Dear Readers,

I am currently enjoying preparing a “Magic Maths” show packed with super-fun puzzles for my colleagues and students at Musica Mundi School, Waterloo. For inspirational sources of material, I use lots of good things from what is still beautiful in the world around us. Also, I reflect on all that I have read and learned from very fine books linked to the Cambridge Maths courses that my students are following.

One treasure-trove source of ideas is “Approaches to learning and teaching Mathematics, a toolkit for international teachers”, a new book from Cambridge University Press working in collaboration with Cambridge International Examinations and the NRICH project. The ISBN is: 978-1-108-40697-0.

I like the book so much that I decided to now feature my own puzzle based on several ideas from it. Particularly important is the idea of LTHC, standing for Low Threshold High Ceiling, and meaning that it’s good to provide students with activities which they can begin to access quite readily (Low Threshold), but which also offer scope to investigate much further according to their various ability levels (High Ceiling).

__THE ISBN PUZZLE__

Consider the 13-digit ISBN number 9781108406970.

- Begin by adding up all the 13 digits.

- Is it possible to remove just one digit from the original 13-digit number so that the sum total of the remaining 12 digits is an exact multiple of 12?

- Which two digits should be removed from the original 13-digit number if we want the sum total of the remaining 11 digits to be an exact multiple of 11? Can you find more than one solution?

- Is it possible to remove eleven digits from the original 13-digit number so that the sum total of the remaining 2 digits is an exact multiple of 11?

- Is the original 13-digit number actually a multiple of 12? Can you prove your answer without using a calculator?

- The 13-digit number 9781108406970 has numerous factors from 1 to 9781108406970 itself. However, curiously, it has only one 4-digit factor, which has a sneaky connection with Waterloo, Belgium!

Given that 47 is the 2nd-largest 2-digit factor of 9781108406970, find the largest 2-digit factor.

Also find the only 4-digit factor.

- As a wee joke to finish, what is the connection between the 4-digit factor and Waterloo, Belgium? Hint: Don’t send your answers by post!

__SOLUTIONS TO PUZZLES__

- 9+7+8+1+1+0+8+4+0+6+9+7+0=
**60**. **Removing a 0**from the original number keeps the digit sum at 60, which equals 5 x 12.**Removing 4 and a 1**from the original number lowers the digit sum to 55, which equals 5 x 11.

**Alternatively, removing a 9 and a 7 or 8 and 8** from the original number lowers the digit sum to 44, which equals 4 x 11.

**Removing all the digits except for 4 and a 7**from the original number lowers the digit sum to 11.- The original number 9781108406970 is clearly a multiple of 3 (as its digit sum is 60, a multiple of 3). However, it is not a multiple of 4 because the last two digits, 70, do not make a multiple of 4. Therefore,
**9781108406970 cannot be a multiple of 12**. - We were given that the odd number 47 is the 2nd-largest 2-digit factor of 9781108406970, but since 9781108406970 is an even number, it also has 2 as a factor. So,
**the largest 2-digit factor must be 47 x 2=94**.

Knowing already that 3, 47 and *10* are all factors of 9781108406970 (*as it ends with a 0)*, we can deduce that 3 x 47 x 10=**1410** must also be a factor, and in fact it is the only 4-digit factor.

**Coincidentally, 1410 is the area postal code for Waterloo, Belgium!**

Wherever you are, I wish you a very happy weekend now.