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Summer maths puzzle pack for A Level Maths

A take-home pack for the summer: number puzzles, logic puzzles and counting puzzles that need thinking more than formulas, and a short proof to finish. Print the student sheet; the answers are in a separate PDF for after the holiday.

Level
A Level Maths (Year 12 and Year 13)
Time
40 minutes, plus a 10-minute extension
Topics
Number theory: patterns and factorials; Algebra: forming equations; Counting and probability; Proof (extension)
Equipment
No calculator needed.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A12 minNumber puzzles
Main: task B12 minLogic puzzles
Main: task C11 minCounting and chance
Extension10 minFast finishers or homework

Starter (5 minutes)

Try these together before the pack goes home.

  1. Find the next term: 1, 1, 2, 3, 5, 8, …
  2. A brick weighs 1 kg plus half a brick. How much does a brick weigh?
  3. Work out 1 + 3 + 5 + … + 99.

Main activity (35 minutes)

Task A: Number puzzles (12 min)

Look for a pattern before you calculate.

  1. What is the last digit of 7100?
  2. How many zeros are there at the end of 25! (25 × 24 × … × 1)?
  3. Find every positive whole number n for which n2 − 1 is prime.

Task B: Logic puzzles (12 min)

Write an equation, or try and improve.

  1. Ana is 3 times as old as her brother. In 5 years she will be twice as old as him. How old are they now?
  2. A farm has chickens and goats: 20 heads and 56 legs. How many goats are there?
  3. Three consecutive whole numbers add to 252. Find the largest.

Task C: Counting and chance (11 min)

Count carefully: list a few cases first.

  1. Ten friends each shake hands once with every other friend. How many handshakes is that?
  2. Two fair dice are rolled. Find the probability that they show the same number.
  3. How many different arrangements are there of the letters of LEVEL?
  4. How many squares of any size are there on an 8 × 8 chessboard?

Extension (10 minutes)

For fast finishers.

  1. Prove that the sum of any three consecutive integers is a multiple of 3.

For teachers

Teacher notes and full worked answers
  • Take-home: print the student sheet for the last lesson before the summer and set it as optional holiday puzzles. Share the answers PDF in the first week back, or put it up after a week so families can check.
  • Do the starter together in class so everyone knows how to start a puzzle: draw it, try a small case, look for a pattern.
  • Good bridging work between Year 12 and Year 13, or for students about to start the course.

Starter

  1. 13
    • Add the two before: 5 + 8
  2. 2 kg
    • Half a brick weighs 1 kg.
  3. 2500
    • 50 odd numbers: 502 = 2500

Task A: Number puzzles

  1. 1
    • Last digits cycle 7, 9, 3, 1.
    • 100 = 4 × 25, so the last digit is 1.
  2. 6
    • Each zero needs a 5 (2s are plentiful).
    • Multiples of 5 up to 25: 5 of them, and 25 gives an extra 5: 6
  3. n = 2 only
    • n2 − 1 = (n − 1)(n + 1).
    • For a prime, the smaller factor must be 1, so n = 2 (giving 3).

Task B: Logic puzzles

  1. Ana 15, her brother 5
    • a = 3b and a + 5 = 2(b + 5)
    • b = 5
  2. 8 goats
    • c + g = 20 and 2c + 4g = 56
    • 2g = 16
  3. 85
    • (n − 1) + n + (n + 1) = 3n = 252, so n = 84

Task C: Counting and chance

  1. 45
    • 10C2 = 45
  2. 1/6
    • 6 doubles out of 36
  3. 30
    • 5!/(2! × 2!) = 120/4 = 30
  4. 204
    • 12 + 22 + … + 82 = 204

Extension

  1. (n − 1) + n + (n + 1) = 3n
    • Call the middle integer n.
    • The sum is 3n, which is a multiple of 3 for every integer n.

The same answers are in the answers PDF. Every answer was recomputed by computer before publishing.

Practise the topics

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