Final answer:
To determine how many 4-digit mysterious numbers exist, we analyze the number of even and odd digits that can be placed in each position of a four-digit number while ensuring that consecutive digits sum to an odd number. An even thousands place digit limits the first digit to four options, while each subsequent digit has five options leading to a total of 500 mysterious numbers.
Step-by-step explanation:
To find the number of 4-digit mysterious numbers, where the sum of each pair of consecutive digits is odd, we have to consider that an odd sum means one digit is even and the other is odd. This rule applies to each pair of digits within a 4-digit number. Therefore, starting with the thousands place, we have two possibilities: the first digit (thousands place) can either be even or odd.
- If it's even, then the next digit (hundreds place) must be odd, the following (tens place) must be even, and the last (units place) must be odd.
- If it's odd, then the sequence for the digits must be odd-even-odd-even.
Since there are five choices for an odd digit (1, 3, 5, 7, 9) and five for an even digit (0, 2, 4, 6, 8), except the first digit which cannot be 0, we need to subtract one option from the even set for the first digit. This gives 4 options for the first digit, followed by 5 options for the second digit, 5 options for the third digit, and 5 options for the fourth digit.
The total number of mysterious numbers is the product of these options: 4 (thousands place) x 5 (hundreds place) x 5 (tens place) x 5 (units place), which equals 500 mysterious 4-digit numbers.