### Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

### N000ughty Thoughts

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in 10 000! and 100 000! or even 1 000 000!

### Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

# Multiplication Equation Sudoku

##### Stage: 4 and 5 Challenge Level:

By Henry Kwok
Rules of Multiplication Equation Sudoku
Like the standard sudoku, this sudoku variant has two basic rules:
1. Each column, each row and each box ($3\times3$ subgrid) must have the numbers $1$ through $9$.
2. No column, row or box can have two squares with the same number.
The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid. At the bottom and right side of the $9\times9$ grid are numbers, each of which is the product of a column or row of unknown digits marked by asterisks. Altogether a set of 18 equations can be formed from the columns and rows of unknown digits and constants.

For example, in the first and second columns beginning from the left of the $9\times9$ grid, the following equations can be formed:
$*\times* = 40$
and
$*\times*\times*\times* = 576$.

In the fourth and fifth rows beginning from the top of the $9\times9$ grid, the following equations can be formed:
$*\times*\times*\times* = 60$
and
$*\times* = 27$.

After solving all the equations, the puzzle is solved by the usual sudoku technique and strategy.