### Geoboards

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

### Polydron

This activity investigates how you might make squares and pentominoes from Polydron.

If you had 36 cubes, what different cuboids could you make?

# Family Tree

##### Stage: 2 Challenge Level:

We had several solutions to this problem. Michael explained his reasoning as below. You will probably find it easiest to follow the solution if you fill in the people on a copy of the tree as you read it.

G must be $1$ or $2$ as she mentions a granddaughter; and she is female so she is number $2$. We also know that she is a non-mathematician. Her granddaughter, B, must be $9$ or $10$.

E, as the youngest male, must be $8$ or $11$.

J is $6$ or $3$ (if we allow 'brother-in-law' to describe her husband's sister's husband).

F has exactly one brother, so must be $5$, $7$, $8$ or $10$.

C has a husband, and says her father is a mathematician, so C is $5$. Therefore $1$ is a mathematician, and as the oldest mathematician in the family can be identified as I. F can no longer be $5$.

D, having cousins in the tree, must be one of the third generation.

There is now no option but to assume that J is $6$. [Michael hasn't quite explained why: he probably noticed that $6$ is C's husband, and therefore not a mathematician, and so J can't be $3$.] From here we can deduce that $4$ is a mathematician, and that E and B are his children, since they have each said that their father is a mathematician.

We can now rule out F being $7$ or $8$, and so F is $10$. F tells us that her parents (C and J) are not mathematicians, and her brother ($11$) is.

H is a mother of two mathematicians, so she is $3$, and either $7$ or $9$ is a mathematician (we were told that E is). This means that for D to have just one mathematician cousin, D must be $7$ and F must be a non-mathematician.

Finally, A must be $11$, as his father is a mathematician, so K is $11$. We still haven't said which of $7$ and $9$ is the mathematician, but since we haven't yet got a female mathematician, it must be B.

Other people who sent in correct solutions (but mostly without such detailed explanation) were Claire , Sian and Fiona of Stamford High School; Ankaru; Samantha, Jacqui, Sandy and Claire of The Mount School, York; Chong Ching, Clement Goh, Chen Wei and Ng Yan; Emma and Monica of Hethersett High School, Norwich.