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Problem-solving Pack
MathematicsYear 7 · 7.1 Positive Integers
Problem-solving Pack
Name: _________________________________
Date: _________________ Class: ___________
These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.
1Problem 1 of 12
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?
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?
Working space
2Problem 2 of 12
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.)
Working space
3Problem 3 of 12
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 , tens digit , and units digit where .
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 , tens digit , and units digit where .
Working space
4Problem 4 of 12
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.
Show your reasoning clearly.
Working space
5Problem 5 of 12
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 people.
(d) At a conference there were 190 handshakes in total. How many people attended?
(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 people.
(d) At a conference there were 190 handshakes in total. How many people attended?
Working space
6Problem 6 of 12
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)?
- 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)?
Working space
7Problem 7 of 12
Unknown digits. In the multiplication below, A and B represent single digits (0–9):
Find the values of A and B. Show your working.
Find the values of A and B. Show your working.
Working space
8Problem 8 of 12
Sequence puzzle. Here is a sequence: 2, 6, 12, 20, 30, …
Find the 10th term, and write a rule for the th term.
Find the 10th term, and write a rule for the th term.
Working space
9Problem 9 of 12
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.
- divisible by 7, and
- has a digit sum of 9.
Show how you checked your answer.
Working space
10Problem 10 of 12
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.
Hint: the magic sum can be found from the diagonal containing 2, 5, 8.
The grid below has three numbers already placed. Find all nine entries.
Hint: the magic sum can be found from the diagonal containing 2, 5, 8.
Working space
11Problem 11 of 12
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?
What are the dimensions that give the largest possible area? What is that area?
Working space
12Problem 12 of 12
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?
It is set correctly at 6:00 am. What time does the clock show at 6:00 pm the same day?
Working space