every $4 \times 1$ rectangle respectively in the grid has at least one cell coloured in?
<image1>
Answer: 6
Problem: <image>Question: The diagram shows a grid made of vertical and horizontal lines. Which part was cut from the grid? <image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: Which of the following shapes cannot be cut into two trapeziums with one single straight line? <image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: A dark disc with two holes is placed on top of a dial of a watch as shown. The dark disc is now rotated so that the number 8 can be seen through one of the holes. Which of the numbers could one see through the other hole now? <image1>
Choices:
A. 4 and 12
B. 1 and 5
C. 1 and 4
D. 7 and 11
E. 5 and 12
Answer: A
Problem: <image>Question: Kristina has a piece of see-through foil on which some points and lines are drawn. She folds the foil along the dotted line. <image1> What can she see now?
Choices:
A. $2\vdots6\vdots9$
B. $2\vdots6\vdots6$
C. $5\vdots6\vdots6$
D. $2\vdots8\vdots6$
E. $5\vdots8\vdots9$
Answer: C
Problem: <image>Question: A $4\times 6$ grid should be cut along the black lines into several identical shapes. No piece is to be left over. Into which of the following shapes is it not possible to cut this grid in this way? <image1> <image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: The diagram shows the starting position, the direction and the distance covered within 5 seconds by four bumper cars. Which two cars will first crash into each other? <image1>
Choices:
A. A and B
B. A and C
C. A and D
D. B and C
E. C and D
Answer: B
Problem: <image>Question: Werner wants to label each side and each corner point of the rhombus shown with exactly one number. He wants the number on each side to be equal to the sum of the numbers on the corner points of that sides. Which number is he going to write in the place of the question mark? <image1>
Answer: 12
Problem: <image>Question: Anna has five circular discs that are all of different sizes. She wants to build a tower using three discs where a smaller disc always has to lie on top of a bigger disc. How many ways are there for Anna to build the tower? <image1>
Answer: 10
Problem: <image>Question: Evita wants to write the numbers from 1 to 8 with one number in each field. The sum of the numbers in each row should be equal. The sum of the numbers in the four columns should also be the same. She has already written in the numbers 3, 4 and 8 (see diagram). <image1> Which number does she have to write in the dark field?
Answer: 7
Problem: <image>Question: The diagram shows 5 equally big semicircles and the length of 5 distances. How big is the radius of one semicircle? <image1>
Answer: 18
Problem: <image>Question: Some edges of a cube are coloured in red so that each sides of the cube has at least one red edge. What is the minimum number of red edges that the cube has? <image1>
Answer: 3
Problem: <image>Question: The digits 0 to 9 can be formed using matchsticks (see diagram). How many different positive whole numbers can be formed this way with exactly 6 matchsticks? <image1>
Answer: 6
Problem: <image>Question: Tom, John and Lily have each shot 6 arrows on a disc with three sections (see diagram). The number of points of a hit depends on the section that has been hit. Tom has 46 points and John has 34 points. How many points did Lily get? <image1>
Answer: 40
Problem: <image>Question: Two rays starting in $S$ form a right angle. More rays starting in $S$ are drawn within the right angle so that each angle $10^{\circ}, 20^{\circ}, 30^{\circ}, 40^{\circ}, 50^{\circ}, 60^{\circ}, 70^{\circ}$ and $80^{\circ}$ is enclosed by two rays. What is the minimum number of rays that have to be drawn? <image1>
Answer: 3
Problem: <image>Question: The diagram shows a grey rectangle that lies within a bigger rectangle which sides it touches. Two corner points of the grey rectangle are the midpoints of the shorter sides of the bigger rectangle. The grey rectangle is made up of three squares that each have an area of $25 \mathrm{~cm}^{2}$. How big is the area of the bigger rectangle in $\mathrm{cm}^{2}$ ? <image1>
Answer: 150
Problem: <image>Question: The shown triangle $A B C$ is isosceles with $\measuredangle A B C=40^{\circ}$. The two angles indicated $\measuredangle E A B$ and $\measuredangle D C A$ are equally big. How big is the angle $\measuredangle C F E$ ? <image1>
Choices:
A. $55^{\circ}$
B. $60^{\circ}$
C. $65^{\circ}$
D. $70^{\circ}$
E. $75^{\circ}$
Answer: D
Problem: <image>Question: An ant walks along the sides of an equilateral triangle (see diagram). Its velocity is $5 \mathrm{~cm} / \mathrm{min}$ along the first side, $15 \mathrm{~cm} / \mathrm{min}$ along the second and $20 \mathrm{~cm} / \mathrm{min}$ along the third. With which average velocity in $\mathrm{cm} / \mathrm{min}$ does the ant walk once around the entire triangle? <image1>
Choices:
A. $10$
B. $\frac{80}{11}$
C. $\frac{180}{19}$
D. $15$
E. $\frac{40}{3}$
Answer: C
Problem: <image>Question: Elisabeth wants to write the numbers 1 to 9 in the fields of the diagram shown so that the product of the numbers of two fields next to each other is no greater than 15. Two fields are called „next to each other“ if they share a common edge. How many ways are there for Elisabeth to label the fields? <image1>
Answer: 16
Problem: <image>Question: Several mice live in three houses. Last night every mouse left their house and moved directly to one of the other two houses. The diagram shows how many mice were in each house yesterday and today. How many mice used the path that is indicated with an arrow? <image1>
Answer: 11
Problem: <image>Question: Bart wrote the number 1015 as a sum of numbers that are made up of only the digit 7 . In total he uses the digit 7, 10 times. Now he wants to the write the number 2023 as a sum of numbers that are made up of only the digit 7. He uses the digit 7, 19 times in total. How often does he have to use the number 77? <image1>
Answer: 6
Problem: <image>Question: A regular hexagon is split into four quadrilaterals and a smaller regular hexagon. The ratio $\frac{\text { Area of the dark sections }}{\text { Area of the small hexagon }}=\frac{4}{3}$. How big is the ratio $\frac{\text { Area of the small hexagon }}{\text { Area of the big hexagon }}$ ? <image1>
Choices:
A. $\frac{3}{11}$
B. $\frac{1}{3}$
C. $\frac{2}{3}$
D. $\frac{3}{4}$
E. $\frac{3}{5}$
Answer: A
Problem: <image>Question: Jakob wrote six consecutive numbers on six little pieces of white paper, one number per piece of paper. He stuck those six pieces of paper on the front and back of three coins. Then he threw the coins three times. After the first throw the numbers 6, 7, 8 were on top (see diagram) which Jakob then coloured in red. After the second throw the sum of the numbers on top was 23 and after the third throw the sum was 17. How big is the sum of the numbers on the three white pieces of paper? <image1>
Answer: 18
Problem: <image>Question: $15 \%$ of a round cake is cut as shown in the figure. How many degrees is the angle denoted by 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, three strips of the same horizontal width $a$ are marked 1,2,3. These strips connect the two parallel lines. Which strip has the biggest area?
<image1>
Choices:
A. All three strips have the same area
B. Strip 1
C. Strip 2
D. Strip 3
E. Impossible to answer without knowing $a$
Answer: A
Problem: <image>Question: The area of the wooden square equals $a$. The area of each wooden circle equals $b$. Three circles are lined up as shown in the picture. If we tie together the three circles with a thread as short as possible, without moving them, what is the area inside the thread?
<image1>
Choices:
A. 3b
B. 2a + b
C. a + 2b
D. 3a
E. a + b
Answer: C
Problem: <image>Question: In this addition each of the letters $X, Y$ and $Z$ represents a different non-zero digit. The letter $X$ will then have to stand for
<image1>
Answer: 8
Problem: <image>Question: A rectangular parallelepiped was composed of 4 pieces, each consisting of 4 little cubes. Then one piece was removed (see picture) Which one?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: In the picture there are four overlapping squares with sides 11, 9, 7 and 5 long. How much greater is the sum of the two grey areas than the sum of the two black areas?
<image1>
Answer: 64
Problem: <image>Question: The diagram shows four semicircles with radius 1. The centres of the semicircles are at the mid-points of the sides of a square. What is the radius of the circle which touches all four semicircles?
<image1>
Choices:
A. $\sqrt{2}-1$
B. $\frac{\pi}{2}-1$
C. $\sqrt{3}-1$
D. $\sqrt{5}-2$
E. $\sqrt{7}-2$
Answer: A
Problem: <image>Question: The graph of the function $f(x)$, defined for all real numbers, is formed by two half-lines and one segment, as illustrated in the picture. Clearly, -8 is a solution of the equation $f(f(x))=0$, because $f(f(-8))=f(-4)=0$. Find all the solutions of the equation $f(f(f(x)))=0$.
<image1>
Choices:
A. -4 ; 0
B. -8 ;-4 ; 0
C. -12 ;-8 ;-4 ; 0
D. -16 ;-12 ;-8 ;-4 ; 0
E. No solutions
Answer: C
Problem: <image>Question: The rectangle $A B C D$ has area 36. A circle with center in point $O$ is inscribed in the triangle $A B D$. What is the area of the rectangle $O P C R$?
<image1>
Answer: 18
Problem: <image>Question: A rectangular sheet of paper with measures $6 \times 12$ is folded along its diagonal. The shaded parts sticking out over the edge of the overlapping area are cut off and the sheet is unfolded. Now it has the shape of a rhombus. Find the length of the side of the rhombus.
<image1>
Answer: 7.5
Problem: <image>Question: Unit squares of a squared board $2 \times 3$ are coloured black and white like a chessboard (see picture). Determine the minimum number of steps necessary to achieve the reverse of the left board, following the rule: in each step, we must repaint two unit squares that have a joint edge, but we must repaint a black square with green, a green square with white and a white square with black.
<image1>
Answer: 6
Problem: <image>Question: Some angles in the quadrilateral $A B C D$ are shown in the figure. If $B C=A D$, then what is the angle $A D C$?
<image1>
Choices:
A. $30^{\circ}$
B. $50^{\circ}$
C. $55^{\circ}$
D. $65^{\circ}$
E. $70^{\circ}$
Answer: D
Problem: <image>Question: In a square $2003 \times 2003$, the squares $1 \times 1$ on the diagonals are colored (like in the picture, where the square is $7 \times 7$). How many white squares are there?
<image1>
Choices:
A. $2002^{2}$
B. $2002 \times 2001$
C. $2001^{2}$
D. $2003 \times 2002$
E. $2003^{2}-2004$
Answer: A
Problem: <image>Question: The dartboard shown consists of an inner black circle and 2 rings around it. The width of each ring is equal to the radius of the black circle. How many times greater is the area of the grey ring than the area of the inner black circle?
<image1>
Answer: 5
Problem: <image>Question: The circles with centers $C$ and $D$ meet at the points $A$ and $B$, as shown. Angle $A C B=60^{\circ}$ and angle $A D B=90^{\circ}$. How many times longer is the radius of the larger circle than the radius of the smaller circle?
<image1>
Choices:
A. $\frac{4}{3}$
B. $\sqrt{2}$
C. $\frac{3}{2}$
D. $\sqrt{3}$
E. 2
Answer: B
Problem: <image>Question: In tank I, whose base has an area of $2 \mathrm{dm}^{2}$ and whose height is $10 \mathrm{~cm}$, the water is $5 \mathrm{~cm}$ high. An empty tank II with a base of area $1 \mathrm{dm}^{2}$ and a height of $7 \mathrm{~cm}$ is placed in tank I. The water of tank I rises, of course, and spills over into tank II. What level does the water reach in tank II?
<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: Three semi-circles, the diameters of two of which are equal to 4 and of the third to 8, are arranged as seen in the picture. What is the distance from the center $S$ of the bigger semi-circle to the tangent point $T$ of the smaller semi-circles?
<image1>
Choices:
A. $6.$
B. $\sqrt{32}$
C. 5.7
D. $\sqrt{40}$
E. 5
Answer: B
Problem: <image>Question: Caroline wants to write the numbers $1,2,3,4$ in the square $4 \times 4$ in such a way that every row and every column has each of the numbers. You see how she started. In how many different ways can she finish?
<image1>
Answer: 4
Problem: <image>Question: The diagram shows two tangential circles with radii in the ratio 1:2. The smaller circle rolls around the inside of the large circle. Which of the following is the path traced out by the point $P$ of the smaller circle?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: In a rectangle we draw both diagonals and the segment which joins a vertex with the midpoint of one of the sides, as shown in the picture. What is the result of dividing the length of the diagonal by the length of segment $O P$?
<image1>
Answer: 6
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, in $\mathrm{cm}^{2}$, is the area of the square?
<image1>
Answer: 72
Problem: <image>Question: A rectangle is divided into 4 triangles as shown in the figure. Of the following possibilities for the areas of the triangles at most one can be true. Which one is it?
<image1>
Choices:
A. $4,5,8,9$
B. $3,5,6,7$
C. $5,6,7,12$
D. $10,11,12,19$
E. $5,6,8,10$
Answer: A
Problem: <image>Question: The area of the shaded shape is equal to $2 \pi$ (see the picture). What is the value of the chord $A B$?
<image1>
Answer: 4
Problem: <image>Question: What is the sum of the 10 angles marked in the picture?
<image1>
Choices:
A. $720^{\circ}$
B. $600^{\circ}$
C. $450^{\circ}$
D. $360^{\circ}$
E. $300^{\circ}$
Answer: A
Problem: <image>Question: A rectangle with length $24 \mathrm{~m}$ and width $1 \mathrm{~m}$ is cut into smaller rectangles, each with width $1 \mathrm{~m}$. There are four pieces with length $4 \mathrm{~m}$, two pieces with length $3 \mathrm{~m}$ and one piece with length $2 \mathrm{~m}$. These smaller rectangles are put together to form another rectangle. What is the smallest possible perimeter of the new rectangle?
<image1>
Choices:
A. $14 \mathrm{~m}$
B. $20 \mathrm{~m}$
C. $22 \mathrm{~m}$
D. $25 \mathrm{~m}$
E. $28 \mathrm{~m}$
Answer: B
Problem: <image>Question: In the pyramid $S A B C$ all plane angles with vertex $S$ are equal to $90^{\circ}$. The areas of the lateral faces $S A B, S A C$ and $S B C$ are 3, 4 and 6, respectively. Find the volume of $S A B C$.
<image1>
Answer: 4
Problem: <image>Question: The sum the dots on opposite faces of a die always equals 7. A die rolls as shown below. At the starting point $(A)$ the top face is 3. Which will be the face at the end point $(B)$?
<image1>
Answer: 6
Problem: <image>Question: Two pieces of land are separated by the borderline $A B C D$, as shown in the figure. The line segments $A B, B C$ and $C D$ are parallel to the sides of the rectangle and have lengths $30 \mathrm{~m}, 24 \mathrm{~m}$ and $10 \mathrm{~m}$, respectively. We want to straighten the borderline by replacing it with a line $A E$, such that the areas of the two pieces of land do not change. How far from $D$ must be $E$?
<image1>
Choices:
A. $8 \mathrm{~m}$
B. $10 \mathrm{~m}$
C. $12 \mathrm{~m}$
D. $14 \mathrm{~m}$
E. $16 \mathrm{~m}$
Answer: C
Problem: <image>Question: Ten matches are used to make this fish-shaped figure. The piece of string is placed on the shape as shown. The area of the whole shape is 24. What is the area of the shaded triangle?
<image1>
Answer: 2
Problem: <image>Question: How many ways are there to choose a white square and a black square from an $8 \times 8$ chess-board so that these squares lie neither in the same row nor in the same column?
<image1>
Answer: 768
Problem: <image>Question: Three squares are placed together as shown. The lines $A E$ and $C H$ intersect at point $P$. What is the angle $\angle C P E$?
<image1>
Choices:
A. $30^{\circ}$
B. $45^{\circ}$
C. $60^{\circ}$
D. $50^{\circ}$
E. $40^{\circ}$
Answer: B
Problem: <image>Question: A flag consists of three stripes of equal width, which are divided into two, three and four equal parts, respectively. What fraction of the area of the flag is coloured grey?
<image1>
Choices:
A. $\frac{1}{2}$
B. $\frac{2}{3}$
C. $\frac{3}{5}$
D. $\frac{4}{7}$
E. $\frac{5}{9}$
Answer: E
Problem: <image>Question: A circle is divided into four arcs of length 2, 5, 6, $x$. Find the value of $x$, if the arc of length 2 subtends an angle of $30^{\circ}$ at the centre.
<image1>
Answer: 11
Problem: <image>Question: Each letter represents a different digit, and each digit a different letter. What digit could G represent?
<image1>
Answer: 1
Problem: <image>Question: Two identical equilateral triangles with perimeters 18 are overlapped with their respective sides parallel. What is the perimeter of the resulting hexagon?
<image1>
Answer: 12
Problem: <image>Question: A square of area $125 \mathrm{~cm}^{2}$ was divided into five parts of equal area - four squares and one L-shaped figure as shown in the picture. Find the length of the shortest side of the L-shaped figure.
<image1>
Choices:
A. 1
B. 1.2
C. $2(\sqrt{5}-2)$
D. $3(\sqrt{5}-1)$
E. $5(\sqrt{5}-2)$
Answer: E
Problem: <image>Question: Two squares have side 1. What is the area of the black quadrangle?
<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
Problem: <image>Question: The numbers 1, 2, 3 are written on the circumference of a circle. Then the sum of each pair of neighbouring numbers is written between them, so 6 numbers are obtained (1,3,2,5,3 and 4). This operation is repeated 4 more times, resulting in 96 numbers on the circle. What is the sum of these numbers?
<image1>
Answer: 486
Problem: <image>Question: A square with sides of length 10 is rolled without slipping along a line. The rolling stops when $P$ first returns to the line. What is the length of the curve that $P$ has travelled?
<image1>
Choices:
A. $10 \pi$
B. $5 \pi+5 \pi \sqrt{2}$
C. $10 \pi+5 \pi \sqrt{2}$
D. $5 \pi+10 \pi \sqrt{2}$
E. $10 \pi+10 \pi \sqrt{2}$
Answer: C
Problem: <image>Question: Points $M$ and $N$ are arbitrarily chosen on the sides $A D$ and $D C$, respectively, of a square $A B C D$. Then the square is divided into eight parts of areas $S_{1}, S_{2}, \ldots, S_{8}$ as shown in the diagram. Which of the following expressions is always equal to $S_{8}$?
<image1>
Choices:
A. $S_{2}+S_{4}+S_{6}$
B. $S_{1}+S_{3}+S_{5}+S_{7}$
C. $S_{1}+S_{4}+S_{7}$
D. $S_{2}+S_{5}+S_{7}$
E. $S_{3}+S_{4}+S_{5}$
Answer: A
Problem: <image>Question: How many routes are possible for the king to travel from the top left square to the bottom right square of the grid with the minimum number of moves. (The king can move to any adjacent square, including that adjacent diagonally.)
<image1>
Answer: 4