The whole numbers from 1 to 100 are written out one after another. How many times does the digit 7 get written?

Count the two columns separately. A 7 in the units place happens in 7, 17, 27, …, 97 — ten times. A 7 in the tens place happens in 70, 71, 72, …, 79 — another ten times. The total is 20. The number 77 must NOT be trimmed to a single count: it genuinely contributes one 7 to each column, so it is counted twice on purpose. Treating 77 as a single 7 gives the near-miss answer 19, and looking at only one column gives 10.