Wednesday, May 16, 2018
(in North America and South America)
Thursday, May 17, 2018
(outside of North American and South America)
©2017 University of Waterloo
Since
Answer: (B)
Reading from the pie chart,
Since
Answer: (D)
There are 30 minutes between 8:30 a.m. and 9:00 a.m.
There are 5 minutes between 9:00 a.m. and 9:05 a.m.
Therefore, the length of the class is
Answer: (C)
The side length of a square having an area of 144 cm
Answer: (D)
The cost of nine $1 items and five $2 items is
The correct answer is (C).
(We may check that each of the remaining four answers gives a cost that is less than $18.)
Answer: (C)
Converting each of the improper fractions to a mixed fraction, we get
Of the five answers given, the number that lies between 3 and 4 on a number line is
Answer: (D)
Exactly 2 of the
Therefore, the probability that Carrie chooses a sunflower seed is
Answer: (A)
Since
Answer: (A)
The sum of the three angles in any triangle is
If one of the angles in an isosceles triangle measures
Since the triangle is isosceles, then two of the angles in the triangle have equal measure.
If the two unknown angles are equal in measure, then they each measure
However,
If the measure of one of the unknown angles is equal to the measure of the given angle,
Therefore, the measures of the other angles in this triangle could be
Answer: (C)
Moving 3 letters clockwise from
Moving 1 letter clockwise from the letter
Therefore, the letter that is 4 letters clockwise from
Moving 4 letters clockwise from
Moving 4 letters clockwise from
The ciphertext of the message
Answer: (C)
Every cube has exactly 8 vertices, as shown in the diagram.
Answer: (E)
The area of the 2 cm by 2 cm base of the rectangular prism is
The top face of the prism is identical to the base and so its area is also 4 cm
Each of the 4 vertical faces of the prism has dimensions 2 cm by 1 cm, and thus has area
Therefore the surface area of the rectangular prism is
Answer: (E)
Since 11 410 kg of rice is distributed into 3260 bags, then each bag contains
Since a family uses 0.25 kg of rice each day, then it would take this family
Answer: (D)
Since Dalia’s birthday is on a Wednesday, then any exact number of weeks after Dalia’s birthday will also be a Wednesday.
Therefore, exactly 8 weeks after Dalia’s birthday is also a Wednesday.
Since there are 7 days in each week, then
Since 56 days after Dalia’s birthday is a Wednesday, then 60 days after Dalia’s birthday is a Sunday (since 4 days after Wednesday is Sunday).
Therefore, Bruce’s birthday is on a Sunday.
Answer: (E)
Solution 1:
Since each emu gets 2 treats and each chicken gets 4 treats, each of Karl’s 30 birds gets at least 2 treats .
If Karl begins by giving his 30 birds exactly 2 treats each, then Karl will have given out
Since Karl has 100 treats to hand out, then he has
However, each emu has already received their 2 treats (since all 30 birds were given 2 treats).
So the remaining 40 treats must be given to chickens.
Each chicken is to receive 4 treats and has already received 2 treats.
Therefore, each chicken must receive 2 more treats.
Since there are 40 treats remaining, and each chicken receives 2 of these treats, then there are
(We may check that if there are 20 chickens, then there are
Solution 2:
Using a trial and error approach, if Karl had 5 emus and
Since Karl hands out 100 treats, we know that Karl has more emus than 5 (and fewer chickens than 25).
We show this attempt and continue with this approach in the table below.
Number of Emus | Number of Chickens | Number of Emu Treats | Number of Chicken Treats | Total Number of Treats |
---|---|---|---|---|
5 | ||||
7 | ||||
10 |
Therefore, Karl has 20 chickens.
Answer: (D)
Solution 1:
The integers 1 to 32 are spaced evenly and in order around the outside of a circle.
Consider drawing a first straight line that passes through the centre of the circle and joins any one pair of these 32 numbers.
This leaves
Since this first line passes through the centre of the circle, it divides the circle in half.
In terms of the remaining 30 unpaired numbers, this means that 15 of these numbers lie on each side of the line drawn between the first pair.
Let the number that is paired with 12 be
If we draw the line through the centre joining 12 and
Beginning at 12 and moving in the direction of the 13, the 15 numbers that lie between 12 and
Therefore, the next number after 27 is the number
The number paired with 12 is 28.
Solution 2:
We begin by placing the integers 1 to 32, spaced evenly and in order, clockwise around the outside of a circle.
As in Solution 1, we recognize that there are 15 numbers on each side of the line which joins 1 with its partner.
Moving in a clockwise direction from 1, these 15 numbers are
Since 2 is one number clockwise from 1, then the partner for 2 must be one number clockwise from 17, which is 18.
Similarly, 12 is 11 numbers clockwise from 1, so the partner for 12 must be 11 numbers clockwise from 17.
Therefore, the number paired with 12 is
Answer: (A)
We may begin by assuming that the area of the smallest circle is 1.
The area of the shaded middle ring is 6 times the area of the smallest circle, and thus has area 6.
The area of the unshaded outer ring is 12 times the area of the smallest circle, and thus has area 12.
The area of the largest circle is the sum of the areas of the smallest circle, the shaded middle ring, and the unshaded outer ring, or
Therefore, the area of the smallest circle is
Note: We assumed the area of the smallest circle was 1, however we could have assumed it to have any area. For example, assume the area of the smallest circle is 5 and redo the question. What is your final answer?
Answer: (E)
For the product of two integers to equal 1, the two integers must both equal 1 or must both equal
Similarly, if the product of six integers is equal to 1, then each of the six integers must equal 1 or
For the product of six integers, each of which is equal to 1 or
That is, there must be zero, two, four or six
We summarize these four possibilities in the table below.
Number of |
Product of the six integers | Sum of the six integers |
---|---|---|
0 | ||
2 | ||
4 | ||
6 |
Of the answers given, the sum of such a group of six integers cannot equal 0.
Answer: (C)
Since the heights of the 4 athletes on the team are all different, then if Laurissa’s height is different than each of these, there is no single mode height.
Therefore, Laurissa’s height must be equal to the height of one of the 4 athletes on the team for there to be a single mode.
If Laurissa’s height is 135 cm, then the median height of the 5 athletes is 160 cm which is not possible, since the median does not equal the mode.
Similarly, if Laurissa’s height is 175 cm, then the median height of the 5 athletes is 170 cm which is not possible.
Therefore, Laurissa’s height must equal 160 cm or 170 cm, since in either case the median height of the 5 athletes will equal Laurissa’s height, which is the mode.
If Laurissa’s height is 170 cm, then the mean height of the 5 athletes is
If Laurissa’s height is 160 cm, then the mean height of the 5 athletes is
When Laurissa’s height is 160 cm, the heights of the 5 athletes (measured in cm) are: 135, 160, 160, 170, 175.
In this case, each of the mode, median and mean height of the 5 athletes equals 160 cm.
Answer: (B)
Consider the following diagram.
Since
Answer: (A)
The figure formed by combining a pair of adjacent small parallelograms, is also a parallelogram.
For example, each of the two figures shown is a parallelogram.
The reason for this is that opposite sides of these new figures are equal in length and they are parallel. We use the notation
Similarly, more than 2 small parallelograms can be combined to form new parallelograms.
In addition to the small parallelogram (
In the table below, the number of parallelograms of each of the different sizes is shown.
Size | ||||||||
---|---|---|---|---|---|---|---|---|
Number of Parallelograms | 8 | 6 | 4 | 4 | 2 | 3 | 2 | 1 |
The number of parallelograms appearing in the figure is
Answer: (B)
Solution 1:
The number of dimes in the jar is one more than the number of nickels.
If we remove one dime from the jar, then the number of coins remaining in the jar is
Also, the number of dimes remaining in the jar is now equal to the number of nickels remaining in the jar, and the number of nickels remaining in the jar is three times the number of quarters remaining in the jar.
That is, for every 1 quarter remaining in the jar, there are 3 nickels and 3 dimes.
Consider groups consisting of exactly 1 quarter, 3 nickels and 3 dimes.
In each of these groups, there are 7 coins whose total value is
Since there are 49 coins having a value of $4.90 remaining in the jar, then there must be 7 such groups of 7 coins remaining in the jar (since
(We may check that 7 such groups of coins, with each group having a value of $0.70, has a total value of
Therefore, there are 7 quarters in the jar.
Solution 2:
To find the number of quarters in the jar, we need only focus on the total number of coins in the jar, 50, or on the total value of the coins in the jar, $5.00.
In the solution that follows, we consider both the number of coins in the jar as well as the value of the coins in the jar, to demonstrate that each approach leads to the same answer.
We use a trial and error approach.
Suppose that the number of quarters in the jar is 5 (the smallest of the possible answers given).
The value of 5 quarters is
Since the number of nickels in the jar is three times the number of quarters, there would be
The value of 15 nickels is
Since the number of dimes in the jar is one more than the number of nickels, there would be
The value of 16 dimes is
If there were 5 quarters in the jar, then the total number of coins in the jar would be
Similarly, if there were 5 quarters in the jar, then the total value of the coins in the jar would be
Since the value of the coins in the jar is $5.00 or 500¢ then the number of quarters in the jar is greater than 5.
We summarize our next two trials in the table below.
Number of Quarters | Value of Quarters | Number of Nickels | Value of Nickels | Number of Dimes | Value of Dimes | Total Value of Coins |
---|---|---|---|---|---|---|
6 | 150¢ | 18 | 90¢ | 19 | 190¢ | 430¢ |
7 | 175¢ | 21 | 105¢ | 22 | 220¢ | 500¢ |
When there are 7 quarters in the jar, there are
When there are 7 quarters in the jar, the value of the coins in the jar is
In either case, the number of quarters in the jar is 7.
Answer: (A)
In each block
The total number of digits written in each block is
We note that
Since each block contains 45 digits, then 43 blocks contain
Since
Writing out the first 18 digits in a block, we get 122333444455555666, and so the 1953
Answer: (C)
For a positive integer to be divisible by 9, the sum of its digits must be divisible by 9.
In this problem, we want to count the number of six-digit positive integers containing 2018 and divisible by 9.
Thus, we must find the remaining two digits, which together with 2018, form a six-digit positive integer that is divisible by 9.
The digits 2018 have a sum of
Let the remaining two digits be
When the digits
That is, the sum
The smallest that each of
The largest that each of
The only integers between 11 and 29 that are divisible by 9 are 18 and 27.
Therefore, either
If
If
In this case, there are 6 possible six-digit positive integers.
Similarly, if
Likewise, if
If
In this case, there are 4 possible six-digit positive integers.
Therefore, for the case in which the sum of
Finally, we consider the case for which the sum of the digits
If
If
If
Therefore, for the case in which the sum of
We note that all of these 31 six-digit positive integers are different from one another, and that they are the only six-digit positive integers satisfying the given conditions.
Therefore, there are 31 six-digit positive integers that are divisible by 9 and that contain the digits 2018 together and in this order.
Answer: (C)
We label the unknown numbers in the circles as shown:
Since the sum of the numbers along each side of the triangle is
This means that
Therefore,
Looking at the possible numbers that can go in the circles, the smallest that
Looking at the possible numbers that can go in the circles, the largest that
So which of the values
The following diagrams show ways of completing the triangle with
Coming up with these examples requires a combination of reasoning and fiddling.
For example, consider the case when
Since
In the given example, we have
Since the bottom row (
We note that there are other possible combinations of
In a similar way, we can determine examples like those shown with
To complete the solution, we show that
Suppose that
In this case,
The only integers from the list
Consider the bottom row, which should have
Since
Since the maximum number in the triangle is 8, then
This makes
This means that we cannot build a triangle with
Suppose that
In this case,
There are several possible sets of values for
Consider the bottom row again, which should have
Here we have
Since
This means that we cannot build a triangle with
In summary, the possible values of
The sum of these values is
Answer: (E)
Since the cost of 1 melon is $3, then the cost of 6 melons is
Answer: (C)
The number line shown has length
The number line is divided into 10 equal parts, and so each part has length
The
(Similarly, we could note that
Answer: (D)
Following the correct order of operations, we get
Answer: (B)
Since Lakshmi is travelling at 50 km each hour, then in one half hour (30 minutes) she will travel
Answer: (C)
Exactly 2 of the
Therefore, the probability that Evgeny randomly chooses a tulip is
Answer: (E)
The range of the students’ heights is equal to the difference between the height of the tallest student and the height of the shortest student.
Reading from the graph, Emma is the tallest student and her height is approximately 175 cm.
Kinley is the shortest student and her height is approximately 100 cm.
Therefore, the range of heights is closest to
Answer: (A)
Solution 1:
The circumference of a circle,
Since the circle has a diameter of 1 cm, then its circumference is
Since
Solution 2:
The circumference of a circle,
Since the circle has a diameter of 1 cm, then its radius is
Since
Answer: (B)
The ratio of the amount of cake eaten by Rich to the amount of cake eaten by Ben is
Thus, if the cake was divided into 4 pieces of equal size, then Rich ate 3 pieces and Ben ate 1 piece or Ben ate
Converting to a percent, Ben ate
Answer: (D)
Moving 3 letters clockwise from
Moving 1 letter clockwise from the letter
Therefore, the letter that is 4 letters clockwise from
Moving 4 letters clockwise from
Moving 4 letters clockwise from
The ciphertext of the message
Answer: (C)
The smallest of 3 consecutive even numbers is 2 less than the middle number.
The largest of 3 consecutive even numbers is 2 more than the middle number.
Therefore, the sum of 3 consecutive even numbers is three times the middle number.
To see this, consider subtracting 2 from the largest of the 3 numbers, and adding 2 to the smallest of the 3 numbers.
Since we have subtracted 2 and also added 2, then the sum of these 3 numbers is equal to the sum of the original 3 numbers.
However, if we subtract 2 from the largest number, the result is equal to the middle number, and if we add 2 to the smallest number, the result is equal to the middle number.
Therefore, the sum of any 3 consecutive even numbers is equal to three times the middle number.
Since the sum of the 3 consecutive even numbers is 312, then the middle number is equal to
If the middle number is 104, then the largest of the 3 consecutive even numbers is
(We may check that
Answer: (B)
If
Answer: (E)
The time in Vancouver is 3 hours earlier than the time in Toronto.
Therefore, when it is 6:30 p.m. in Toronto, the time in Vancouver is 3:30 p.m..
Answer: (C)
Solution 1:
Mateo receives $20 every hour for one week.
Since there are 24 hours in each day, and 7 days in each week, then Mateo receives
Sydney receives $400 every day for one week.
Since there are 7 days in each week, then Sydney receives
The difference in the total amounts of money that they receive over the one week period is
Solution 2:
Mateo receives $20 every hour for one week.
Since there are 24 hours in each day, then Mateo receives
Sydney receives $400 each day, and so Mateo receives
Since there are 7 days in one week, then the difference in the total amounts of money that they receive over the one week period is
Answer: (A)
Since
Note: In the question, we are given that 2018 has exactly two divisors that are prime numbers, and since 2 is a prime divisor of 2018, then 1009 must be the other prime divisor.
Answer: (B)
The first place award can be given out to any one of the 5 classmates.
Once the first place award has been given, there are 4 classmates remaining who could be awarded second place (since the classmate who was awarded first place cannot also be awarded second place).
For each of the 5 possible first place winners, there are 4 classmates who could be awarded second place, and so there are
Once the first and second place awards have been given, there are 3 classmates remaining who could be awarded the third place award (since the classmates who were awarded first place and second place cannot also be awarded third place).
For each of the 5 possible first place winners, there are 4 classmates who could be awarded second place, and there are 3 classmates who could be awarded third place.
That is, there are
Answer: (B)
For the product of two integers to equal 1, the two integers must both equal 1 or must both equal
Similarly, if the product of six integers is equal to 1, then each of the six integers must equal 1 or
For the product of six integers, each of which is equal to 1 or
That is, there must be zero, two, four or six
We summarize these four possibilities in the table below.
Number of |
Product of the six integers | Sum of the six integers |
---|---|---|
0 | ||
2 | ||
4 | ||
6 |
Of the answers given, the sum of such a group of six integers cannot equal 0.
Answer: (C)
Solution 1:
Each translation to the right 5 units increases the
Similarly, each translation up 3 units increases the
After 1 translation, the point
After 2 translations, the point
After 3 translations, the point
After 4 translations, the point
After 5 translations, the point
After 6 translations, the point
After these 6 translations, the point is at
Solution 2:
Each translation to the right 5 units increases the
Similarly, each translation up 3 units increases the
Therefore, each translation of point
After 6 of these translations, the sum
The sum of the
After these 6 translations, the value of
Answer: (D)
Solution 1:
The volume of any rectangular prism is given by the product of the length, the width, and the height of the prism.
When the length of the prism is doubled, the product of the new length, the width, and the height of the prism doubles, and so the volume of the prism doubles.
Since the original prism has a volume of 30 cm
When the width of this new prism is tripled, the product of the length, the new width, and the height of the prism is tripled, and so the volume of the prism triples.
Since the prism has a volume of 60 cm
When the height of the prism is divided by four, the product of the length, the width, and the new height of the prism is divided by four, and so the volume of the prism is divided by four.
Since the prism has a volume of 180 cm
Solution 2:
The volume of any rectangular prism is given by the product of its length,
When the length of the prism is doubled, the length of the new prism is
Similarly, when the width is tripled, the new width is
Therefore, the volume of the new prism is the product of its length,
That is, the volume of the new prism is
Since the original prism has a volume of 30 cm
Answer: (E)
The mean height of the group of children is equal to the sum of the heights of the children divided by the number of children in the group.
Therefore, the mean height of the group of children increases by 6 cm if the sum of the increases in the heights of the children, divided by the number of children in the group, is equal to 6.
If 12 of the children were each 8 cm taller, then the sum of the increases in the heights of the children would be
Thus, 96 divided by the number of children in the group is equal to 6.
Since
Answer: (A)
Solution 1:
We begin by constructing a line segment
Since
In
Since the sum of the angles in a triangle is
In
Similarly,
Since
That is,
Solution 2:
We begin by extending line segment
Since
In
Since the sum of the angles in a triangle is
Since
That is,
Solution 3:
We begin by constructing a line segment
Since
Similarly, since
Since
Answer: (D)
Solution 1:
We begin by assuming that there are 100 marbles in the bag.
The probability of choosing a brown marble is 0.3, and so the number of brown marbles in the bag must be 30 since
Choosing a brown marble is three times as likely as choosing a purple marble, and so the number of purple marbles in the bag must be
Choosing a green marble is equally likely as choosing a purple marble, and so there must also be 10 green marbles in the bag.
Since there are 30 brown marbles, 10 purple marbles, and 10 green marbles in the bag, then there are
Choosing a red marble is equally likely as choosing a yellow marble, and so the number of red marbles in the bag must equal the number of yellow marbles in the bag.
Therefore, the number of red marbles in the bag is
Of the 100 marbles in the bag, there are
The probability of choosing a marble that is either red or green is
Solution 2:
The probability of choosing a brown marble is 0.3.
The probability of choosing a brown marble is three times that of choosing a purple marble, and so the probability of choosing a purple marble is
The probability of choosing a green marble is equal to that of choosing a purple marble, and so the probability of choosing a green marble is also 0.1.
Let the probability of choosing a red marble be
The probability of choosing a red marble is equal to that of choosing a yellow marble, and so the probability of choosing a yellow marble is also
The total of the probabilities of choosing a marble must be 1.
Therefore,
The probability of choosing a red marble is 0.25 and the probability of choosing a green marble is 0.1, and so the probability of choosing a marble that is either red or green is
Answer: (C)
The area of square
Each of the 5 regions has equal area, and so the area of each region is
The area of
The area of
The area of
Let the base of
The height of
The area of
The area of
In
In
By the Pythagorean Theorem,
Therefore,
Of the answers given,
Answer: (B)
Solution 1:
In the table, we determine the value of the product
From the table, we see that
In general, because
A positive integer is a multiple of
Next, we make a table that lists the units digits of
Units digit of |
Units digit of |
Units digit of |
---|---|---|
1 | 2 | 3 |
2 | 3 | 4 |
3 | 4 | 5 |
4 | 5 | 6 |
5 | 6 | 7 |
6 | 7 | 8 |
7 | 8 | 9 |
8 | 9 | 0 |
9 | 0 | 1 |
0 | 1 | 2 |
From the table, one of the three factors has a units digit of 0 or 5 exactly when the units digit of
This means that 6 out of each block of 10 values of
We are asked for the
Note that
This means that, in the first
We need to count two more integers along the list.
The next two integers
This means that
Solution 2:
In the table below, we determine the value of the product
From the table, we see that the value of
Similarly, we see that the value of
That is, if we consider groups of 5 consecutive integers beginning at
Will this pattern continue?
Since 5 is a prime number, then for each value of
(We also note that for each value of
For what values of
When
When
Finally, when
We also note that when
Similarly, when
We have shown that the value of
We have also shown that the value of
Since every positive integer is either a multiple of 5, or 1, 2, 3, or 4 less than a multiple of 5, we have considered the value of
In the first group of 5 positive integers from 1 to 5, there are exactly 3 integers
Similarly, in the second group of 5 positive integers from 6 to 10, there are exactly 3 integers
As was shown above, this pattern continues giving 3 values for
When these positive integers,
Since
The next two integers, 3361 and 3362, do not give values for
The next two integers, 3363 and 3364, do give values for
Therefore, the 2018
Answer: (E)
Let
These four digits can be arranged in 24 different ways to form 24 distinct four-digit numbers.
Consider breaking the solution up into the 3 steps that follow.
Step 1: Determine how many times each of the digits
If one of the 24 four-digit numbers has thousands digit equal to
That is, there are exactly 6 four-digit numbers whose thousands digit is
If the thousands digit of the four-digit number is
Similarly, there are 6 four-digit numbers whose thousands digit is
The above reasoning can be used to explain why there are also 6 four-digit numbers whose hundreds digit is
In fact, we can extend this reasoning to conclude that among the 24 four-digit numbers, each of the digits
Step 2: Determine
Since each of the digits
Similarly, since each of the digits
Continuing in this way for the hundreds digits and the thousands digits, we get
Step 3: Determine the largest sum of the distinct prime factors of
Writing 6666 as a product of prime numbers, we get
So then
That is, to determine the largest sum of the distinct prime factors of
Since
If
If
(See if you can convince yourself that all other possible values of
Therefore, the largest sum of the distinct prime factors of
Answer: (D)
Since the grid has height 2, then there are only two possible lengths for vertical arrows: 1 or 2.
Since all arrows in any path have different lengths, then there can be at most 2 vertical arrows in any path.
This means that there cannot be more than 3 horizontal arrows in any path. (If there were 4 or more horizontal arrows then there would have to be 2 consecutive horizontal arrows in the path, which is forbidden by the requirement that two consecutive arrows must be perpendicular.)
This means that any path consists of at most 5 arrows.
Using the restriction that all arrows in any path must have different lengths, we now determine the possible combinations of lengths of vertical arrows and of horizontal arrows to get from
Once we have determined the possible combinations of vertical and horizontal arrows independently, we then try to combine and arrange them.
First, we look at vertical arrows.
The grid has height 2, and
We use “U” for up and “D” for down.
The possible combinations are:
U1 (up arrow with length 1)
D1, U2 (down arrow with length 1, up arrow with length 2)
Next, we look at horizontal arrows.
The grid has width 12, and
We use “R” for right and “L” for left.
Many of these combinations of arrows can be re-arranged in different orders. We will deal with this later.
We proceed by looking at combinations of 1 arrow, then 2 arrows, then 3 arrows.
We note that every combination of vertical arrows includes an arrow with length 1 so we can ignore any horizontal combination that uses an arrow of length 1.
Also, any combination of 3 horizontal arrows must be combined with a combination of 2 horizontal arrows, which have lengths 1 and 2.
Thus, we can ignore any combination of 3 horizontal arrows that includes either or both of an arrow of length 1 and length 2.
R9
R2, R7
R3, R6
R4, R5
L2, R11
L3, R12
L3, R4, R8
L3, R5, R7
L4, R3, R10
L4, R5, R8
L4, R6, R7
L5, R3, R11
L5, R4, R10
L5, R6, R8
L6, R3, R12
L6, R4, R11
L6, R5, R10
L6, R7, R8
L7, R4, R12
L7, R5, R11
L7, R6, R10
L8, R5, R12
L8, R6, R11
L8, R7, R10
L9, R6, R12
L9, R7, R11
L9, R8, R10
L10, R7, R12
L10, R8, R11
L11, R8, R12
There is only one combination of 1 horizontal arrow.
The combinations of 2 horizontal arrows are listed by including those with two right arrows first (in increasing order of length) and then those with left and right arrows (in increasing order of length).
The combinations of 3 arrows are harder to list completely.
There are no useful combinations that include either 3 right arrows or 2 left arrows, since in either case an arrow of length 1 or 2 would be required.
Here, we have listed combinations of 2 right arrows, then those with “L3” (left arrow of length 3), then those with “L4”, and so on.
Now we combine the vertical and horizontal combinations to get the paths.
Each combination of arrow directions and lengths can be drawn to form a path.
Vertical combination U1 can only be combined with horizontal paths a through f, since it cannot be combined with 3 horizontal arrows.
There are 2 paths: U1/R9 or R9/U1.
There are 2 paths: R2/U1/R7 or R7/U1/R2.
Again, there are 2 paths.
Again, there are 2 paths.
There is 1 path: L2/U1/R11. This is because the arrows must alternate horizontal, vertical, horizontal and we cannot end with a left arrow.
Again, there is 1 path.
This is 10 paths so far.
Vertical combination D1, U2 can be combined with horizontal paths of lengths 1, 2 or 3.
There is 1 path: D1/R9/U2. This is because we cannot end with a down arrow.
Not possible because this would include two arrows of length 2.
There are 4 paths: R3/D1/R6/U2, R6/D1/R3/U2, D1/R3/U2/R6, D1/R6/U2/R3. We can interchange R3 and R6 as well as picking whether to start with a vertical or horizontal arrow.
Again, there are 4 paths.
Not possible because this would include two arrows of length 2.
There are 2 paths: L3/D1/R12/U2 and D1/L3/U2/R12.
There are 4 paths: L3/D1/R4/U2/R8, R4/D1/L3/U2/R8, L3/D1/R8/U2/R4,
R8/D1/L3/U2/R4. Each such combination must start with a horizontal arrow, must end with a right arrow, and must have the down arrow before the up arrow.
Again, there are 4 paths.
There is 1 path: R3/D1/L4/U2/R10. We cannot begin with R10 or L4 since either would take the path off of the grid, and we must end with an arrow to the right.
There are 2 paths: R5/D1/L4/U2/R8 and R8/D1/L4/U2/R5.
Again, there are 2 paths.
As with i, there is 1 path.
Again, there is 1 path.
As with j, there are 2 paths.
As with i, there is 1 path.
Again, there is 1 path.
Again, there is 1 path.
As with j, there are 2 paths.
In each of the remaining 12 cases (s. to ad.), there is 1 path as with i.
Including the previously counted 10 paths that use U1 only, we have
Answer: (B)