7.2 Directed Numbers

**Diophantine puzzle.** Find all pairs of positive integers $(x, y)$ that satisfy $3x + 5y = 47$. List every solution. Explain why there are finitely many.

**Number line puzzle.** Three points A, B, C lie on a number line. A is at −14. B is exactly halfway between A and C. The distance from A to C is 22. (a) What is the coordinate of C? (b) What is the coordinate of B? (c) What is B × A?

**Temperature number line investigation.** The temperatures (°C) recorded at six cities at midnight are: Oslo: −18, Moscow: −23, Rome: 4, Cairo: 12, Reykjavik: −7, Seoul: −11. (a) Write the cities in order from coldest to warmest. (b) What is the range of temperatures? (c) A city X has a temperature exactly halfway between Moscow and Cairo. What is X's temperature? (d) If every temperature rises by 15°C, how many cities are then above 0°C?

**Magic square with negatives.** Complete the 3 × 3 magic square below so that every row, column, and diagonal has the same sum. Some entries are given. $$\begin{array}{|c|c|c|}\hline {-5} & \square & {3} \\\hline \square & {-1} & \square \\\hline {1} & \square & {-3} \\\hline\end{array}$$

**Frog jumping puzzle.** Frogs sit on lily pads in a row. Red frogs face right; blue frogs face left. They need to swap sides. The rules: - A frog can move one space forward onto an empty pad. - A frog can jump over exactly one frog of the **opposite** colour onto an empty pad. - No frog may move backwards. (a) With 1 red and 1 blue frog (3 pads: R _ B → B _ R), what is the minimum number of moves? (b) With 2 red and 2 blue frogs (5 pads), what is the minimum number of moves? (c) With 3 red and 3 blue frogs (7 pads), what is the minimum number of moves? (d) Spot the pattern and predict the minimum moves for $n$ red and $n$ blue frogs.

**Sign rules investigation.** Without using a calculator, decide whether each expression is positive or negative. Then calculate the exact value. (a) $(−3) × (−4) × (−2)$ (b) $(−2)^5$ (c) $(−1)^{100}$ (d) $\dfrac{(−6)^2}{(−3)}$

**Factor and sign puzzle.** Two integers satisfy all three conditions: - Their product is **+36** - Their sum is **negative** - Both integers are negative List all possible pairs. Which pair has the smallest (most negative) sum?

**Prime factorisation and directed numbers.** The prime factorisation of a number $n$ is $2^3 \times 3^2$. (a) Write down the value of $n$. (b) Write down all the factors of $n$ that are greater than 10. (c) A temperature starts at $-n$°C. It rises by $\sqrt{n}$°C. What is the new temperature? (Leave your answer in terms of a square root if $n$ is not a perfect square.)

**Directed number patterns.** Here is a pattern of directed numbers: Row 1: $−1$ Row 2: $−2, −1$ Row 3: $−3, −2, −1$ Row 4: $−4, −3, −2, −1$ (a) What is the sum of the numbers in Row 4? (b) Find a formula for the sum of Row $n$. (c) For which row is the sum equal to $-55$?

**Overdrawn accounts.** Three friends have the following bank balances: Alice £−24, Ben £−15, Chris £48. (a) Who has the least money? Who has the most? (b) Chris transfers enough money to Alice and Ben so that all three balances are equal. How much does each person end up with? (c) How much did Chris transfer in total?

**Connecting HCF and negative numbers.** Two numbers $p$ and $q$ satisfy: - $p × q = -180$ - $\text{HCF}(|p|, |q|) = 6$ - $p > 0$ and $q < 0$ - $|p| < |q|$ Find all possible pairs $(p, q)$.

**Temperature data.** A class records the daily high temperature (°C) for one week: $$−3, \ 1, \ −5, \ 2, \ −1, \ 4, \ −2$$ (a) What is the range of temperatures? (b) What is the mean temperature? (Give your answer as a fraction if not exact.) (c) On how many days was the temperature below the mean? (d) The following week every temperature is double the previous week's. Does the mean double? Does the range double? Explain.