Strange Numbers

PROBLEM STATEMENT

Alice is very fond of digits. While she was playing with digits from 0 to 9, she noticed that when digits 0, 1, 6, 8, 9 are rotated 180 degrees, they become 0, 1, 9, 8, 6, respectively, and when digits 2, 3, 4, 5, and 7 are rotated 180 degrees, they become invalid.

After noticing such a property of digits, she started considering some numbers as strange numbers. According to her, a strange number is a number that, when rotated 180 degrees in a clockwise fashion, becomes a different number with each digit valid. As she is busy playing with digits, she gave you an integer ‘N’ and asked you to find the number of strange numbers from 1 to ‘N’ inclusive.

Note :

The rotated number can be greater(or smaller) than the original number.

Input Format :

The first line of the input contains an integer ‘T’ representing the number of test cases.

The first line of each test case contains a single integer ‘N’ denoting the integer given by Alice.

Output Format :

For each test case, print the number of strange numbers between 1 and ‘N’ inclusive.

The output of each test case will be printed in a separate line.

Note :

You do not need to print anything, it has already been taken care of. Just implement the given function.

Constraints :

1 <= T <= 5
1 <= N <= 10 ^ 5

Time Limit: 1sec

Approach 1

The idea here is to perform a breadth-first search over all the numbers from 1 to ‘N’ inclusive with 0, 1, 6, 8, and 9 as their digits. Initially, we will insert all the numbers between 1 to ‘N’ inclusive having a single - digit with 0 or 1 or 6 or 8 or 9 as their digit into a queue. We will gradually increase the length(i.e., number of digits) of the numbers present in the queue and insert it into the queue if the number lies between 1 to ‘N’ inclusive. Refer to the below diagram for a clear understanding.

For all the numbers present in the queue, we will check if the current number is a strange number.

Algorithm:

Declare an integer, say ‘cntStrange’ to count the total number of strange numbers in a given range, and initialize it with 0.
Declare an integer, say ‘currNumber’, to keep track of the current number during our BFS traversal.
Declare a queue of integers.
Run a loop for ‘digit’ in [0, 1, 6, 8, 9]:
- If ‘digit’ lies between 1 and ‘N’ inclusive
  - Insert ‘digit’ into the queue
Run a loop until the queue is not empty:
- Store the front of the queue in ‘currNumber’ and remove it from the queue.
- Call the ‘isStrangeNumber’ function to check if ‘currNumber’ is a strange number or not.
- If ‘currNumber’ is a strange number:
  - Increment ‘cntStrange’ i.e., do cntStrange = cntStrange + 1.
- Append digits 0, 1, 6, 8, and 9 to ‘currNumber’
- Run a loop for ‘digit’ in [0, 1, 6, 8, 9]:
  - Declare an integer, say ‘appendedNumber’
  - Update ‘appendedNumber’ as ‘appendedNumber’ = currNumber * 10 + digit
  - If ‘appendedNumber’ lies between 1 and ‘N’, inclusive
    - Insert ‘appendedNumber’ into the queue.
Return ‘cntStrange’.

Description of ‘isStrangeNumber’ function

This function is used to check whether the current number is a strange number or not.

This function will take one parameter:

currNumber: An integer denoting the current number.

bool isStrangeNumber(currNumber):

Declare a variable ‘rotatedNumber’ to store the value of ‘currNumber’ after rotating 180 degrees.
Declare a variable ‘dummy’ and initialize it with ‘currNumber’.
Run a loop while ‘dummy’ is greater than 0.
- Declare a variable ‘currDigit’ and initialize it with the rightmost digit of ‘dummy’, i.e., do currDigit = dummy % 10.
- Remove the rightmost digit of ‘dummy’, i.e., do dummy = dummy / 10
- If ‘currDigit’ is 6, then make it 9, or if the ‘currDigit is 9, make it 6, i.e., rotate the ‘currDigit’ by 180 degrees.
- Append the ‘currDigit’ to ‘rotatedNumber’ i.e. do rotatedNumber = rotatedNumber * 10 + currDigit.
If ‘currNumber’ is not equal to ‘rotatedNumber’ then return true, as the current number follows properties of a strange number.
Else return false, as the current number does not follow properties of a strange number as after rotating, the number must be different.

Try Problem

Approach 2

The idea here is to perform a depth-first search over all the numbers from 1 to ‘N’ inclusive with 0, 1, 6, 8, and 9 as their digits. Initially, we will start the depth-first search from all the numbers between 1 to ‘N’ inclusive having a single - digit with 0 or 1 or 6 or 8 or 9 as their first digit. We will gradually increase the length(i.e., number of digits) of the numbers and recursively do a depth-first search if the number lies between 1 to ‘N’ inclusive. For every number that we encounter during the depth-first search, we will check if it is a strange number or not. If it is a strange number, then we will increment our answer.

Algorithm:

Declare an integer, say ‘cntStrange’ to count the total number of strange numbers in a given range.
Call the ‘depthFirstSearch’ function with ‘N’ and 0 as arguments and store its returned value into ‘cntStrange’
Return ‘cntStrange’

Description of ‘isStrangeNumber’ function

This function is used to check whether the current number is a strange number or not.

This function will take one parameter:

currNumber: An integer denoting the current number.

bool isStrangeNumber(currNumber):

Declare a variable ‘rotatedNumber’ to store the value of ‘currNumber’ after rotating 180 degrees.
Declare a variable ‘dummy’ and initialize it with ‘currNumber’.
Run a loop while ‘dummy’ is greater than 0.
- Declare a variable ‘currDigit’ and initialize it with the rightmost digit of ‘dummy’, i.e., do currDigit = dummy % 10.
- Remove the rightmost digit of ‘dummy’, i.e., do dummy = dummy / 10
- If ‘currDigit’ is 6, then make it 9, or if the ‘currDigit is 9, make it 6, i.e., rotate the ‘currDigit’ by 180 degrees.
- Append the ‘currDigit’ to ‘rotatedNumber’ i.e. do rotatedNumber = rotatedNumber * 10 + currDigit.
If ‘currNumber’ is not equal to ‘rotatedNumber’ then return true, as the current number follows properties of a strange number.
Else return false, as the current number does not follow properties of a strange number as after rotating, the number must be different.

Description of ‘depthFirstSearch’ function

This function will take two parameters:

N: An integer given in the problem.
currNumber: An integer denoting the current number in depth-first search.

int depthFirstSearch(N, currNumber):

Declare a variable ‘answer’ to count the total number of strange numbers in a given range and initialize it with 0.
Call the ‘isStrangeNumber’ function to check if ‘currNumber’ is a strange number or not.
If ‘currNumber’ is a strange number:
- Increment ‘answer’ i.e., do answer = answer + 1.
Append digits 0, 1, 6, 8, and 9 to ‘currNumber’
Run a loop for ‘digit’ in [0, 1, 6, 8, 9]:
- Declare an integer, say ‘appendedNumber’
- Make appendedNumber = currNumber * 10 + digit
- If ‘appendedNumber’ lies between 1 and ‘N’, inclusive
  - Declare an integer, say ‘currAnswer’
  - Call the ‘depthFirstSearch’ function with ‘N’ and ‘appendedNumber’ as arguments and store its returned value into ‘currAnswer’
  - Add ‘currAnswer’ to the ‘answer’ i.e. do answer = answer + currAnswer
Return ‘answer’

Try Problem