7.1 Positive Integers

**Locker puzzle.** 100 lockers are numbered 1 to 100, and they are all closed. 100 students walk past them in order. - Student 1 opens every locker. - Student 2 closes every **2nd** locker (lockers 2, 4, 6, …). - Student 3 changes the state of every **3rd** locker. - This continues until student 100. How many lockers are open at the end? Which lockers are they?

**Coin combinations.** Using only 5¢, 10¢, and 20¢ coins, in how many different ways can you make exactly 50¢? (Order does not matter — {3 × 10¢, 4 × 5¢} is one way.)

**The 1089 trick.** Take any 3-digit number where the first digit is at least 2 more than the last digit (e.g. 731). Step 1: Reverse the digits (137). Step 2: Subtract the smaller from the larger (731 − 137 = 594). Step 3: Reverse your result (495). Step 4: Add the result from Step 2 to its reverse (594 + 495). (a) Try the trick with 731. What do you get? (b) Try it with 852. What do you get? (c) Does it always give 1089? Explain why by using a general 3-digit number with hundreds digit $a$, tens digit $b$, and units digit $c$ where $a > c$.

**Calendar logic.** 1st January is a Wednesday. What day of the week is 1st March in the same (non-leap) year? Show your reasoning clearly.

**Handshake problem.** At a party, every person shakes hands exactly once with every other person. (a) If there are 5 people, how many handshakes are there in total? (b) If there are 10 people, how many handshakes are there? (c) Find a formula for the number of handshakes when there are $n$ people. (d) At a conference there were 190 handshakes in total. How many people attended?

**Three bells.** Three bells toll at the start of school assembly. After that: - Bell A tolls every 8 minutes. - Bell B tolls every 12 minutes. - Bell C tolls every 18 minutes. (a) After how many minutes will all three bells first toll together again? (b) How many times does Bell A toll in the first 2 hours (not counting the start)? (c) Between the start and the first time all three bells toll together, how many times does Bell B toll on its own (not at the same time as any other bell)?

**Unknown digits.** In the multiplication below, **A** and **B** represent single digits (0–9): $$A3 \times B = 161$$ Find the values of A and B. Show your working.

**Sequence puzzle.** Here is a sequence: 2, 6, 12, 20, 30, … Find the 10th term, and write a rule for the $n$th term.

**Number theory.** Find the smallest three-digit number that is: - divisible by 7, **and** - has a digit sum of 9. Show how you checked your answer.

**Magic square.** In a 3 × 3 magic square, every row, every column, and both main diagonals have the same sum (the "magic sum"). The grid below has three numbers already placed. Find all nine entries. $$\begin{array}{|c|c|c|}\hline 2 & \square & \square \\\hline \square & 5 & \square \\\hline \square & \square & 8 \\\hline\end{array}$$ Hint: the magic sum can be found from the diagonal containing 2, 5, 8.

**Optimisation.** A farmer has exactly 120 m of fencing. He wants to enclose a rectangular field using all of it. What are the dimensions that give the **largest possible area**? What is that area?

**Modular reasoning.** A clock loses exactly 4 minutes every hour. It is set correctly at **6:00 am**. What time does the clock **show** at **6:00 pm** the same day?