Ascending Decimal Digit Puzzle

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In summary, there are a total of 502 ascending numbers in the decimal (base 10) system. These are positive decimal integers with at least 2 digits, where all digits are strictly ascending and there are no leading zeroes. This can also be seen as the total number of ways to climb a 10-step stairway, assuming you can skip as many steps as you want, and each step combination can be represented by a unique ascending number. The formula to calculate this is 2^n - n - 1, where n is the number of usable digits (9 in this case).
  • #1
K Sengupta
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A positive decimal (base 10) integer G is called an ascending number if the number of digits in G is at least 2, all the digits of G are strictly ascending, and G does not contain any leading zeroes.

For example, the number 1236789 is an ascending number. However, the number 1222333555666 is not an ascending number, since the digits corresponding to the said number are not strictly ascending. Similarly, 00123456 is not an ascending number, since it contains leading zeroes.

Determine the total number of ascending numbers in the decimal (base 10) system.
 
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  • #2
Consider a stairway with 10 steps. Assume you can skip as many steps as you want. One choice of step combination to climb this step can be labeled by a unique ascending number. Reciprocally, any ascending number can be uniquely associated to such a choice.

I thought the stairway riddle has already been posted. Otherwise, you can solve it with an arbitrary number of steps. The answer equals the number of ascending numbers in the corresponding base :smile:
 
  • #3
Let N be the number of ascending numbers using the digits 1 through 8. Tack a 9 onto the end of each one of them. They are also ascending numbers and are all of them except for 19, 29, ..., 89. So the answer is 2N + 8.
Let M be the number of ascending numbers using the digits 1 through 7. Tack an 8 onto the end of each one of them.
Etc.
 
  • #4
Is it 502?

The formula for this would be:

2^n - n - 1

where n = the number of USABLE digits, which in this case is 9 (since the digit 0 is useless in this situation)
 

1. What is the Ascending Decimal Digit Puzzle?

The Ascending Decimal Digit Puzzle is a mathematical puzzle in which a set of decimal digits is given and the goal is to arrange them in ascending order to form the largest possible number.

2. How many decimal digits are typically used in the puzzle?

The number of decimal digits used in the puzzle can vary, but it is typically between 3 and 9 digits.

3. What is the strategy for solving the Ascending Decimal Digit Puzzle?

The strategy for solving the puzzle is to first identify the largest digit in the set and place it in the leftmost position. Then, continue to place the remaining digits in descending order from left to right.

4. Are there any shortcuts or tricks for solving the puzzle?

Yes, there are some shortcuts and tricks for solving the puzzle. One common strategy is to look for repeating digits and place them next to each other to form a larger number. Another trick is to identify the second largest digit and place it in the rightmost position, as this will often result in a larger number.

5. Can the puzzle be solved with any set of decimal digits?

Yes, the puzzle can be solved with any set of decimal digits. However, the more digits there are, the more challenging it may be to find the correct solution.

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