A dog is standing at the bottom left corner of a grid of 46×46 streets. The dog is trying to get to the top right corner of the grid, where it knows there is some food. As the dog runs, between the corners, it will only ever run up and to the right. Any time the dog runs to the right, it runs at least 4 consecutive blocks to the right, and any time it runs up, it runs at least 12 consecutive blocks up. How many different intersections are unreachable for the dog by following these rules?
Details and assumptions
The last stretch that the dog runs must also satisfy the condition on the minimum number of consecutive blocks.
An intersection is reachable if the dog runs through it. It doesn't matter if the dog can change direction at that intersection.
All of them.
Since there are 46x46 streets, there are 45x45 blocks.
The dog has to run 4 blocks in a row in the horizontal direction, and 12 blocks in a row in the vertical direction, both of which are even numbers, it will never reach the food, in which direction it tries.
then wats the answer
I guess I did not interpret the question correctly the first time around.
"At least 4 blocks" means that the 2nd, 3rd, 4th streets (north-south) are not accessible.
Similarly, the 2nd to the 12th avenues (east-west) are not accessible.
So near the starting point there are 46×11 intersections not accessible, in addition to the 2nd, 3rd and 4th streets of the first avenue,
for a total of
46×11+3=509 intersections.
Similarly, for the end point (top-right), the same number applies.
The total number of inaccessible intersections is therefore 2×509=1018.
To find the number of unreachable intersections, we can break down the problem into sub-problems.
Let's consider the number of unreachable intersections within a single row or column. For a row, the dog can only reach the intersection points where it makes a right turn. Similarly, for a column, the dog can only reach the intersection points where it makes an upward turn.
Now, let's count the number of intersections in a single row. Since the dog must run at least 4 consecutive blocks to the right, it will make a right turn every 4 blocks. So, the number of right turns in a single row will be 1/4th of the total number of blocks in the row. Similarly, the number of intersections in a single column will be 1/12th of the total number of blocks in the column.
In a 46x46 grid, there are 47 rows and 47 columns. Therefore, the number of unreachable intersections within a single row will be (46/4) = 11.5 (rounded down to the nearest whole number) and within a single column will be (46/12) = 3.83 (rounded down to the nearest whole number).
To find the total number of unreachable intersections, we need to consider both rows and columns. Since the dog can only reach an intersection if it has both a right turn and an upward turn, we multiply the number of unreachable intersections in a row with the number of unreachable intersections in a column.
Total number of unreachable intersections = (number of unreachable intersections in a row) * (number of unreachable intersections in a column) = 11 * 3 = 33
Therefore, there are 33 different intersections that are unreachable for the dog by following these rules.