### Sums of Squares

Prove that 3 times the sum of 3 squares is the sum of 4 squares. Rather easier, can you prove that twice the sum of two squares always gives the sum of two squares?

### Double Time

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

 Process Graph Equation Explanation 1 6 H In a food-limited environment bacteria tend to maximum number after initial exponential growth (following lag phase) 2 5 B Concentration will exponentially decay as drug is metabolised 3 9 E Pendulum will have an sinusodial motion with a decaying amplitude due to damping from air resistance 4 7 I Ideal gas equation $pV = nRT$ therefore pressure and volume are in a reciprocal relationship 5 2 A Ball is small and heavy, therefore assume air resistance is negligible at first and so acceleration is constant. $s = ut + \frac{1}{2}at^2$ therefore for $u=0;\ a=g \Rightarrow s=\frac{1}{2}gt^2$ (qualitatively vertical speed represented by gradient of graph is increasing) 6 1 F As concentration of reagent increases so does reaction rate, however as increase continues concentration of the catalyst becomes the limiting factor (saturation) 7 4 G When not food limited, bacteria follow exponential growth after initial lag phase 8 3 D Hours of daylight varies from a maximum at mid-summer to minimum at mid-winter, with a mean value of 12 9 8 C Earth's orbit is not perfectly circular therefore small oscillations about mean distance (1 AU) with period of one year (note non-zero origin on vertical axis)