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.)

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  1. 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?

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  2. 6+7+4+1+3=21

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