### Counting Counters

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

### Cuisenaire Rods

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

### Doplication

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

# Lost Books

##### Age 7 to 11 Challenge Level:

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?