Hello! I am Pip. Puzzles are like building a nest: add one twig at a time, and the whole thing fits together.
Pip’s tip: List possibilities in a clear order. If you work backwards, undo the last step first.
Choose one of three sandwiches and one of two drinks shown above. How many different pairs are possible?
List water and milk for each sandwich: 2 + 2 + 2 = 6.
Answer: 6.
There are 4 fruit choices and 3 yoghurt choices. Choose one of each. How many pairs are possible?
12. 4 × 3 = 12.
A number plus 9, then doubled, gives 38. Find the number.
10. 38 ÷ 2 = 19; 19 − 9 = 10.
Zara is 11 and Eli is 7. What will their age difference be in 6 years?
4 years. Both ages rise by 6, so the difference stays 11 − 7 = 4.
Skills: PROB-01. Stretch: usually taught in Year 7, but some 11+ tests include it.
Probability tells us how likely something is. Write it as a fraction: the number of ways you want, over the total number of equally likely ways. A chance of 0 is impossible and 1 is certain. Be careful: some outcomes are more likely than others, as the dice grid shows.
7 equal sections, 3 of them red. The chance of red is 3 out of 7, or 3/7.
A bag has 4 red, 3 blue and 3 green counters. What is the chance of picking a red counter?
Answer: 2/5
Skills: PROB-02
Systematic means in an organised order, so you do not miss anything or list something twice. Keep one thing fixed while you change the next, then move on to the next fixed thing. A tree diagram or a table helps.
3 sandwiches × 2 drinks = 6 different meals.
A café has 3 sandwiches (cheese, ham, egg) and 2 drinks (juice, milk). A meal is one sandwich and one drink. How many different meals are there?
Answer: 6 meals
Skills: PROB-03
In a logic puzzle, every clue rules something out. Draw a grid and cross off what cannot be true. When only one choice is left in a row or column, it must be right. Optimisation means finding the best, such as the fewest coins.
A tick means “yes”, a cross means “no”. Each row and column has exactly one tick.
Ali, Bo and Cy each have one pet: a cat, a dog or a fish. Bo’s pet lives in water. Ali does not have the dog. Who has which pet?
Answer: Ali has the cat, Bo has the fish, Cy has the dog
Skills: PROB-04
When you know the end result, work backwards. Start from the answer and undo each step in reverse order, using the opposite operation. For age puzzles, remember that everybody gets older by the same amount.
In 5 years’ time Tom will be 12. How old was Tom 3 years ago?
Answer: 4 years old
Level 1 is the easiest. Level 5 is a real challenge. Try each one yourself before you read the steps.
A shop has 2 hats (a woolly hat and a sun hat) and 2 scarves (red and blue). You choose one hat and one scarf. List every pair. How many are there?
Answer: 4 pairs
A school offers 2 clubs on Monday (chess or art) and 3 clubs on Tuesday (choir, football or drama). A child picks one club each day. How many different pairs of clubs are possible?
Answer: 6 pairs
How many different 3-digit numbers can you make using the digits 3, 6 and 9 once each? How many of them are more than 600?
Answer: 6 numbers, and 4 of them are more than 600
I think of a number, multiply it by 3, subtract 4, then halve it. The answer is 13. What was my number?
Answer: 10
Nina is twice as old as Omar. Four years ago she was three times as old as he was. How old are they now?
Answer: Omar is 8 and Nina is 16
Mistake code: PROB_EQUIPROBABILITY
The mistake: Saying that with two dice a total of 2 is as likely as a total of 7 because “they are both just one total”.
Why it is wrong: A total of 2 can only happen one way (1 and 1). A total of 7 can happen 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1).
How to avoid it: Write out all the possible results in a grid first. Count the ways, then compare.
Mistake code: TECH_STOPPED_AT_STEP_1
The mistake: In the red counter question, working out the total of 10 counters and giving 10 as the answer.
Why it is wrong: The total is only the first step. The question asks for a chance, so you must go on and write the fraction.
How to avoid it: Write “total = …” first, then finish with the fraction and simplify.
The mistake: Writing outfits or numbers in a random order and counting 5 when there are 6.
Why it is wrong: Without an order it is easy to forget one or write one twice.
How to avoid it: Be systematic. Fix the first choice, list every second choice, then change the first choice.
The mistake: A machine does +5 and then ×2, and the answer is 30. Working back by doing 30 + 5, or by undoing the +5 first.
Why it is wrong: To undo a step you need its opposite, and you must undo the last step first.
How to avoid it: Write the steps in a row, then go from right to left using opposites.
Five questions, easy to hard. Use paper and have a go before you open the answer.
A fair spinner has 5 equal sections numbered 1 to 5. What is the chance of landing on 4?
1/5
One section is a 4, and there are 5 equal sections.
A bag has 1 red, 2 yellow and 7 green counters. What is the chance of picking a counter that is not green?
3/10
Total = 1 + 2 + 7 = 10. Not green = 1 + 2 = 3. So the chance is 3/10.
A pizza comes in 2 sizes with a choice of 4 toppings. Each pizza has one size and one topping. How many different pizzas are there?
8
2 sizes × 4 toppings = 8. You can list them to check.
A number machine multiplies by 3 and then subtracts 7. The answer is 29. What was the starting number?
12
Backwards: 29 + 7 = 36, then 36 ÷ 3 = 12. Check: 12 × 3 = 36, 36 − 7 = 29.
You have 1p, 2p, 5p, 10p, 20p and 50p coins. What is the fewest coins you need to make 63p?
4 coins
Biggest first: 50p + 10p + 2p + 1p = 63p. That is 4 coins, and no way uses fewer.