### Modular Fractions

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

Decipher a simple code based on the rule C=7P+17 (mod 26) where C is the code for the letter P from the alphabet. Rearrange the formula and use the inverse to decipher automatically.

### Double Time

Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.

# Function Pyramids

##### Stage: 5 Challenge Level:

Arti from the USA found a way to get 1 and 5 in the top cell:

To get 1 at the top cell use 2, 2, 1 for the three bottom cells values. To get 5 at the top cell use 16, 16, 1 for the three bottom cells values.

Arti, along with James from Newmarket College and Felipe from St Pauls School in Brazil all worked out the function. Here is Felipe's explanation:

The function for calculating the number above is given by $log_2$ of the product of the two numbers below it. This can be found by constructing a table where you put all the whole numbers you can make. A pattern will soon appear and you will notice that the values you input for a and b to form these whole numbers are in fact a logarithmic scale of base 2 and the obtained value is the sum of the $log_2$ of these numbers. Therefore the expression for calculating the number above is: $log_2(a) + log_2(b)$ , which simplified is: $log_2(ab)$

Another solver from St Pauls explained how he found the answer:

$16= 2^4$ and in the next cell $4= 2^2$ will give the answer $6$ in the space above both these numbers.

To get negative numbers you need to have decimals such as $0.5$ which is $2^-1$

To get a negative number on the top bracket you need to have numbers such as 1.1 in the spaces given as these will give decimals which will then give negative numbers in the top space.

Christopher from Sale Grammar School sent this picture to illustrate his answer:

Finally, Daniel from Savile Park school send us this clear and well-explained answer .