### Rationals Between...

What fractions can you find between the square roots of 65 and 67?

### Root to Poly

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

### Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

# Napier's Location Arithmetic

##### Stage: 4 Challenge Level:

The final part of this problem is very challenging, but students can gain much that is valuable just from Part One. Validating the process as a reliable method of multiplication involves some clear reasoning and communication skills.

The second and third parts of this problem are included as a challenge for abler students and intended to draw them into a deeper understanding of the structure.

One possible approach for classroom use could be to ask what factors could produce particular products. Students could, for example, create products and challenge each other to find possible factors.

There are also patterns that suggest structure. For example $111 .\ldots111111$ in binary sometimes has a factor and sometimes does not, depending on the length of the line of ones. $1111$ is the product of $11$ and $101$ but $11111$ has no factors.

Teachers may wish to offer students, as a conjecture to explore, the suggestion that $2^{ab} - 1$ has $2^a - 1$ as a factor.

For example : $2^6 - 1$ is $63$ and its factors include $2^2 - 1$ ($3$) and $2^3 - 1$ ($7$)