In how many positive four-digit integers that are not multiples of 1111 do the digits form an arithmetic sequence? (The digits must form an arithmetic sequence, in order. For example, the number 5137 does not count.)

so, start with 1xxx. With various values of d, we get

d=1: 1234, 2345, 3456, 4567, 5678, 6789
d=2: 1357, 2468, 3579
d=3: 147? bzzt

Now, with negative d, we have
d=-1: 9876, 8765, 7654, 6543, 5432, 4321, 3210
d=-2: 9753, 8642, 7531, 6420
d=-3: 9630

So, of the numbers listed, how many are not multiples of 1111?

6+7+4+1+3=21

STEVE YOU MSUT HAVE POOP FOR BRAINS!

Agree that the answer is 21

Yes, I checked. The answer is 21.

To find how many positive four-digit integers form an arithmetic sequence and are not multiples of 1111, we can use the following steps:

Step 1: Determine the possible common differences.
Since we want a four-digit integer, the common difference (d) should be greater than 0, but less than or equal to 9, as each digit ranges from 0 to 9.

Step 2: Count the number of four-digit integers for each common difference.
For each possible common difference, we count the number of four-digit integers that can be formed.

Step 3: Sum all the counts to get the final answer.
Add up the counts obtained in the previous step to find the total number of four-digit integers that form an arithmetic sequence.

Let's go through each step in detail:

Step 1: Determine the possible common differences.
An arithmetic sequence can have common differences ranging from 1 to 9 since we are working with four-digit numbers (ranging from 1000 to 9999).

Step 2: Count the number of four-digit integers for each common difference.
For each possible common difference, we can find the number of arithmetic sequences by considering the range of possibilities for the first digit and the remaining three digits based on that.

For a given common difference d, we can calculate the number of four-digit integers using the formula:
Count = 9 - (d - 1)

Step 3: Sum all the counts to get the final answer.
Add up the counts from each possible common difference to find the total number of four-digit integers that form an arithmetic sequence and are not multiples of 1111.

Let's calculate the counts for each possible common difference:

For d = 1, the count is 9 - (1 - 1) = 9.
For d = 2, the count is 9 - (2 - 1) = 8.
For d = 3, the count is 9 - (3 - 1) = 7.
For d = 4, the count is 9 - (4 - 1) = 6.
For d = 5, the count is 9 - (5 - 1) = 5.
For d = 6, the count is 9 - (6 - 1) = 4.
For d = 7, the count is 9 - (7 - 1) = 3.
For d = 8, the count is 9 - (8 - 1) = 2.
For d = 9, the count is 9 - (9 - 1) = 1.

Now, add up all the counts:
9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45

Therefore, there are 45 positive four-digit integers that form an arithmetic sequence and are not multiples of 1111.