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'Lost Books' printed from https://nrich.maths.org/
Here is one way to think about this
numbering problem. This came from James and Kirsty. There are many
other ways to look for patterns (although drawing a table can
help).
For the first part (with the sheet with 8 and 21), the book had a
total of 28 pages on 7 sheets of paper.
First I noticed that with four sheets, page 8 is next to page 9 as
in the picture. Then I noticed that if you add a sheet, there are
an extra four pages in between page 8 and the page opposite, which
will be page 13. I then made this table:
| Number of sheets |
Number of page next to page 8 |
| 4 |
9 |
| 5 |
13 |
| 6 |
17 |
| 7 |
21 |
| 8 |
25 |
For the second part, I used a similar pattern. I started by
noticing that with 9 sheets, page 36 will be the back page and will
be opposite page 1. I then considered adding pages to the outside
of the book. Each time you do this, the number of the page opposite
page 36 increases by 4.
| Number of sheets |
Number of page next to page 36 |
| 9 |
1 |
| 10 |
5 |
| 11 |
9 |
| 12 |
13 |
| 13 |
17 |
| 14 |
21 |
| 15 |
25 |
Therefore the page with 25 and 36 on it came from a book with
15 sheets and a total of 60 pages.
Arthur looked carefully at the last
book and noticed something strange.
For the first half of the book odd pages are on the back and
even pages are on the front, and for the second half of the book
odd pages are on the front and even pages are on the back. This
means that if we look at any sheet taken from the book (from the
front or from the back), it will have an even number on the left
and an odd number on the right. However, the page with 59 and 14 is
the other way round, so must have come from a book with a different
numbering method.
Well done for noticing this! One way toget
this numbering would be to start numbering on the inside
cover.
Why not try making your own numbering
puzzles?