The Newspaper Puzzle | Solution

Did you come up with pages 19, 20, 23, 37, 38, and 42? If not, then here’s an explanation of that answer.

This problem involves using your visual and conceptual thinking. The first insight is to not think of this as a stack of 60 pages, but as a newspaper. Drawing on the familiar concept of how a newspaper is put together, we can move forward.

Newspapers are always printed on sheaves, from the outside in. In our example, if there are 60 pages, then when you pick up this paper, page 1 is on the top, and page 60 on the bottom, but they are all part of the same sheaf: page 1 and 2 (top) and page 59 and 60 (bottom). The next sheaf is inserted inside the first. It contains: pages 3 and 4 (top), and pages 57 and 58 (bottom). And the paper is put together this way, from the outside in.

Now, if pages 24 and 41 are missing, then we need to think about which sheaves are missing. Following the pattern, from outside in, we have pages 1 and 2, 3 and 4, etc. In other words, we always move forward with an odd page number. So, that means we’ll continue this counting pattern 1 and 2, 3 and 4, …, all the way to 23 and 24.

This means that the sheaf containing page 24 also must contain page 23. So, page 23 is missing too.

Now, this sheaf is missing, which means there are two other pages on the bottom side missing. To figure this out, we must count backward from the bottom the same amount of times we counted forward from the top. 1 and 2, 3 and 4, …, 23 and 24 involves counting forward eleven times—i.e., the eleventh sheaf is missing. To figure out which pages on the bottom half are missing, we must count backward eleven times from 59 and 60: 59 and 60, 57 and 58, …, 37 and 38.

Our sheaf missing page 24, then, is the eleventh sheaf, which contains pages 23 and 24, and 37 and 38.

The process is the same for our other missing page, 41.

To do this, though, we’ll approach it in the opposite way. This is because the newspaper is 60 pages wide, so in the middle is page 30 and 31 (and remember, thinking conceptually, this is the innermost sheaf, so pages 29 and 30 are on the top, 31 and 32 on the bottom).

Let’s find the sheaf that contains page 41 by counting backward from 59 and 60: 59 and 60, 57 and 58, …, 41 and 42. This involved counting backward nine times, so it’s the ninth sheaf that’s missing. (And it means we know page 42 is missing.)

To locate the other two missing pages, we’ll count forward nine times from the top: 1 and 2, 3 and 4, …, 19 and 20. And there we have it: If page 41 is missing, that means sheaf 9 is missing, which means pages 19, 20, 41, and 42 are missing.

And putting it all together, if pages 24 and 41 are missing, then this means that sheaves 9 and 11 are missing, which means pages 19, 20, 23, 24, 37, 38, 41, and 42 are missing.

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