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?