**Published on** April 15th, 2013 |
*by Jo*

# Puzzle Challenge 3

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**1. HOT HEXAGONS**

The hexagon $ABCDEF$ with vertices $A(0,0)$, $B(10, 0)$, $C(10,11)$, $D(9,11)$, $E(9,1)$ and $F(0,1)$ has been partitioned into 20 unit squares with vertices at the lattice points. Find the number of paths from $A$ to $C$ along grid lines, passing through every grid node no more than once.

**2. DIAL A DANCER**

In a country dance, the dancers are standing in two rows, $n$ boys facing $n$ girls. Each dancer gives his/her left hand to the person opposite or to his/her left neighbour or to the person opposite the left neighbour. The analagous rule applies to right hands. Nobody gives both hands to the same person. Find the number of possible configurations.

**3. TINY ODDITIES**

Determine the smallest odd number $N$ such that $N^2$ is the sum of an odd number (greater than 1) of squares of successive positive integers.

**4. TETRAHEDRON TERROR**

Consider all tetrahedra $ABCD$ such that the area of faces $ABD$, $ACD$ and $BCD$ does not exceed 1. Find all those tetrahedra which satisfy these conditions and have the maximum volume.

Puzzles contributed by Dr Michael Evans. Email your solutions to *mpe@amsi.org.au *[subscribe2]