1. Suppose we write the number 1 as many times as we wish (but at least once), then we write the number 2 as many times as we wish (but at least once), and so on up to the number n. Can the concatenation of all of these numbers (e.g., 1122234444455) be a palindrome? (Assume n>1.)

2. Let a_1 < a_2 < a_3 < … be an increasing sequence of natural numbers. Let b_n be the concatenation of the first n entries of that sequence. If there are infinitely many n such that b_n is a palindrome, does it follow that the sequence (a_i) has density approaching zero?

I know the answer to #1, but #2 should be a real test for your overconfident friends’ intuition. ðŸ™‚

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