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Joint Distribution of Base-2 and Base-3 Digit Sums

Why making an integer's binary digit-sum equal (or proportionally equal) to its ternary digit-sum is hard and yet happens infinitely often. Students learn that the pair of base-2 and base-3 sum-of-digits values attains almost every point of the integer plane in the sense of asymptotic density, generalizing the resolution of Hildebrand's collision conjecture, and see how iterated van der Corput differencing and exponential-sum bounds force two independently normal statistics to align.