Wednesday, April 15, 2020
(in North America and South America)
Thursday, April 16, 2020
(outside of North American and South America)
©2020 University of Waterloo
The number of letters in each row after the first is twice the number of letters in the previous row.
Since Row 4 has 8 letters, then Row 5 has
Alternatively, we can continue the pattern to Row 6 as shown.
Row 1 | |
---|---|
Row 2 | |
Row 3 | |
Row 4 | |
Row 5 | |
Row 6 |
If the pattern consists of 6 rows, the total number of letters is
Solution 1
If the total number of letters in the pattern is 63, then there are 6 rows in the pattern (as we saw in part (b)).
Counting, we get that there are
Solution 2
Notice that in Row 2 there are twice as many
Further, the rows alternate between
Thus, if there are an even number of rows in the pattern, then the total number of
If the total number of letters in the pattern is 63, then there are 6 rows in the pattern (as we saw in part (b)), and so the number of
Solution 1
We begin by determining the number of rows in the pattern given that the total number of letters is 4095.
We may do this by counting the number of
Row Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Number of |
1 | 0 | 4 | 0 | 16 | 0 | 64 | 0 | 256 | 0 | 1024 | 0 |
Number of |
0 | 2 | 0 | 8 | 0 | 32 | 0 | 128 | 0 | 512 | 0 | 2048 |
Number of Letters | 1 | 3 | 7 | 15 | 31 | 63 | 127 | 255 | 511 | 1023 | 2047 | 4095 |
If the pattern has 12 complete rows, there are a total of 4095 letters, of which
Thus, if there are 4095 letters in the pattern, the difference between the number of
Solution 2
We begin by determining the number of rows in the pattern given that the total number of letters is 4095.
Since
Since 12 is an even number of rows, we may use the result from Solution 2 in part (c) to determine that the pattern has
Thus, if there are 4095 letters in the pattern, the difference between the number of
(Alternatively, we may have concluded that if
The surface area of a rectangular prism is given by the formula
Thus, the rectangular prism with length 2 cm, width 5 cm, and height 9 cm has surface area
The volume of a rectangular prism is given by the formula
If the rectangular prism has a square base, then
Substituting
Therefore, the side length of the square base of a rectangular prism with height 10 cm and volume 160 cm
If a rectangular prism has a square base, then
Since the area of the base is 36 cm
If the surface area of this prism is 240 cm
Thus, the volume of the prism is
Substituting into the formula for volume, we get
Substituting into the formula for surface area, we get
Equating the two expressions that are each equal to
The original product is
If we add 1 to each of 2 and 8, we get 3 and 9 and a new product of
The ones digit of this new product is 1 more than the ones digit of the original product (
The tens digit of this new product is 1 more than the tens digit of the original product (
Since the positive integer
The original product is
If we add 1 to each of 3 and 6, we get 4 and 7 and a new product of
The ones digit of this new product is not 1 more than the ones digit of the original product (
If we add 2 to each of 3 and 6, we get 5 and 8 and a new product of
The ones digit of this new product is not 2 more than the ones digit of the original product (
Adding 3 to each of 3 and 6, we get 6 and 9 and a new product of
The ones digit of this new product is not 3 more than the ones digit of the original product (
Adding 4 to each of 3 and 6, we get 7 and 10 and a new product of
The ones digit of this new product is not 4 more than the ones digit of the original product (
Adding 5 to each of 3 and 6, we get 8 and 11 and a new product of
The ones digit of this new product is not 5 more than the ones digit of the original product (
Adding 6 to each of 3 and 6, we get 9 and 12 and a new product of
Since this new product is not a two-digit integer, then
All integer values of
Solution 1
In part (b) we showed that
If
For each of the remaining values,
We summarize this work in the table that follows.
Original Product |
New Product | RadPair? | ||
---|---|---|---|---|
12 | 3 | Yes, since |
||
24 | 1 | Yes, since |
||
30 | No |
|||
36 | No |
Therefore, the positive integers
Solution 2
If
That is,
Further, if
the product
the ones digit of the product
the tens digit of the product
The second bullet gives that the ones digit of the product
The third bullet gives that the tens digit of the product
Since the product
Expanding each side of this equation and using the earlier fact that
If
That is, the condition
We saw an example of this in part (b) when we showed that
If
However
As we saw in Solution 1,
If
If
If
If
We can verify that each of
The positive integers
As in part (c) Solution 2, we will proceed algebraically to determine a condition that is required for
To then obtain an accurate count of the number of RadPairs, we will need to show that a specific value for
If
That is,
Further, if
the product
the ones digit of the product
the tens digit of the product
The second bullet gives that the ones digit of the product
The third bullet gives that the tens digit of the product
Therefore, the product
Expanding each side of this equation and using the earlier fact that
If
Again, this condition
When
Thus, there are no RadPairs
When
If
Thus, there are no RadPairs
In the table below, we use the condition
We may exclude checking some pairs by recalling that:
we have previously shown that
Check | ||||
---|---|---|---|---|
4 | 10 | |||
3 | 12 | |||
2 | 14 | |||
1 | 16 | |||
4 | 12 | |||
3 | 15 | |||
1 | 21 | |||
3 | 16 | |||
2 | 20 | |||
1 | 24 | |||
1 | 25 |
These are the only possibilities satisfying the given requirements, and so there are exactly 11 RadPairs.
Throughout the solution to this question, we will represent a path as a sequence of moves, down (
Every path in a
Thus, every path can be represented by a sequence containing at least 2
We begin by determining the number of paths in a
There are 6 such paths, as shown, and these paths can be represented by the sequences of moves
We note that these are the only possible arrangements of exactly 2
Can a path in a
If in addition to the required 2 moves down and 2 moves right, a path contained a move up, then this move would need to be “undone” by a move in the opposite direction, down.
That is, the net result of every path must be 2
To summarize, every path in a
Thus, every path in a
A path of length 5 is not possible.
Next, we determine the number of paths whose length is 6.
Each of these paths contains 2
How many arrangements are there of 3
To begin, consider the number of arrangements of the 3
The
Thus, there are 2 possible arrangements of these 4 letters, namely
Next, we count the number of possible placements of the 2
A path must pass through each vertex at most once, and so the
For this reason, 1
Thus, there are 2 possible arrangements of 3
A similar argument can be made for the arrangement of 3
These 4 paths, respectively represented by the sequences of moves
Finally, we determine the number of paths whose length is 8.
There are 2 such paths, as shown, and these paths can be represented by the sequences of moves
Can you justify why these are the only paths of length 8?
Is a path of length 10 or greater possible?
The first edge of a path touches two vertices, and each additional edge touches one new vertex.
Thus, a path of length 8 passes through
However, a path must pass through each vertex at most once, and since a
In a
Following the reasoning in part (a), a path from
Further, any additional moves must occur in opposite pairs (
Solution 1
In a
Thus, each path of length 10 has these 8 moves and exactly one additional pair of opposite moves, either left paired with right or up paired with down.
That is, each path has exactly 4 vertical moves (
By symmetry, each of these two cases will give an equal number of paths from
Assume that a path from
Each of the 4 vertical moves must be down.
The 6 horizontal moves must be 5 right and 1 left (4 moves right and 1 left/right opposite pair).
Thus, each path may be given by a string of 10 letters (4
To begin, consider the number of arrangements of the 5
Since
Next, we count the number of possible placements of the 4
A path must pass through each vertex at most once, and so the
For this reason, 1
In each of these sequences, there are 2
The 2
How many ways can the 2
The 2
Note that placing the 2
Thus, there are
How many ways can the 2
One of the
These 2
Therefore, there are
In total, there are
In a
A similar argument can be made for the arrangement of 4
Thus, the total number of paths of length 10 from
Solution 2
In a
Thus, each path of length 10 has these 8 moves and exactly one additional pair of opposite moves, either left paired with right or up paired with down.
That is, each path has exactly 4 vertical moves (
By symmetry, each of these two cases will give an equal number of paths from
Assume that a path from
Each of the 4 vertical moves must be down.
The 6 horizontal moves must be 5 right and 1 left (4 moves right and 1 left/right opposite pair).
Thus, each path may be given by a string of 10 letters (4
A path must pass through each vertex at most once, and so the
Further,
Thus the
How many arrangements of 5
We treat
However, the 5
Similarly, the 2
The total number of arrangements of 5
Are each of these 168 paths possible?
A sequence of edges that revisit a vertex is not permitted, however forcing the path to contain the moves
A sequence of edges that leaves the grid is also not permitted.
Since each sequence contains exactly 4
However, each sequence of moves contains 5
Similarly, if the
Since each of these is possible within the arrangements counted above, some of the 168 arrangements are not possible.
We can count the number of such arrangements which leave the grid and subtract this number from the total number 168.
There are exactly 2 ways that a path leaves the grid: all 5
By symmetry, each of these two cases gives an equal number of paths, and thus we count one case and double the result.
How many of the 168 sequences have 5
We can count these paths by considering the following three cases.
Case (i): the 5
The number of such paths is equal to the number of arrangements of 5
Case (ii): the 5
The number of such paths is equal to the number of arrangements of 5
Case (iii): the 5
In this case, there is only 1 only such path.
Thus, there are a total of
In a
Since there are an additional 112 paths having 4 horizontal moves and 6 vertical moves, the total number of paths of length 10 from